tele-0521199255

www.cambridge.org/9780521199254

Order Statistics in Wireless Communications

Diversity, Adaptation, and Scheduling in MIMO and OFDM Systems

Covering fundamental principles through to practical applications, this self-containedguide describes indispensable mathematical tools for the analysis and design of advancedwireless transmission and reception techniques in MIMO and OFDM systems. Theanalysis-oriented approach develops a thorough understanding of core concepts, anddiscussion of various example schemes shows how to apply these concepts in practice.The book focuses on techniques of advanced diversity combining, channel adaptivetransmission, and multiuser scheduling, the foundations of future wireless systems forthe delivery of highly spectrum-efficient wireless multimedia services. Bringing togetherconventional and novel results from a wide variety of sources, it will teach you toaccurately quantify trade-offs between performance and complexity for different designoptions so that you can determine the most suitable design choice based on specificpractical implementation constraints.

Hong-Chuan Yang is an Associate Professor in the Electrical and Computer EngineeringDepartment at the University of Victoria, Canada. He has developed several mathemat-ical tools for accurate performance evaluation of advanced wireless transmission tech-nologies in fading environments, and his current research focuses on channel modeling,diversity techniques, system performance evaluation, cross-layer design, and energy-efficient communications.

Mohamed-Slim Alouini is a Professor of Electrical Engineering at King Abdullah Uni-versity of Science and Technology (KAUST), Thuwal, Makkah Province, Saudi Arabia.A Fellow of the IEEE, he is a co-recipient of numerous best paper awards, includingawards from ICC, Globecom, VTC, and PIMRC. His research interests include designand performance analysis of diversity combining techniques, MIMO techniques, multi-hop/cooperative communications, cognitive radio, and multi-resolution, hierarchical,and adaptive modulation schemes.

Order Statistics in WirelessCommunicationsDiversity, Adaptation, and Schedulingin MIMO and OFDM Systems

HONG-CHUAN YANGUniversity of Victoria, British Columbia, Canada

MOHAMED-SLIM ALOUINIKing Abdullah University of Science and Technology (KAUST), Saudi Arabia

CAMBRIDGE UNIVERSITY PRESS

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Cambridge University PressThe Edinburgh Building, Cambridge CB2 8RU, UK

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To

Our Families

Contents

Preface page xiNotation xiv

1 Introduction 11.1 Order statistics in wireless system analysis 21.2 Diversity, adaptation, and scheduling 31.3 Outline of the book 4

2 Digital communications over fading channels 72.1 Introduction 72.2 Statistical fading channel models 7

2.2.1 Path loss and shadowing 82.2.2 Multipath fading 102.2.3 Frequency-flat fading 132.2.4 Channel correlation 15

2.3 Digital wireless communications 162.3.1 Linear bandpass modulation 162.3.2 Performance analysis over fading channels 202.3.3 Adaptive transmission 23

2.4 Diversity combining techniques 262.4.1 Antenna reception diversity 262.4.2 Threshold combining and its variants 302.4.3 Transmit diversity 35

2.5 Summary 372.6 Bibliography notes 38

3 Distributions of order statistics 403.1 Introduction 403.2 Basic distribution functions 40

3.2.1 Marginal and joint distributions 403.2.2 Conditional distributions 41

3.3 Distribution of the partial sum of largest orderstatistics 42

viii Contents

3.3.1 Exponential special case 433.3.2 General case 44

3.4 Joint distributions of partial sums 463.4.1 Cases involving all random variables 463.4.2 Cases only involving the largest random variables 49

3.5 MGF-based unified analytical framework for jointdistributions 533.5.1 General steps 543.5.2 Illustrative examples 55

3.6 Limiting distributions of extreme order statistics 613.7 Summary 633.8 Bibliography notes 63

4 Advanced diversity techniques 724.1 Introduction 724.2 Generalized selection combining (GSC) 72

4.2.1 Statistics of output SNR 734.3 GSC with threshold test per branch (T-GSC) 75

4.3.1 Statistics of output SNR 764.3.2 Average number of combined paths 78

4.4 Generalized switch and examine combining (GSEC) 784.4.1 Statistics of output SNR 804.4.2 Average number of path estimations 814.4.3 Numerical examples 82

4.5 GSEC with post-examining selection (GSECps) 844.5.1 Statistics of output SNR 854.5.2 Complexity analysis 894.5.3 Numerical examples 90


5 Adaptive transmission and reception 975.1 Introduction 975.2 Output-threshold MRC 98

5.2.1 Statistics of output SNR 1005.2.2 Power saving analysis 103

5.3 Minimum selection GSC 1045.3.1 Mode of operation 1055.3.2 Statistics of output SNR 1065.3.3 Complexity savings 113

5.4 Output-threshold GSC 1155.4.1 Complexity analysis 1185.4.2 Statistics of output SNR 122

Contents ix

5.5 Adaptive transmit diversity 1275.5.1 Mode of operation 1285.5.2 Statistics of received SNR 130

5.6 RAKE finger management over the soft handoff region 1355.6.1 Finger management schemes 1365.6.2 Statistics of output SNR 1375.6.3 Complexity analysis 140

5.7 Joint adaptive modulation and diversity combining 1445.7.1 Power-efficient AMDC scheme 1465.7.2 Bandwidth-efficient AMDC scheme 1485.7.3 Bandwidth-efficient and power-greedy AMDC scheme 1505.7.4 Numerical examples 154


6 Multiuser scheduling 1626.1 Introduction 1626.2 Multiuser diversity 163

6.2.1 Addressing fairness 1646.2.2 Feedback load reduction 166

6.3 Performance analysis of multiuser selection diversity 1686.3.1 Absolute SNR-based scheduling 1686.3.2 Normalized SNR-based scheduling 170

6.4 Multiuser parallel scheduling 1716.4.1 Generalized selection multiuser scheduling (GSMuS) 1726.4.2 On–off based scheduling (OOBS) 1746.4.3 Switched-based scheduling (SBS) 1766.4.4 Numerical examples 180

6.5 Power allocation for SBS 1826.5.1 Power reallocation algorithms 1836.5.2 Performance analysis 1846.5.3 Numerical examples 186


7 Multiuser MIMO systems 1937.1 Introduction 1937.2 Basics of MIMO wireless communications 194

7.2.1 MIMO channel capacity 1947.2.2 Multiuser MIMO systems 196

7.3 ZFBF-based system with user selection 1987.3.1 Zeroforcing beamforming transmission 1987.3.2 User selection strategies 200

x Contents

7.3.3 Sum-rate analysis 2017.3.4 Numerical examples 205

7.4 RUB-based system with user selection 2067.4.1 User selection strategies 2087.4.2 Asymptotic analysis for BBSI strategy 2097.4.3 Statistics of ordered-beam SINRs 2107.4.4 Sum-rate analysis 212

7.5 RUB with conditional best-beam index feedback 2227.5.1 Mode of operation and feedback load analysis 2237.5.2 Sum-rate analysis 225

7.6 RUB performance enhancement with linear combining 2297.6.1 System and channel model 2317.6.2 M beam feedback strategy 2327.6.3 Best-beam feedback strategy 234


References 245Index 255

Preface

Order statistics is an important sub-discipline of statistical theory and findsapplications in a vast variety of fields, with life science as the most notableexample [1]. Over the years, order statistics has made an increasing number ofappearances in design and analysis wireless communication systems, primarilybecause of the simple but effective engineering principle – “pick the best”. Forexample, the diversity combining technique is an effective solution to improve theperformance of wireless communication systems operating over fading channelsby generating differently faded replicas of the same information-bearing signal.Selection combining (SC) [2, 3], which selects the replica with the best qualityfor further processing, is an attractive practical combining scheme and has beenresearched extensively in the literature. The performance analysis of the SCscheme entails the distribution functions of the largest random variables amongmultiple ones, which is available in conventional order statistics literature.

More recently, order statistics has also found application in the analysis anddesign of many emerging wireless transmission and reception techniques, suchas advanced diversity combining techniques, channel adaptive transmission tech-niques, and multiuser scheduling techniques. These techniques are becoming theessential building blocks of future wireless systems for the delivery of multimediaservices with high spectrum efficiency [4]. In particular, order statistics resultshave allowed for the accurate quantification of the trade-off of performance versuscomplexity among different design options, which will greatly facilitate the appli-cations of these technologies in future wireless systems. At the same time, theseapplications to wireless system analysis provide new incentives for the furtherdevelopment of order statistic theory. In fact, the study of advanced diversitycombining techniques has led to some new order statistics results [5,6], in termsof the joint distribution functions of linear functions of ordered random variables.

The primary goal of this book is to provide a comprehensive and coherenttreatment of the general subject of order statistics in wireless communications.By collecting the relevant results in the literature in a unified fashion, the bookwill serve as a useful resource for students and researchers to further exploitthe potential of order statistics in the analysis and design of advanced wire-less transmission technologies. It is our sincere hope that the book will build asolid foundation for readers to further explore the potential of order statistics inadvanced wireless communication research. We also believe that the new wireless

xii Preface

communication research problems will in turn stimulate the future evolution oforder statistics theory, which will benefit the solution of research problems inother research fields.

Diversity, adaptation, and scheduling are becoming three essential conceptsin wireless communications [4]. Diversity combining can effectively mitigate thedeleterious fading effect through the generation and exploration of differentlyfaded replicas of the same information signal. Channel adaptation can acheivehighly spectral and power-efficient transmission over fading channels by matchingthe transceiver parameters properly with the prevailing fading channel condition.Multiuser scheduling can explore the multiuser diversity gain inherent in mul-tiuser systems for overall capacity benefit. These fundamental design principlesmanifest themselves in various transmission and reception technologies for bothnarrowband and wideband systems with single-antenna or multiple-antenna sce-narios. In particular, both multiuser multiple-input–multiple-output (MIMO)systems and orthogonal frequency division multiple access (OFDMA) systemscan apply user scheduling techniques to improve the overall system throughput.

Another goal of this book is to provide an in-depth analysis-oriented expo-sition of diversity combining, link adaptation, and user scheduling techniques.The uniqueness of our approach lies in the fact that for each design optionof different techniques, we obtain the exact analytical expression of importantperformance metrics. Such analytical results build on accurate understandingof scheme design, careful statistical reasoning, and proper application of orderstatistics. Combined with the associated complexity quantification, the book willhelp foster a deep understanding of the underlying design principles and trade-offs of different techniques. The readers will also benefit from the analyticalmethodologies, which may help them solve their specific research problem.

This book is intended for senior graduate students, researchers, and practisingengineers in the field of wireless and mobile communications. Refs [4,7] may serveas suitable background references for this book. The material of this book hasbeen used in a term-long graduate-level course on advanced wireless communi-cations at the University of Victoria, Canada, and an intensive three-week shortcourse at the Tsinghua University, China. It has proven to be an ideal venue forstudents to enhance their analytical skills as well as to expand their knowledgeof advanced wireless transmission technologies.

The authors would like to acknowledge their current and past students, post-docs, and collaborators for their contributions in the works that have beenincluded in this book. Specifically, the authors would like to thank Prof. SeyeongChoi, Prof. David Gesbert, Prof. Young-chai Ko, Prof. Geir Oien, Prof. Khalid A.Qaraqe, Dr. Peng Lu, Dr. Sung-Sik Nam, Dr. Ki-Hong Park, Dr. Lin Yang, Mr.Seung-Sik Eom, Mr. Noureddine Hamdi, Mr. Bengt Holter, and Mr. Le Yang.The authors want to thank Dr. Phil Meyler at Cambridge University Press whoseremarkable talent identified the potential of this project. Finally, the valuablesupport from Ms. Sarah Finlay and Ms. Sabine Koch at Cambridge UniversityPress during the preparation of the book is much appreciated.

Preface xiii

References

[1] H. A. David, Order Statistics. New York, NY: John Wiley & Sons, Inc., 1981.[2] W. C. Jakes, Microwave Mobile Communications. New York: Wiley, 1974.[3] G. L. Stuber, Principles of Mobile Communications, 2nd ed. Norwell, MA: Kluwer

Academic Publishers, 2000.[4] A. J. Goldsmith, Wireless Communications. New York, NY: Cambridge University

Press, 2005.[5] H.-C. Yang, “New results on ordered statistics and analysis of minimum-selection gen-

eralized selection combining (GSC),” IEEE Trans. Wireless Commun., vol. TWC-5,no. 7, pp. 1876–1885, July 2006.

[6] Y.-C. Ko, H.-C. Yang, S.-S. Eom and M.-S. Alouini, “Adaptive modulation with diver-sity combining based on output-threshold MRC,” IEEE Trans. Wireless Commun., vol.TWC-6, no. 10, pp. 3728–3737, October 2007.

[7] M. K. Simon and M.-S. Alouini, Digital Communications over Generalized Fading

Channels: A Unified Approach to Performance Analysis. New York, NY: John Wiley& Sons, 2000.

Notation

General notation

AH Hermitian transpose of matrix ABER−1

n (·) the inverse BER operation< BER > overall bit error rate of adaptive transmission systemBERn average error rate of modulation mode n

E[·] statistical expectationE1(·) exponential integral function of first orderFγ (·) cumulative distribution function of γ

I0(·) modified Bessel function of zero orderJ0(·) Bessel function of zero orderLs

−1(·) inverse Laplace transform with respect to s

Mγ (·) moment generating function of γ

pX,Y (·) joint probability density function of X and Y

pX |Y =y (x) conditional probability density function of X given Y is equal to y

pγ (·) probability density function of γ

Pb(E) bit error ratePb average bit error ratePE (γ) instantaneous error probability for given SNR γ

PE average error ratePL(d) dB path loss at distance d in dBPout outage probabilityPr[·] probability of an eventPs(E) symbol error ratePs average symbol error rateQ(·) Gaussian Q-functionQ1(·, ·) Marcum Q-function· real part of a complex numbers∗ complex conjugate of s

tr· trace of a matrixU(·) unit step functionx vector x (bold face lowercase letter)‖x‖2 norm square of vector x

Notation xv

X matrix X (bold face capital letter)α time derivative of the process α

γl:L lth largest one among total L SNR valuesγ average received SNRγ normalized SNR by its averageΓi sum of the first i largest ordered SNRsΓi:j sum of the i largest SNRs among j onesΓ(·) Gamma functionΓ(·, ·) incomplete Gamma functionΩ short-term average channel power gain

Abbreviations

AAP average access probabilityAAR average access rateAAT average access timeABA-CBBI adaptive beam activation based on CBBIAFL average feedback loadAMDC joint adaptive modulation and diversity combiningASE average spectrum efficiencyASK amplitude shift keyingAT-GSC absolute threshold GSCAWGN additive white Gaussian noiseAWT average waiting timeBBI best beam indexBBSI best beam SINR and indexBER bit error rateBS base stationCBBI conditional best beam index feedbackCDF cumulative distribution functionCLT central limit theoremCSI channel state informationDPC dirty paper codingEDF exceedance distribution functionEGC equal gain combiningGEV generalized extreme-valueGSC generalized selection combiningGSEC generalized switch and examine combiningGSECps GSEC with post-examine selectionGSMuS generalized selection multiuser schedulingGWC-ZFBF greedy weight clique ZFBFi.i.d. independent and identically distributed

xvi Notation

i.n.d. independent and non-identical distributedISI intersymbol interferenceLCR level crossing rateLOS line of sightMEC-GSC minimum estimation and combining GSCMGF moment generation functionMIMO multiple-input–multiple-outputMISO multiple-input–single-outputMRC maximum ratio combiningMSE mean square errorMS-GSC minimum selection GSCNT-GSC normalized threshold GSCOC optimum combiningOFDM orthogonal frequency division multiplexingOFDMA orthogonal frequency division multiple accessOOBS on–off based schedulingOT-MRC output threshold MRCOT-GSC output threshold GSCPDF probability density functionPMF probability mass functionPSD power spectral densityPSK phase shift keyingQAM quadrature amplitude modulationQBBSI BBSI schemes with quantized SINR feedbackQPSK quadrature phase-shift keyingRMS root mean squareRUB random unitary beamformingSBS switched-based schedulingSC selection combiningSEC switch and examine combiningSECps switch and examine combining with post-examining selectionSHO soft handoverSINR signal to interference plus noise ratioSNR signal-to-noise ratioSSC switch and stay combiningSTBC space-time block codeSUP-ZFBF successive projection ZFBFTDD time division duplexingTDMA time division multiple accessT-GSC GSC with threshold test per branchUWB ultra widebandWCDMA wideband code division multiple accessZFBF zero-forcing beamforming

1 Introduction

The wireless communication industry has been and is still experiencing an excit-ing era of rapid development. New technologies and designs emerge on a regularbasis. The timely adoption of these technologies in real-world systems relies heav-ily on the accurate prediction of their performance over general wireless fadingchannels and the associated system complexity. Theoretical performance andcomplexity analysis become invaluable in this process, because they can helpcircumvent the time-consuming computer simulation and expensive field testcampaigns. These analytical results, usually in the form of elegant closed-formsolutions, will also bring important insight into the dependence of the perfor-mance as well as complexity measures on system design parameters and, assuch, facilitate the determination of the most suitable design choice in the faceof practical implementation constraints.

Mathematical and statistical tools play a critical rule in the performance anal-ysis of digital wireless communication systems over fading channels [1]. In fact,the proper utilization of these tools can help either simplify the existing results,which do not allow for efficient numerical evaluation, or render new analyticalsolutions that were previously deemed infeasible. One popular example is theapplication of moment generation function (MGF) in the performance analy-sis of digital communication system over fading channels [2]. With the unifiedMGF-based analytical framework, the error probability expressions, which usu-ally involve an infinite integration of Gaussian Q-function, are simplified to asingle integral of elementary functions with finite limits and, as such, facilitateconvenient and accurate numerical evaluation.

Order statistics is an important sub-discipline of statistics theory [3]. Overthe years, order statistics has made an increasing number of appearances in thedesign and analysis of wireless communication systems. Specifically, order statis-tics have been proven to be valuable tools during the performance analysis ofadvanced diversity techniques, adaptive transmission techniques, and multiuserscheduling techniques, where the simple but effective engineering principle of“selecting the best” frequently applies. This book aims at providing a coherentand systematic presentation of the applications of order statistics in wirelesscommunication system analysis, which will prepare readers to further explorethe potentials of ordered statistics in advanced wireless communication research.

2 Chapter 1. Introduction

1.1 Order statistics in wireless system analysis

The most common figure of merit governing the performance of a communicationsystem is the signal-to-noise ratio (SNR). (For wireless transmission subject tocertain interference, the modified matric signal to interference plus noise ratio(SINR) is often used.) The receiver output SNR, which can be directly relatedto the instantaneous error rate performance, serves as an excellent indicator ofthe fidelity of the detection process. As the result of the fading effect associatedwith multipath wireless channels, the receiver output SNR, usually denoted byγ, will randomly vary over time. In this scenario, the average performance mea-sures should be employed, the evaluation of which would require the statisticaldistribution of γ. For example, the so-called outage probability performance mea-sure, defined as the probability that the channel quality is too poor for reliabletransmission, is calculated as the probability that γ falls below a certain specificthreshold γth , i.e.

Pout =∫ γt h

0pγ (γ)dγ, (1.1)

where pγ (·) is the probability density function (PDF) of γ. The average errorrate of a wireless link can be calculated as

PE =∫ ∞

0PE (γ)pγ (γ)dγ, (1.2)

where PE (γ) is the instantaneous error rate of the modulation scheme of interestfor a given SNR value γ.

The PDF of γ for the conventional narrowband single-antenna link can be eas-ily derived based on the adopted fading channel models. With the introduction ofdiversity, adaptation, and scheduling techniques, the statistics of the receiver out-put SNR become more challenging to obtain. Quite often, such analysis mandatessome order statistics results, as the practical best-selection engineering principlefrequently applies in the designs. As a classical example, the selection diversitycombining technique uses the signal replica with the best quality, i.e. the high-est instantaneous SNR, among several available ones for data detection [1]. Assuch, the receiver output SNR becomes the largest one of several branch SNRs,i.e. γmax = maxγ1 , γ2 , · · · , the statistics characterization of which is readilyavailable in traditional order statistics literature [4]. Specifically, the PDF of thelargest one among L independent and identically distributed (i.i.d.) branch SNRsis given by

pγm a x (x) = L[Fγ (x)]K−1pγ (x), (1.3)

where Fγ (x) and pγ (x) are the common PDF and CDF of γ.Recently, generalized selection combining (GSC) was proposed as an attrac-

tive combining scheme for broadband wireless systems (see for example [5–9]).The basic idea is to select Lc best diversity branches out of a total L available

Diversity, adaptation, and scheduling 3

ones, where Lc < L, and combine them in the optimal maximum ratio combining(MRC) fashion. The combiner output SNR with GSC becomes the sum of theLc largest random variables among a total of L ones, the statistics of which werenot immediately available over the general fading channel models. Furthermore,the analysis of adaptive combining schemes, where the receiver tries to utilize aminimum amount of combining operations to satisfy a certain threshold require-ment, further requires the joint statistics of the partial sums of ordered randomvariables [10–12]. Such joint statistics also find application in the analysis ofmultiuser scheduling schemes for OFDMA and MIMO systems. In this book, wewill provide a comprehensive treatment of the application of order statistics inthe analysis of wireless communication systems and focus in particular on howconventional and new order statistics results help obtain the desired statisticsof received output SNR. These statistical results can then be readily utilizedto calculate various performance metric of interest by following the standardanalytical procedure in [1, 2].

1.2 Diversity, adaptation, and scheduling

Many technologies have been developed to improve the quality and efficiencyof wireless communication systems. While providing a coherent presentation oforder statistics in wireless communications, we focus on three general classes ofwireless techniques, namely diversity, adaptation, and scheduling, all of which areessential building blocks of current and emerging wireless systems and, as such,have received a significant amount of interest from both academia and industry.We adopt a unique analysis-oriented approach in presenting these technologies.Specifically, we present several practical designs for each technology and attacktheir exact performance and complexity analysis over general fading environ-ment with the help of order statistics results. The primary objective is to accu-rately quantify the trade-off of performance versus complexity among differentdesign options, which help foster a thorough and in-depth understanding of eachtechnology.

Diversity combining techniques can effectively improve the performance of awireless communication system operating over a fading environment. Conven-tional combining schemes have been well-documented in various textbooks onwireless communications [1, 13]. Over the past decade, there have been signif-icant developments in the field of advanced diversity combining techniques foremerging broadband wireless systems. Motivated by the fact that a large num-ber of available paths may exist whereas the system can only afford to process alimited number of paths due to the complexity and cost constraint, the commongoal of these newly proposed combining schemes is to efficiently select a subset ofstrong diversity paths and combine them in the MRC fashion from those that areavailable [14–16]. The trade-off of performance versus complexity among differentschemes mandates the exact analysis on each scheme. This book will provide a


comprehensive treatment of several representative advanced diversity-combiningschemes.

Channel adaptive transmission and reception techniques can achieve high spec-trum and/or power efficiency by effectively exploring the time-varying natureof wireless fading channels. Conventional adaptive transmission techniques varyvarious transmission parameters with the prevailing channel conditions to achievehighly spectral-efficient transmission while satisfying a certain error rate require-ment [13]. With the recent development of adaptive diversity combining tech-niques, the amount of combining operation will also vary with the fading channelcondition. The main objective of the new class of diversity techniques is to adap-tively utilize the diversity combiner resource, in terms of active diversity branchesand channel estimations, to achieve an overall complexity saving while satisfyinga certain output performance requirement [10, 11]. The idea of adaptive com-bining can be applied to receive diversity, transmit diversity, and diversity ina multicell environment. Furthermore, receiver adaptive combining can also bedesigned jointly with adaptive transmission to reach a highly efficient transceiverdesign. This book will provide an analysis-oriented presentation of such designsand their underlying design trade-offs.

Multiuser scheduling can explore the diversity benefit inherent in multiuserwireless systems. The idea is to explore the independent variation of multiuserchannels and schedule the users with the best channel conditions to transmit[17,18]. Multiuser scheduling has been shown to be able to acheive great through-put performance. The idea of multiuser scheduling has been incorporated intothe emerging cellular system standards. This book will cover some recent devel-opments in multiuser scheduling, including multiuser parallel scheduling [19],and scheduling in multiuser MIMO systems [20, 21]. We focus on those prac-tical designs with low implementation complexity and which, as such, can bedirectly applied to practical wireless systems. Again, special effort will be madeto quantify the various design trade-offs involved, which will benefit both aca-demic research and practicing engineers.

1.3 Outline of the book

The primary focus of this book is the application of ordered statistics in theperformance and complexity analysis of various wireless communication tech-nologies. The book also covers three general classes of wireless technologies –namely, advanced diversity techniques, channel adaptive transmission, and recep-tion techniques and multiuser scheduling techniques in the context of emergingMIMO–OFDM systems.

The book is organized as follows. We first summarize the basics of digitalwireless communications over fading channels in Chapter 2, which provides thenecessary background for the subsequent chapters. Then, the statistical results,more specifically the conventional and new results on the distribution functions

References 5

of random variables involving order statistics, are presented together with theirderivations in Chapter 3. Chapter 4 concentrates on the analysis and design ofadvanced diversity combining techniques, whereas channel adaptive transmissionand reception techniques are presented in Chapter 5. The concept of multiuserscheduling is explored in Chapter 6 and Chapter 7, in the context of OFDMAsystems and multiuser MIMO system, respectively.

We strive to achieve an ideal balance between theory and practice. The detailedmathematical derivations for order statistics are grouped into one chapter so thatother chapters will focus on the practical design insights. Such an arrangementwill allow easy reading and convenient future referencing. Special emphasis willbe placed on the important trade-off of performance versus complexity through-out the presentation. Whenever deemed necessary, the associated complexitymeasures are quantified and plotted together with the performance for cleartrade-off illustration.

References

[1] G. L. Stuber, Principles of Mobile Communications, 2nd ed. Norwell, MA: KluwerAcademic Publishers, 2000.



[3] N. Balakrishnan and C. R. Rao, Handbook of Statistics 17: Order Statistics: Applica-

tions, 2nd ed. Amsterdam: North-Holland Elsevier, 1998.[4] H. A. David, Order Statistics. New York, NY: John Wiley & Sons, Inc., 1981.[5] N. Kong and L. B. Milstein, “Average SNR of a generalized diversity selection combin-

ing scheme,” IEEE Commun. Letters, vol. 3, no. 3, p. 5759, March 1999, see also Proc.

IEEE Int. Conf. on Commun. (ICC’98), Atlanta, Georgia, pp. 1556–1560, June 1998.[6] Y. Roy, J.-Y. Chouinard and S. A. Mahmoud, “Selection diversity combining with mul-

tiple antennas for MM-wave indoor wireless channels,” IEEE J. Select. Areas Commun.,vol. SAC-14, no. 4, pp. 674–682, May 1998.

[7] M. Z. Win and J. H. Winters, “Virtual branch analysis of symbol error probability forhybrid selection/maximal-ratio combining in Rayleigh fading,” IEEE Trans. Commun.,vol. COM-49, no. 11, pp. 1926–1934, November 2001.

[8] A. Annamalai and C. Tellambura, “Analysis of hybrid selection/maximal-ratio diversitycombiner with Gaussian errors,” IEEE Trans. Wireless Commun., vol. TWC-1, no. 3,pp. 498–512, July 2002.

[9] Y. Ma and S. Pasupathy, “Efficient performance evaluation for generalized selec-tion combining on generalized fading channels,” IEEE Trans. Wireless. Commun.,vol. TWC-3, no. 1, pp. 29–34, January 2004.

[10] S. W. Kim, D. S. Ha and J. H. Reed, “Minimum selection GSC and adaptive low-powerRAKE combining scheme,” in Proc. of IEEE Int. Symp. on Circuits and Systems.

(ISCAS’03), Bangkok, Thailand, vol. 4, May 2003, pp. 357–360.[11] H.-C. Yang, “New results on ordered statistics and analysis of minimum-selection gen-

eralized selection combining (GSC),” IEEE Trans. Wireless Commun., vol. TWC-5,no. 7, pp. 1876–1885, July 2006.


[12] Y.-C. Ko, H.-C. Yang, S.-S. Eom and M. -S. Alouini, “Adaptive modulation with diver-sity combining based on output-threshold MRC,” IEEE Trans. Wireless Commun.,vol. TWC-6, no. 10, pp. 3728–3737, October 2007.

[13] A. Goldsmith, Wireless Communications. Cambridge: Cambridge University Press,2005.

[14] A. I. Sulyman and M. Kousa, “Bit error rate performance of a generalized diversityselection combining scheme in Nakagami fading channels,” in Proc. of IEEE Wireless

Commun. and Networking Conf. (WCNC’00), Chicago, Illinois, September 2000, pp.1080–1085.

[15] M. K. Simon and M.-S. Alouini, “Performance analysis of generalized selection combin-ing with threshold test per branch (T-GSC),” IEEE Trans. Veh. Technol., vol. VT-51,no. 5, pp. 1018–1029, September 2002.

[16] H.-C. Yang and M.-S. Alouini, “Generalized switch and examine combining (GSEC):A low-complexity combining scheme for diversity rich environments,” IEEE Trans.

Commun., vol. COM-52, no. 10, pp. 1711–1721, October 2004.[17] R. Knopp and P. Humblet, “Information capacity and power control in single-cell mul-

tiuser communications,” in Proc. IEEE Int. Conf. Commun. (ICC’95), Seattle, WA,vol. 1, pp. 331–335, June 1995.

[18] D. N. C. Tse,“Optimal power allocation over parallel Gaussian channels”, in Proc. Int.

Symp. Inform. Theory (ISIT’97), Ulm, Germany, p. 27, June 1997.[19] Y. Ma, J. Jin, and D. Zhang, “Throughput and channel access statistics of generalized

selection multiuser scheduling,” IEEE Trans. Wireless Commun., vol. TWC-7, no. 8,pp. 2975–2987, August 2008.

[20] K. K. J. Chung, C.-S. Hwang and Y. K. Kim, “A random beamforming techniquein MIMO systems exploiting multiuser diversity,” IEEE J. Select. Areas Commun.,vol. SAC-21, no. 5, pp. 848–855, June 2003.

[21] M. Sharif and B. Hassibi, “On the capacity of MIMO broadcast channels with partialside information,” IEEE Trans. Inform. Theory, vol. IT-51, no. 2, pp. 506–522, February2005.

2 Digital communications overfading channels

2.1 Introduction

We present a brief summary of digital wireless communications in this chapter.The material will serve as a useful background for the advanced wireless tech-nologies in later chapters. We first review the statistical fading channel modelscommonly used in wireless system analysis. After that, we discuss digital modu-lation schemes and their performance analysis over fading channels, including thewell-known moment generating function (MGF)-based approach [1]. The basicconcept of adaptive modulation and diversity combining will also be presented.While most of the materials of this chapter are reviews of classical results, whichcan be found in other wireless textbooks, this chapter provides for the first time athorough treatment of various conventional threshold-based combining schemes,a class of combining scheme enjoying even lower complexity than selection com-bining (SC). The chapter concludes with a brief discussion of the transmit diver-sity technique. The discussion of this chapter is by no means comprehensive.The main objective is to introduce some common notation and system modelsfor later chapters. For a thorough treatment of these subjects, the reader mayrefer to [1, 2].

2.2 Statistical fading channel models

Wireless channels rely on the physical phenomenon of electromagnetic wave prop-agation, due to the pioneering discoveries of Maxwell and Hertz. Radio wavespropagate through several mechanisms, including direct line of sight (LOS),reflection, diffraction, scattering, etc. As such, there usually exist multiple propa-gation paths between the transmitters and the receivers, as illustrated in Fig. 2.1.In general, when the LOS path exists, as in microwave systems and certain indoorapplications, the transmitted radio signal experiences less attenuation. On theother hand, if the LOS path does not exist, the radio signal can still reach thereceiver through other mechanisms, but with severe attenuation.

The complicated propagation environment and the unpredictable nature ofthe propagation process make the modeling of wireless channels very challeng-ing, especially considering the mobility of the transmitter and/or the receiver. To

8 Chapter 2. Digital communications over fading channels

Line of sight (LOS) path

αi

Mobilev

Building

Building

BS

Figure 2.1 Multipath propagation.

avoid the modeling complication associated with the detailed propagation pro-cess, the wireless channel is usually characterized by three major effects: (i) pathloss, for the general trend of power dissipation as propagation distance increases;(ii) shadowing, for the effects of large objects, such as buildings and trees, alongthe propagation path; and (iii) fading, resulting from the random superpositionof signals from different propagation paths at the receiver. The received signalpower variation due to these three effects is demonstrated in Fig. 2.2. In general,path loss and shadowing are called large-scale propagation effects, whereas fad-ing is referred to as the small-scale effect, since the fading effect manifests itselfin a much smaller time/spatial scale.

2.2.1 Path loss and shadowing

Path loss characterizes the general trend of power loss as the propagation dis-tance increases. The linear path loss is defined as the ratio of the transmittedsignal power Pt versus the received signal power Pr , i.e. PL = Pt/Pr . Among thedifferent types of path loss models, the log-distance model is the most convenientfor high-level system analysis [3]. Specifically, under the log-distance model, pathloss at a distance d in dB scale is predicted using the following formula

PL(d)dB = PL(d0)dB + 10β log10(d/d0), (2.1)

where d0 is the reference distance, usually set to 1–10 m for indoor applicationsand 1 km for outdoor applications, PL(d0)dB is the path loss at d0 and β is thepath loss exponent, which can be estimated by minimizing the mean square error(MSE) between the measurement data and the model. Note that the path lossmodel only captures the general trend of power dissipation, ignoring the effectof specific surrounding objects or multipaths.

Shadowing characterizes the blockage effect of large objects in the propaga-tion environment. As the size, positioning, and properties of such objects are in

Statistical fading channel models 9

Fading

Path loss

Shadowing

Received power

Distance

vMobile

BS

Figure 2.2 Received signal power.

general unknown, the shadowing effect has to be described in the statisticalsense. The most popular shadowing model is the log-normal model, which hasbeen emperically confirmed [4]. With the log-normal model, the path loss in dBscale at distance d, denoted by ψdB , is modeled as a Gaussian random variablewith mean value PL(d)dB, given by appropriate path loss model, and varianceσ2

dB , which varies with the environment. As such, the PDF of ψdB is given by

pψd B (x) =1√

2πσdB

exp(− (x − PL(d)dB)2

2σ2dB

). (2.2)

The log-normal model is so named as the path loss in linear scale ψ is a log-normal random variable, with PDF given by

pψ (x) =10/ ln(10)√

2πxσdB

exp(− (10 log10 x − PL(d)dB)2

2σ2dB

). (2.3)

With the path loss and shadowing model, we can address some interestingsystem design problems, e.g. for a given transmitting power and target servicearea, what is the percentage of coverage after considering the path loss andshadowing effects. Assuming a location is covered if the received signal powerafter shadowing is above the threshold Pmin , the percentage of coverage canbe calculated by averaging the probability that the received signal power at a


distance r is less than Pmin over the target area. Mathematically, we have

C =1

πR2

∫ 2π

0

∫ R

0Pr[Pr (r) > Pmin ]rdrdθ, (2.4)

where R is the radius of the cell. Noting that the coverage probability under thecombined log-distance path loss model and log-normal shadowing model is givenby

Pr[Pr (r) > Pmin ] =∫ Pt −Pm in

−∞pψd B (x)dx, (2.5)

the coverage percentage C can be shown to be given by

C = Q(a) + exp(

2 − 2ab

b2

)Q

(2 − ab

b

)(2.6)

where

a =Pmin − Pt + PL(d0)dB + 10γ log10(R/d0)

σdB, b =

10γ log10(e)σdB

,

and Q(·) is the Gaussian Q-function1, defined as

Q(x) =1√2π

∫ ∞

x

e−t 22 dt. (2.7)

2.2.2 Multipath fading

Fading characterizes the effect of random superposition of signal copies arrivingat the receiver from different propagation paths. These signal replicas may addtogether constructively or cancel one another, which leads to a large variation inreceived signal strength. To better demonstrate the process, let us assume thefollowing bandpass signal is transmitted over a wireless channel

s(t) = Reu(t)ej2πfc t, (2.8)

where u(t) is complex baseband envelope. Due to multipath propagation, thereceived signal becomes

r(t) = Re

⎧⎨⎩

N (t)∑n=0

αn (t)u(t − τn (t))ej2πfc (t−τn (t))+φD n (t)

⎫⎬⎭ . (2.9)

where N(t) is the number of paths, τn (t) is the delay, αn (t) is the amplitude, andφDn

(t) is the phase shift, all for the nth path at time t. Note that the phase shiftφDn

(t) is related to the Doppler frequency shift as φDn(t) =

∫t 2πfDn

(t)dt. Aftersome manipulations while focusing on the complex baseband input and output

1 The Gaussian Q-function is related to the complementary error function erfc(·) by erfc(x) =

2Q(√

2x) and Q(x) = 12 erfc

(x√2

).


c(τ, t)

τ

t1

t2

τ1(t1) τ2(t1) τ3(t1) τ4(t1)

t

τ1(t2)τ2(t2) τ3(t2)

τ4(t2)

Figure 2.3 Impulse response of linear time variant channel.

relationship, the impulse response of the complex baseband wireless channel canbe obtained as

c(τ, t) =N (t)∑n=0

αn (t)e−jφn (t)δ(τ − τn (t)), (2.10)

where φn (t) = 2πfcτn (t) − φDn(t). As such, the wireless multipath channel is

modeled as a linear time-variant system. Figure 2.3 illustrates the impulseresponse of the multipath fading channel. It follows that the wireless fadingchannels cause time-domain variation and power delay spread to the transmit-ted signal. Based on the relative severity of these effects with respect to thetransmitted signal, the multipath fading channels can be classified to slow/fastfading and frequency flat/selective fading.

The so-called RMS (root mean square) delay spread, denoted by σT , is com-monly used to quantify power spread along the delay axis introduced by themultipath channel. The definition of σT is based on the power delay profile,which describes the average signal power distribution along the delay axis andis mathematically given by Ac(τ) = ET [c2(τ, T )]. Specifically, RMS delay spreadσT can be calculated as

σT =

√∫∞0 (τ − µT )2Ac(τ)dτ∫∞

0 Ac(τ)dτ=

√∑Nn=0 α2

n (τn − µT )2∑Nn=0 α2

n

(2.11)

where µT is the average delay spread, given by

µT =

∫∞0 τAc(τ)dτ∫∞0 Ac(τ)dτ

=∑N

n=0 α2nτn∑N

n=0 α2n

. (2.12)

If σT is large compared to the symbol period of the transmitted signal, thenthe delay spread will lead to significant intersymbol interference (ISI). Noting


∑

∆τ ∆τ ∆τ

z0(t) z1(t) zi(t) zL−2(t) zL−1(t)

Figure 2.4 Tapped delay line channel model for selective fading.

that time-domain delay spread will translate to frequency selectiveness in thefrequency-domain, the channel coherence bandwidth serves as an alternativemetric for quantifying power spread along the delay axis. By definition, thechannel coherence bandwidth, denoted by Bc , is the bandwidth over whichthe channel frequency response remains highly correlated. In particular, Bc sat-isfies: AC (∆f) = E[C∗(f1 , t)C(f2 , t)] ≈ 1 for all ∆f = |f1 − f2 | < Bc . It can beintuitively expected that Bc ∝ 1/σT . Correspondingly, the wireless channel isconsidered frequency-flat if σT Ts , or equivalently, Bc Bs , where Bs is thesignal bandwidth. Otherwise, the wireless channel introduces selective fading.

Future wireless systems will use increasely wideband channels to provide highdata rate multimedia services. As such, most emerging systems will be operatingin a frequency-selective fading environment. The most popular model for selec-tive fading is the tapped delay line model, which essentially models the wirelesschannel as a discrete-time filter. With this model, the channel impulse responseis given by

c(τ, t) =L−1∑i=0

zi(t)δ(τ − iτ), (2.13)

where L is the total number of delay bins, also known as the length of the chan-nel, τ is the width of the delay bin, and zi(t) is the composite time-varying gainof all paths in the ith bin. Figure 2.4 illustrates the tapped delay line model forselective fading channels. On the other hand, most current wireless technologies,such as diversity combining, multiuser scheduling, and multi-antenna transmis-sion, are still designed based on frequency-flat fading channel models. The basicpremise of such approaches is that the wideband frequency-selective channelcan be converted into multiple parallel frequency-flat fading channels with thewell-known multicarrier transmission/orthogonal frequency division multiplexing(OFDM) technique [2]. We follow the same approach and focus on the statisticalchannel model for a flat fading environment in the following subsections.


2.2.3 Frequency-flat fading

Frequency-flat fading corresponds to the scenario that the delay spread is smallwith respect to the transmit signal symbol period, i.e. σT Ts . In this case, themultipath can be deemed to arrive at the receiver at the same time. The channelimpulse response simplifies to

c(τ, t) =

⎛⎝N (t)∑

n=0

αn (t)e−jφn (t)

⎞⎠ δ(τ) = z(t)δ(τ), (2.14)

which means that the channel introduces a time-varying complex gain of z(t) =∑N (t)n=0 αn (t)e−jφn (t) . Correspondingly, the complex baseband input/output rela-

tion of the channel becomes

r(t) = z(t)u(t) + n(t), (2.15)

where u(t) is the transmitted complex envelope and n(t) is the additive Gaussiannoise. To develop further statistical models for the complex channel gain z(t),we note that z(t) = zI (t) + jzQ (t), where

zI (t) =N (t)∑n=0

αn (t) cos φn (t), zQ (t) =N (t)∑n=0

αn (t) sin φn (t). (2.16)

As such, z(t) can be modeled as a complex Gaussian random process with theapplication of central limit theorem (CLT). Starting from this basic result, we canarrive at the instantaneous statistics of the channel gain depending on whetheran LOS component exists or not.

When there is no LOS component, the random process z(t) can be assumedto have zero mean. With the additional assumption that φn (t) follows a uni-form distribution over [−π, π] and is independent of αn (t), we can show thatzI (t) and zQ (t) are independently Gaussian random variables with zero mean.It follows that the channel amplitude |z(t)| =

√zI (t)2 + zQ (t)2 is Rayleigh dis-

tributed with distribution function

p|z |(x) =x

σ2 exp[− x2

2σ2

], (2.17)

where σ2 is the common variance of zI (t) and zQ (t). The channel phaseθ(t) = arctan(zQ (t)/zI (t)) is uniform distributed over [0, 2π]. We can furthershow that the channel power gain |z(t)|2 , which is proportional to the instanta-neous received signal power, follows an exponential distribution with PDF givenby

pz 2 (x) =1

2σ2 exp[− x

2σ2

], x ≥ 0, (2.18)

where 2σ2 is the average channel power gain, depending upon the pathloss/shadowing effects.


When the LOS component exists, zI (t) and zQ (t) become a Gaussian randomprocess with nonzero mean. In this case, the instantaneous channel amplitude|z| =

√z2I + z2

Q follows a Rician distribution with distribution function

p|z |(x) =x

σ2 exp[−x2 + s2

2σ2

]I0

(xs

σ2

), (2.19)

where s2 = α20 is the channel power gain of the LOS component, 2σ2 is the

average channel power gain of all non-LOS components, and I0(·) is the modifiedBessel function of zeroth order. Alternatively, the Rician distribution functionis given in terms of the so-called Rician fading parameter K = s2/2σ2 and thetotal average power gain Ω = s2 + 2σ2 . It follows that we can show that thechannel power gain |z(t)|2 follows a non-central χ2 distribution with distributionfunction

p|z |2 (x) =K + 1

Ωexp

[−K − (K + 1)x

Ω

]I0

(2

√K(K + 1)x

Ω

). (2.20)

The Nakagami model is another statistical fading model for the LOS scenarioand was developed from experimental measurements. Based on the Nakagamimodel, the channel amplitude is modeled as a random variable with distributionfunction

p|z |(x) =2mm x2m−1

Γ(m)Ωexp

[−mx2

Ω

], (2.21)

where Γ(·) is the Gamma function and m ≥ 1/2 is the Nakagami fading param-eter. It follows that the distribution function of the channel power gain underthe Nakagami model is given by

pz 2 (x) =(m

Ω

)m xm−1

Γ(m)exp

[−mx

Ω

]. (2.22)

With properly selected values for the Nakagami parameter m, the Nakagamimodel can apply to many fading scenarios. Specifically, when m = 1 (or K = 0for the Rician fading model), we have Rayleigh fading. If m approaches ∞ (orK approaches ∞ for the Rician model), then the model corresponds to the nofading environment. The Nakagami model can well approximate Rician fadingwhen m = (K +1)2

2K +1 . Finally, when m < 1, the Nakagami model applies a fadingscenario that is more severe than Rayleigh fading.

An immediate application of these statistical models would be outage perfor-mance evaluation. Outage occurs when the instantaneous received signal power istoo low for reliable information transmission. For transmission over fading chan-nels, since the received signal power is proportional to the channel power gain,the outage probability can be calculated by evaluating the cumulative distribu-tion function (CDF) of the channel power gain at a certain outage threshold, i.e.Pout = Pr[|z(t)|2 < P0 ], where P0 is the normalized power outage threshold. Forexample, the outage probability of a point-to-point link under the Rician fading


model is given by

Pout = 1 − Q1

(√

2K,

√2(1 + K)

ΩP0

), (2.23)

where Q1(·, ·) is the Marcum Q-function, defined as

Q1(α, β) =∫ ∞

β

x exp(−x2 + α2

2

)I0(αx)dx. (2.24)

Similarly, the outage probability under the Nakagami fading is

Pout = 1 −Γ(m, m

Ω P0)

Γ(m), (2.25)

where Γ(·, ·) is the incomplete Gamma function.

2.2.4 Channel correlation

In the design and analysis of different wireless technologies, we are also interestedin the correlation of the wireless channel gains over space or time. For spatialcorrelation, we focus on the autocorrelation of the bandpass channel gain givenby

Rez(t)ej2πfc t = zI (t) cos(2πfct) − zQ (t) sin(2πfct). (2.26)

With the assumption of uniform scattering, i.e. the angle formed by the incidentwave with the moving direction is uniformly distributed over [0, 2π], we canshow that the autocorrelation and cross-correlation function of the in-phase andquadrature components of the complex channel gain satisfy

AzQ(τ) = AzI

(τ) =Pr

2J0(2πfD τ);

AzI ,zQ(τ) = 0, (2.27)

where J0(x) is the zero-order Bessel function. It follows that the autocorrelationfunction of Rez(t)ej2πfc t is

Az (τ) = AzI(τ) cos(2πfcτ), (2.28)

which is approximately zero if fD τ ≥ 0.4, i.e. vτ ≥ 0.4λ. Based on this result, wearrive at a rule of thumb that channel gains become uncorrelated over a distanceof half wavelength.

The time-domain variation of wireless channel gain is mainly due to the relativemotion of transmitters and receivers.We usually use the so-called channel coher-ence time Tc to characterize the rate of such variation. The channel coherencetime is defined as the time duration that the channel response remains highlycorrelated. In terms of the time-domain correlation of the wireless channel fre-quency response, C(f, t + ∆t), Tc should satisfy E[C∗(f, t)C(f, t + ∆t)] ≈ 1 forall ∆t < Tc . It can be shown, as one would intuitively expect, that Tc is inversely


proportional to the maximum Doppler shift fD = v/λ and approximately givenby Tc ≈ 0.4/fD . If Tc is much greater than the symbol period Ts , then we claimthat the transmitted signal experiences slow fading. Otherwise, the fading is con-sidered to be fast. To meet the increasing demand for high data rate applications,most emerging wireless systems will be operating in a slow fading environment.In this context, a block fading channel model is often adopted in the design andanalysis of various wireless technologies. In particular, the channel response isassumed to remain constant for the duration in the order of the channel coher-ence time and it becomes independent afterwards. While serving as an inaccurateapproximation of the reality, the block fading channel model greatly facilitatesthe description and understanding of new wireless technologies, especially thosebased on channel estimation and feedback. We will adopt the same model in thefollowing chapters unless otherwise noted.

2.3 Digital wireless communications

Most emerging wireless systems employ digital modulation schemes. Digital mod-ulation schemes offer the following advantages among many others: (i) facilitatesource/channel coding for efficient transmission and error protection; (ii) pro-vide better immunity to additive noise and interference; and (iii) achieve higherspectrum efficiency with guaranteed error performance through adaptive trans-mission. While the fundamental process of digital modulation is essentially amapping from bit/bit sequences (obtained after source/channel coding and dig-itization if necessary) to different sinusoidal waveforms, i.e.

djnj=1 =⇒ A(i) cos(2πf(i)t + θ(i)), i = 1, 2, · · · , 2n , (2.29)

different modulation schemes lead to different trade-offs among spectrum effi-ciency, power efficiency, error performance, as well as implementation complex-ity. The desired properties of a modulation scheme for wireless systems include:(i) high spectral efficiency to better explore the limited spectrum resource;(ii) high power efficiency to preserve the valuable power resource of the battery-powered mobile terminals; (iii) robustness to the impairments introduced bymultipath fading; and (iv) low implementation complexity to reduce the over-all system cost. Usually, these are conflicting requirements. Therefore, the bestchoice would be that resulting in the most desirable trade-off.

2.3.1 Linear bandpass modulation

In this context, we focus the class of linear bandpass modulation schemes, whichare widely used in current and emerging wireless systems. With these modulationschemes, the information is carried using either the amplitude and/or phase of thesinusoidal waveform. In particular, the modulated symbols over the ith symbol

Digital wireless communications 17

djnj=1

Mapper

PulseShaping

PulseShaping

CarrierGenerator

π/2

∑ s(t)

A(i) sin θ(i)

A(i) cos θ(i)

cos(2πfct)

sin(2πfct)

Figure 2.5 Demodulator structure of linear modulation schemes.

period can be written as

s(t) = A(i) cos(2πfct + θ(i)), (i − 1)Ts ≤ t ≤ iTs, (2.30)

where fc is the carrier frequency, A(i) and θ(i) are information-carrying ampli-tude and phase. Both amplitude-shift keying (ASK), with constant θ(i), andphase-shift keying (PSK), with constant A(i), are special cases of this generalmodulation type. Applying the trigonometric relationship, we can also write themodulation symbol of the linear modulation scheme into the in-phase/quadraturerepresentation as

s(t) = sI (i) cos 2πfct − sQ (i) sin 2πfct, (2.31)

where sI (i) = A(i) cos θ(i) is the in-phase component and sQ (i) = A(i) sin θ(i)is the quadrature component. The in-phase/quadrature representation is conve-nient for the understanding of the actual modulator implementation, the genericstructure of which is shown in Fig. 2.5. Note that different modulation schemeswill differ only in the mapping from bits to in-phase/quadrature components.Such properties will greatly facilitate the implementation of adaptive modula-tion schemes. Finally, the modulation symbol can be written into the complexenvelope format as

s(t) = Re(sI (i) + jsQ (i))ej2πfc t, (2.32)

where sI (i) + jsQ (i) is the complex baseband symbol, which varies from onesymbol period to another.

The signal space concept is useful to understanding the demodulation processof digital modulation schemes. All linear modulation schemes share the samebasis for their modulated symbols, which are given by

φ1(t) =√

2Ts

cos(2πfct), and φ2(t) =√

2Ts

sin(2πfct). (2.33)


2d ∝ Es

√Es

1000

01 11

(a) QPSK (b) 16-QAM

θ(i)φ1(t)

φ2(t)φ2(t)

φ1(t)

Figure 2.6 Sample symbol constellation for linear bandpass modulation.

The modulated symbols become points in the two-dimensional plane definedby these two orthonormal bases. The collection of all possible symbol pointsform a constellation. Different modulation schemes differ by their constellationstructure. As an illustration, the constellations of a square quadrature ampli-tude modulation (QAM) with M = 4 and M = 16 are plotted in Fig. 2.6. Thecoordinates of the constellation points are given by

siI =

√Ts

2A(i) cos θ(i), siQ =

√Ts

2A(i) cos θ(i), (2.34)

i = 1, 2, · · ·M.

Noting that the energy of the ith symbol can be calculated as

Esi=∫ Ts

0s2(t)dt =

A(i)2Ts

2,

the coordinates simplify to

siI =√

Esicos θ(i), siQ =

√Esi

cos θ(i). (2.35)

The demodulator will first calculate the projection of the received signal overone symbol period, r(t), (i − 1)Ts ≤ t ≤ iTs , onto the symbol space defined bythe orthonormal basis and obtain the sufficient statistics for the detection of thetransmit symbols, which are given by

rI =∫ Ts

0r(t)φ1(t)dt, rQ =

∫ Ts

0r(t)φ2(t)dt. (2.36)

The maximum likelihood detection rule is to detect the symbol si if the point(rI , rQ ) is the closest to (siI , siQ ). The structure of the demodulator is shown inFig. 2.7.

Detection error occurs when the additive noise causes the received symbol tobe closer to a symbol different from that which is transmitted. Therefore, theerror performance of digital modulation scheme depends heavily on the ratio of


r(t) = s(t) + n(t)

MatchFilter

CarrierRecovery

π/2

cos(2πfct)

sin(2πfct)

MatchFilter

rQTs

rITs

Find

Closest si

djnj=1

Figure 2.7 Demodulator structure of linear modulation schemes.

received signal power over noise power over signal bandwidth B. For additivewhite Gaussian noise (AWGN) channels, where the received signal r(t) is relatedto the transmitted signal s(t) simply by

r(t) = s(t) + n(t), (2.37)

where n(t) is the Gaussian random process with mean zero and power spectraldensity (PSD) N0/2, the received signal power is Pr = Es/Ts and the noise powerN = N0/2 · 2B = N0B. It follows that the SNR per symbol can be determinedas

SNR =Pr

N=

Es

N0BTs. (2.38)

The product of BTs varies only with the pulse-shaping function used and, assuch, can be assumed to be constant. As such, the main figure of merit is theratio of Es/N0 , which is commonly referred to as received SNR per symbol anddenoted by γs . For example, it can be shown that the error rate of the binaryPSK (BPSK) modulation scheme over AWGN channel is given by

Pb(E) = Q(√

2γs

), (2.39)

where Q(·) is the Gaussian Q-function. In addition, the symbol error probabilityof quadrature phase shift keying (QPSK) or equivalently, 4-QAM, over AWGNchannel can be shown to be given by

Ps(E) = 1 − [1 − Q(√

2γb

)]2 ≈ 2Q

(√2γb

), (2.40)

where γb = Eb/N0 is the received SNR per bit and related to γs as γb =γs/ log M . Finally, the symbol error probability of square M -QAM is

Ps(E) = 1 −[1 − 2(M − 1)

MQ

(√3γs

M 2 − 1

)]2

. (2.41)


2.3.2 Performance analysis over fading channels

The complex baseband channel model for a flat fading channel is given by

r(t) = z(t)s(t) + n(t), (2.42)

where z(t) is the time-varying complex channel gain, the statistical characteri-zation of which was discussed in previous sections. The instantaneous power ofdesired signal at the receiver can be shown to be given by Pr = |z(t)|2 Es

Tsand will

also vary randomly with |z(t)|2 . It follows that the instantaneous SNR becomes

γs = |z(t)|2 Es

N0, (2.43)

which also becomes a random variable. In this context, the instantaneous systemperformance will not reflect the overall system performance. Instead, we need toapply average performance measures, including the outage probability and theaverage error rate.

Performance measuresOutage occurs when the instantaneous received signal power is too low for reli-able information transmission. Over fading channels, the system may experienceoutage even when the average received SNR, after considering path loss and shad-owing effects, is very large. Equivalently, an outage event can also be defined interms of the instantaneous SNR. Mathematically, the outage probability, denotedby Pout , is given by

Pout = Pr[γs < γth ], (2.44)

where γth is the SNR threshold. For the flat fading scenario, the outage proba-bility can be calculated using the CDF of the instantaneous SNR. For example,the outage probability for the Rician fading case is

Pout = 1 − Q1

(√2K,

√2(1 + K)γth

), (2.45)

where Q1(·, ·) is the Marcum Q-function, and for Nakagami fading is

Pout = 1 − Γ (m,mγth)Γ(m)

, (2.46)

where Γ(·, ·) is the incomplete Gamma function.The average error rate performance can be evaluated by averaging the instan-

taneous error rate over the distribution of the SNR. Note that at any timeinstant, the fading channel can be viewed as an AWGN channel with SNR equalto |z(t)|2 Es

N0. Therefore, the average error rate of a modulation scheme over a

flat fading channel can be calculated by averaging the instantaneous error rate,which is the error rate of this modulation scheme over an AWGN channel withSNR γ, over the distribution function of γ. Mathematically, the average error


rate, denoted by PE , is given by

PE =∫ ∞

0PE (γ)pγ (γ)dγ, (2.47)

where PE (γ) is the error rate over the AWGN channel with SNR γ and pγ (γ) isthe distribution function of γ.

As an example, let us consider the average error rate performance of the BPSKmodulation scheme over Rayleigh fading channels. The instantaneous error rateof BPSK is equal to Q

(√2γ). For Rayleigh fading, the PDF of the received SNR

γ can be shown to be given by

pγ (γ) =1γ

exp(−γ

γ

), γ ≥ 0, (2.48)

where γ = 2σ2Es/N0 is the average received SNR. Therefore, the average errorrate of BPSK over Rayleigh fading can be calculated as

Pb =∫ ∞

0Q(√

2γ) 1

γexp

(−γ

γ

)dγ (2.49)

=12

(1 −

√γ

1 + γ

).

For other fading channel models and/or modulation schemes, we need to plugin different distribution functions and/or an instantaneous error rate expressionand then perform the integration. It is worth noting that in most cases, we cannotobtain a close-form expression, partly because the instantaneous rate expressionusually involves the Gaussian Q-function and its square, and must evaluate theintegration through numerical methods. Since the integration involves the wholereal axis, it is challenging to obtain very accurate results with numerical inte-gration, where truncation is always required. In this context, a new analyticalframework based on the MGF of random variables was proposed and extensivelyused in practice to achieve the accurate evaluation of the average error rate overgeneral fading channels [1].

MGF-based approach for error rate analysisThe MGF of a nonnegative random variable γ is defined as

Mγ (s) =∫ ∞

0pγ (γ)esγ dγ, γ ≥ 0, (2.50)

where s is a complex dummy variable and pγ (·) is the PDF. Due to the similaritywith the Laplace transform definition, we have Mγ (−s) = Lpγ (γ). The MGFis so named because we can easily calculate the moments of random variable γ

as

E[γn ] =dn

dsnMγ (s)|s=0 . (2.51)

Fortunately, the MGF of the received SNR for most fading models are readilyavailable in a compact closed form. For example, the MGF of Rayleigh faded


SNR is

Mγs(s) = (1 − sγs)

−1 , (2.52)

whereas those for Rician and Nakagami faded SNRs are given by

Mγs(s) =

1 + K

1 + K − sγs

es γ s K

1 + K −s γ s , (2.53)

and

Mγs(s) =

(1 − sγs

m

)−m

, (2.54)

respectively.The key starting point of the MGF-based approach is the alternative expres-

sion of the Gaussian Q-function and its square, which was obtained by Craig inthe early 1990s, given by

Q (x) =1π

∫ π/2

0exp

[−x2

2 sin2 φ

]dφ, x > 0. (2.55)

Q2 (x) =1π

∫ π/4

0exp

[−x2

2 sin2 φ

]dφ, x > 0. (2.56)

Applying these alternative expressions, we can calculate the average error rateof most modulation schemes of interest over general fading channel modelsthrough the finite integration of basic functions. As an example, let us considera generic class of modulation schemes whose instantaneous error rate takes theform PE (γ) = aQ

(√bγ), where a and b are modulation-specific constants. With

the conventional approach, the average error rate of this modulation schemeshould be calculated as

Ps =∫ ∞

0aQ

(√bγs

)pγs

(γ)dγ. (2.57)

Substituting the alternative expression of the Q-function and changing the orderof integration, we can rewrite the average error rate as

Ps =a

π

∫ π/2

0Mγs

(−b

2 sin2 φ

)dφ, (2.58)

where Mγs(·) is the MGF of the received SNR under the fading model under

consideration. Note that the resulting expression only involves the integrationwith respect to φ over the integral of [0, π/2].

Figure 2.8 plots the average error rate of coherent BPSK and non-coherentBFSK over fading channels as the function of the average received SNR. Forreference, the error rate of these modulation schemes over AWGN channels withreceived SNR equal to the average SNR over the fading case is also plotted on


0 5 10 15 20 25 3010

−6

10−5

10−4

10−3

10−2

10−1

100

Average SNR, γ, dB

Ave

rage

Bit

Err

or R

ate,

Pb(E

)

Faded

Not Faded

Coherent BPSKNon-coherent BFSK

Figure 2.8 Error rate comparison between fading and non-fading environments.

the same figure. As we can see, the error performance degrades dramaticallyover fading channels. In particular, the error rate decreases in a log rate overthe AWGN channel as the SNR increases, but in a linear rate in the fadingcase. As such, to achieve the same average error rate over fading channels, weneed to maintain a much higher average SNR. An intuitive explanation of thisphenomenon is that fading causes the frequent occurrence of very low receivedsignal power due to destructive cancellation of different multipath signals. Itis exactly from this perspective that diversity techniques try to improve theperformance of wireless systems. Diversity has become one of the most essentialconcepts in wireless system design and implementation. We will elaborate moreon it in the following sections.

2.3.3 Adaptive transmission

Adaptive transmission can achieve very high spectral and power efficiency overwireless fading channels with acceptable error rate performance [5–7]. The basicidea of adaptive transmission is to vary the transmission schemes/parameters,such as modulation mode, coding rate, or transmitting power, with the prevailingfading channel conditions. The system will exploit favorable channel conditionswith higher data rate transmission at lower power levels and response to channel


degradation with reduced data rate or increased power level. As a result, the over-all system throughput is maximized with controlled transmit power consumptionwhile maintaining a certain desired error rate. As such, there has recently beena growing interest in this technique in both academia and industry to meet theincreasing demand for highly spectrum-efficient transmission over wireless fad-ing channels. Several adaptive transmission schemes have been incorporated inGSM/CDMA cellular systems and wireless LAN systems.

The availability of a certain channel state information (CSI) at the transmitteris the fundamental requirement of adaptive transmission techniques. It has beenshown that with perfect CSI at the transmitter, there exists an optimal adap-tive transmission scheme, involving continuous rate and power adaptation, thatcan achieve the Shannon capacity over fading channels. However, the provision ofperfect CSI at the transmitter is a very challenging task in reality even for today’smost advanced wireless systems. In addition, implementing continuous rate adap-tation will entail prohibitively high complexity. As a result, while acknowledg-ing a certain performance gap compared to the optimal scheme, most currentwireless standards adopt adaptive transmission schemes employing discrete rateadaptation, which requires only limited CSI at the transmitter, achieved eitherthrough feedback signaling or by exploring the channel reciprocity. Among them,the constant-power variable-rate adaptive M -QAM scheme is of both theoreticaland practical interest.

Constant-power variable-rate adaptive M-QAMWith the constant-power variable-rate adaptive M -QAM scheme, the systemadaptively selects one of N different M -QAM modulation schemes based onthe fading channel condition while using a fixed power level for transmission.Different M -QAM schemes differ by their constellation sizes. For squared M -QAM schemes, the constellation sizes are M = 2n , n = 1, 2, · · · , N , with size2n corresponding to a spectral efficiency of n bps/Hz based on the Nyquistcriterion. The modulation schemes are chosen to achieve the highest spectralefficiency while maintaining the instantaneous error rate below a certain targetvalue. Specifically, the value range of the channel quality indicator, usually thereceived SNR, is divided into N + 1 regions, with threshold values denoted by 0 <

γT1 < γT2 < · · · < γTN< ∞. When the received SNR γ falls into the nth region,

i.e. γTn≤ γ < γTn + 1 , the constellation size 2n will be selected for transmission.

For practical implementation, the modulation mode selection is carried out atthe receiver after estimating the received SNR. The receiver will then feed backthe mode selection result to the transmitter over the control channels, whichentails a feedback load of only log2 N bits.

The threshold values are set such that the instantaneous error rate of thechosen modulation mode is below a certain target value, denoted by BER0. Forexample, instantaneous bit error rate (BER) of 2n -QAM with two-dimensionalGrey coding over AWGN channel with SNR γ can be approximately calculated


Table 2.1 Threshold values in dB for 2n -ary QAM to satisfy 1%, 0.1%, and 0.01% bit errorrate.

n BER0 = 10−2 BER0 = 10−3 BER0 = 10−4

2 4.32 6.79 8.343 7.07 9.65 11.304 7.88 10.52 12.215 10.84 13.58 15.306 11.95 14.77 16.527 15.40 18.01 19.798 16.40 19.38 21.20

as [6, eq. (28)]

BERn (γ) =15

exp(− 3γ

2(2n − 1)

), n = 1, 2, · · · , N. (2.59)

As such, the threshold values can be calculated, for a target BER value of BER0,as

γTn= −2

3ln(5 BER0)(2n − 1); n = 0, 1, 2, · · · , N. (2.60)

Alternatively, we can solve for the threshold values by inverting the exact BERexpression for square M -QAM [8] as

γTn= BER−1

n (BER0), (2.61)

where BER−1n (·) is the inverse BER expression. The threshold values for the

target BER of 10−2 , 10−3 , and 10−4 cases are summarized in Table 2.1.

Performance analysis over fading channelsThe performance of an adaptive M -QAM system can be evaluated in terms of theaverage spectral efficiency and the average error rate. In particular, the averagespectral efficiency of the adaptive M -QAM scheme under consideration can becalculated as [6, eq. (33)]

η =N∑

n=1

n Pn , (2.62)

where Pn is the probability of using the nth constellation size. With the appli-cation of an appropriate fading channel model, Pn can be calculated using thedistribution function of the received SNR as

Pn =∫ γT n + 1

γT n

pγ (x)dx, (2.63)


where pγ (·) denotes the PDF of the received SNR. The average BER of theadaptive modulation system can be calculated as [6, eq. (35)]

< BER >=1η

N∑n=1

n BERn, (2.64)

where BERn is the average error rate of using modulation mode n, given by

BERn =∫ γT n + 1

γT n

BERn (x)pγ (x)dx, (2.65)

where BERn (γ) is the instantaneous BER of modulation mode n with SNR γ,the approximate expression of which was given in (2.60).

2.4 Diversity combining techniques

Diversity combining techniques can effectively improve the performance of wire-less communication systems over fading channels. The basic idea is to somehowreceive the same data signal over multiple independent channels. Since the prob-ability that these independent channels simultaneously experience deep fade islow, by properly combining these replicas together, we can improve the qualityof the received signal and achieve better performance. The independent chan-nels can be implemented in several ways, including using different frequencies,using different time slots, using different antennas, or using different codewords,etc. In general, the antenna diversity and path diversity approaches are moreattractive as they provide diversity benefit without introducing redundancy intothe transmit signal. We will base our following discussion mainly on the antennareception diversity approach.

2.4.1 Antenna reception diversity

The generic structure of the diversity combiner is shown in Fig. 2.9. Specifically,the diversity combiner will generate its output signal by properly combining thesignal replicas received from L diversity paths. Note that the detection will beperformed based on the combiner output signal. As such, the statistics of thecombiner output, denoted by γc , dictates the overall performance of the sys-tem. The design objective of the diversity combiner is to maximize the qualityof the combined signal under a given complexity constraint. Four traditionalcombining schemes are selection combining (SC), threshold combining, maximalratio combining (MRC), and equal gain combining (EGC). There is a trade-offof performance versus complexity among different combining scheme, as eachscheme requires a different amount of channel state information of each diver-sity path and entails a different level of hardware complexity while leading todifferent performance improvement. To accurately quantify these trade-offs, we

Diversity combining techniques 27

γc

Diversity

Combiner

γ1

γ2

γL

Detector

Figure 2.9 Antenna reception diversity system.

need to accurately evaluate the performance of these combining schemes. In thefollowing, we will briefly explain the basic ideas of these combining schemes andelaborate more on how to analyze the performance of the resulting system basedon the statistics of γc .

For the sake of clarity, we assume in the following that the diversity paths expe-rience independent and identically distributed (i.i.d.) flat fading. The complexchannel gains of the ith diversity path, denoted by zi = aie

jθi , vary indepen-dently with one another over time. As such, the received SNR corresponding tothe ith diversity path is given by γi = a2

iEb

N0, which are independent and identi-

cally distributed random variables and can serve as the channel quality indicatorof each path. The performance of the diversity system will depend on the statis-tics of the combiner output SNR γc . In particular, the outage probability, whichnow becomes the probability that γc is smaller than a threshold γth , is given by

Pout = Pr[γc < γth ] = Fγc(γth), (2.66)

where Fγc(·) denotes the CDF of the combined SNR γc . The average error rate

performance of diversity systems can be evaluated by averaging the instantaneousBER over the distribution of γc as

PE =∫ ∞

0PE (γ)pγc

(γ)dγ, (2.67)

where pγc(·) denotes the PDF of the combined SNR γc , respectively. As such, we

need to derive the statistics of the combiner output SNR based on the combinermode of operation as well as the statistics of the individual path SNR.

Selection combining (SC) is the most popular low-complexity combiningscheme. With SC, the diversity path with the highest SNR is used for datadetection. The mode of operation is illustrated in Fig. 2.10. The combinerneeds to estimate the SNR of all available diversity paths and select the bestone. The combiner output SNR with SC is then mathematically given by


SNR

Time

Channel AChannel B

Selected channelChannel C

Figure 2.10 Selection combining over three diversity paths.

γc = maxγ1 , γ2 , · · · , γL. Therefore, the combiner output SNR is the largestone of L different random variables. The statistics of γc can be easily obtainedfrom the basic order statistic results. As an example, the PDF of γc can beobtained, with i.i.d. fading assumption on different diversity paths, as

pγc(γ) = L[Fγ (γ)]L−1pγ (γ), (2.68)

where Fγ (·) and pγ (·) denote the common CDF and PDF of the path SNRs. Theoutage probability can be easily calculated as

Pout = [Fγ (γth)]L . (2.69)

While SC has relative low receiver complexity as it only needs to process thesignal from one diversity path, SC always requires the estimation of all availablediversity paths.

Maximum ratio combining (MRC) is the optimal linear combining scheme ina noise-limited environment. The basic idea is to generate the combiner out-put signal as a linear combination of the received signal from different diversitypaths such that the SNR of the combined signal is maximized. Mathematicallyspeaking, the combined signal is given by

rc(t) =L∑

i=1

wiri(t) =L∑

i=1

wiaiejθi s(t) +

L∑i=1

wini(t), (2.70)

where ri(t) = aiejθi s(t) + ni(t), i = 1, 2, · · · , L is the received signal from the ith

path, and wi represents the weights for the ith path, which need to be cho-sen optimally. It can be shown that the optimal weights wi should be propor-tional to the complex conjugate of the complexity channel gain of the ith path,


Est.

Est.

Est.

∑

h1

h2

hL

h∗L

h∗1

h∗2

γc

Figure 2.11 Structure of an MRC-based diversity combiner.

i.e. aie−jθi , which will lead to the maximum combiner output SNR given by

γc =L∑

i=1

γi. (2.71)

The structure of an MRC combiner is shown in Fig. 2.11.For the performance evaluation of an MRC scheme, we need to determine

the statistics of the sum of L random variables for the subsequent performanceanalysis. When the path SNRs are independent random variables, the MGF-based approach can readily apply. Note that the MGF of the sum of independentrandom variables is the product of the MGFs of individual random variables. Assuch, the MGF of the combined SNR with MRC over independent fading pathscan be written as

Mγc(s) =

L∏i=1

Mγi(s), (2.72)

where Mγi(s) is the MGF of the ith path SNR. For convenience, we summa-

rize in Table 2.2 the distribution functions of the individual path SNRs underthree popular fading channel models. In Table 2.2, γ is the common averageSNR per branch, Γ(·) is the Gamma function [9, sec. 8.31], I0(·) is the modi-fied Bessel function of the first kind with zero order [9, sec. 8.43], Γ(·, ·) is theincomplete Gamma function [9, sec. 8.35], and Q1(·, ·) is the first-order MarcumQ-function [10]. As an example, for the i.i.d. Rayleigh fading scenario, the MGF


Table 2.2 Statistics of the fading signal SNR γ for the three fading models underconsideration.

Model Rayleigh Rice Nakagami-m

Parameter · K ≥0 m ≥ 12

PDF, pγ (x) 1γ e−

xγ

(1+K )γ e−K− 1 + K

γxI0

(2√

1+Kγ Kx

)(m

γ )m xm −1

Γ(m ) e−m x

γ

CDF, Fγ (x) 1 − e−xγ 1 − Q1

(√2K,

√2(1+K )

γ x

)1 −

Γ(

m, mγ

x)

Γ(m )

MGF, Mγ (s) (1 − sγ)−1 1+K1+K−sγ e

s γ K1 + K −s γ

(1 − sγ

m

)−m

of the combined SNR with MRC is given by

Mγc(s) =

L∏i=1

(1 − sγ)−1 , (2.73)

which leads to the PDF of the combined SNR, after proper inverse Laplacetransform, as

pγc(γ) =

γL−1e−γ/γ

γL (L − 1)!. (2.74)

Both results can apply to the average error rate analysis over Rayleigh fadingchannels.

The equal gain combining (EGC) scheme has slightly lower complexity thanthe MRC. With EGC, the combined signal is still the linear combination ofthe signals received from different paths. But the weight for the ith path withEGC becomes e−jθi , which is simpler than that for MRC. As such, the receiverwith EGC needs only to estimate the channel phase of each diversity path. Ingeneral, it is more challenging to estimate the channel phase than the channelamplitude. Therefore, EGC entails higher implementation complexity than SC.Furthermore, SC needs only to process the currently selected path, whereas theMRC and EGC receiver needs to process all L available paths.

2.4.2 Threshold combining and its variants

Threshold combining is another type of combining scheme with even lower com-plexity than SC [11–13]. With threshold combining, the receiver needs only tomonitor the quality of the currently used branch. Dual-branch switch and staycombining (SSC) is the most well-known example of threshold combining [14,15].With SSC, the receiver estimates the SNR of the currently used branch and com-pares it with a fixed threshold, denoted by γT . If the estimated SNR is greater orequal to γT , then the receiver continues to use the current branch. Otherwise, thereceiver will switch to the other branch and use it for data reception, regardless


SNR

γT

Channel 1

Channel 2

Used channel

Time

Figure 2.12 Mode of operation of dual branch switch and stay combining.

of its quality. The mode of operation of SSC is illustrated in Fig. 2.12 and canbe mathematically summarized, assuming γ1 is the SNR of the currently usedbranch, as

γc =

γ1 , γ1 ≥ γT ;

γ2 , γ1 < γT .(2.75)

To derive the statistics of the combined SNR with SSC, we can start by apply-ing the total probability theorem and writing the CDF of the combined SNR as

Fγc(γ) = Pr[γ1 < γ, γ1 ≥ γT ] + Pr[γ2 < γ, γ1 < γT ]. (2.76)

With the i.i.d. fading path assumption, the CDF can be obtained, in terms ofthe common CDF of individual path SNR, as

Fγc(γ) =

Fγ (γT )Fγ (γ) + Fγ (γ) − Fγ (γT ), γ ≥ γT ;

Fγ (γT )Fγ (γ), γ < γT .(2.77)

It follows that the PDF of the combined SNR with SSC is given by

pγc(γ) =

Fγ (γT )pγ (γ) + pγ (γ), γ ≥ γT ;

Fγ (γT )pγ (γ), γ < γT .(2.78)

The generic expression of the average error rate for a certain modulation schemecan be calculated as

PE = Fγ (γT )∫ ∞

0PE (γ)pγ (γ)dγ +

∫ ∞

γT

PE (γ)pγ (γ)dγ. (2.79)


The statistics of the combined SNR for more general scenario with path correla-tion and/or unbalanced paths can also be obtained by applying a Markov chainbased approach [16].

In the multiple branch scenario, SSC was generalized to switch and examinecombining (SEC) [17], which can exploit the diversity benefit of extra paths.The receiver with SEC will examine the quality of the switched-to path andswitch again if it finds that the quality of that path is unacceptable. This processis continued until either an acceptable path is found or all paths have beenexamined. The mode of operation of L-branch SEC can be summarized with thefollowing mathematical relationship, assuming the first branch is the currentlyused branch,

γc =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

γ1 , γ1 ≥ γT ;

γ2 , γ1 < γT , γ2 ≥ γT ;

γ3 , γ1 < γT , γ2 < γT , γ3 ≥ γT ;...

...

γL , γi < γT , i = 1, 2, · · · , L − 1.

(2.80)

It follows with the i.i.d. fading assumption, that the CDF of SEC output SNRis given by

Fγc(x) =

⎧⎨⎩

[Fγ (γT )]L−1Fγ (x), x < γT ;∑L−1j=0 [Fγ (x) − Fγ (γT )][Fγ (γT )]j

+ [Fγ (γT )]L , x ≥ γT .

(2.81)

The PDF and MGF of the combined SNR with L-branch SEC can thenbe routinely obtained and applied to its performance analysis over fadingchannels.

Figure 2.13 compares the average error rate performance of SC, SSC/SEC, andMRC. As we can see, MRC achieves the best performance and the performancegap increases as the number of diversity paths increase. The performance gainof MRC comes with higher hardware complexity and power consumption. Notethat the receiver needs to know the complete complex channel gain, including theamplitude and phase, for each diversity path in order to determine the optimalbranch weights for MRC. With SC, the receiver only requires the amplitude orpower gain of all diversity branches. We can also see that the complexity savingof threshold combining over SC scheme comes at the cost of high average errorrate, even with the optimal value of the switching threshold.

Another variant of threshold combining is the so-called switch and exam-ine combining with post-examining selection (SECps) [18]. Similar to SSC/SECschemes, the receiver with SECps tries to use an acceptable diversity path byexamining as many paths as necessary. However, when no acceptable path is


0 5 10 15 20 25 3010

−6

10−5

10−4

10−3

10−2

10−1

100

Average SNR per Branch, dB

Ave

rage

Err

or P

roba

bilit

y

SSCSECSCMRC

L=4

L=2

Figure 2.13 Average error rate comparison of conventional diversity combiningschemes [17]. c© 2003 IEEE.

SNR

γT

Channel 1

Channel 2

Used channel

Time

Figure 2.14 Mode of operation of SECps scheme over dual diversity branches.

found after examining all available ones, the receiver with SECps will use thebest unacceptable one. The operation of SECps for the dual branch case is illus-trated in Fig. 2.14. Note that compared with the dual SSC scheme, the SECpsscheme will lead to a better output signal when both branches are unacceptable.


−15 −10 −5 0 510

−6

10−5

10−4

10−3

10−2

10−1

100

Normalized Outage Threshold, γth

/γ−, in dB

Out

age

Pro

babi

lity,

Pou

t

No DiversityDual−branch SECps4−branch SEC4−branch SECps4−branch SC

Figure 2.15 Outage probability of L-branch SECps as a function of normalized outagethreshold γth/γ in comparison with SEC and SC (γT /γ = −5 dB) [18]. c© 2006 IEEE.

The mode of operation of L-branch SECps can be summarized as

γc =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

γ1 , γ1 ≥ γT ;

γ2 , γ1 < γT , γ2 ≥ γT ;

γ3 , γ1 < γT , γ2 < γT , γ3 ≥ γT ;...

...

maxγ1 , γ2 , · · · , γL, γi < γT , i = 1, 2, · · · , L − 1.

(2.82)

Consequently, the CDF of the combined SNR with SECps over i.i.d. fadingpaths can be obtained as

Fγc(x) =

⎧⎪⎪⎨⎪⎪⎩

1 −L−1∑i=0

[Fγ (γT )]i [1 − Fγ (x)], x ≥ γT ;

[Fγ (x)]L , x < γT .

(2.83)

which can be applied to the outage performance evaluation. Figure 2.15 showsthe outage performance of SECps in a multi-branch scenario. As we can see,as the number of diversity paths increases from 2 to 4, the outage performanceof SECps improves considerably. While comparing the outage performance of4-branch SECps with 4-branch SEC and SC, we can see that SECps has the sameoutage probability as SC when γth ≤ γT and as SEC when γth > γT . Intuitively,


Table 2.3 Complexity comparison of dual-branch SECps, SSC, and SC in terms of opera-tions needed for combining decision over Rayleigh fading paths [18].

Schemes Average number of Probability ofchannel estimations branch switching

Dual-branch SC 2 0.5

Dual-branch SSC 1 1 − e−γ Tγ ∈ [0, 1)

Dual-branch SECps 2 − e−γ Tγ ∈ [1, 2) 1

2 − 12 e−2 γ T

γ

when γth ≤ γT , the receiver with SECps experiences outage only when the SNRof all diversity paths falls below γT and their maximum is smaller than γth ,which is equivalent to the case of SC. On the other hand, when γth > γT , thereceiver with SECps experiences outage when the SNR of the first acceptablepath is smaller than γth or when the SNR of all diversity paths falls below γT ,which is the same as the case of conventional SEC.

To summarize this section, we quantitatively compare the complexity of SC,SEC and SECps in terms of the average number of path estimations and theprobability of branch switching. Based on its mode of operation, the averagenumber of path estimations needed by SECps is given by

NSECpsE = 1 +

L−1∑i=1

[Fγ (γT )]i =1 − [Fγ (γT )]L

1 − Fγ (γT ). (2.84)

Note that the SC scheme always needs L path estimations, i.e. NSCE = L. The

receiver with conventional SEC needs to estimate at most L − 1 paths becausewhen the first L − 1 paths are found unacceptable, the receiver will use the lastpath for reception. Therefore, SEC requires less path estimations than SECps,i.e. NSEC

E = 1 +∑L−2

i=1 [Pγ (γT )]i . As a result, we have NSECE ≤ NSECps

E ≤ NSCE .

On the other hand, the probability of branch switching with SECps, denoted byP SECps

SW , can be shown to be given by

P SECpsSW = Fγ (γT ) −

∫ γT

0pγ (x)[Fγ (x)]L−1 dx. (2.85)

It is easy to see that the receiver with SC switches paths with a probabil-ity of (L − 1)/L over identically faded diversity paths, i.e. P SC

SW = (L − 1)/L.Meanwhile, the receiver with L-branch SEC switches paths whenever thecurrent path becomes unacceptable (i.e. γ1 ≤ γT ). As such, the switchingprobability with SEC is P SEC

SW = Pγ (γT ). Consequently, we have P SECpsSW < P SEC

SW

and P SECpsSW < P SEC

SW . Table 2.3 summarizes these results for the L = 2 case.

2.4.3 Transmit diversity

When there are multiple transmit antennas and a single receive antenna, as isoften the case in cellular downlink transmission, we can apply transmit diver-sity techniques to obtain diversity benefit. If the appropriate CSI corresponding


w1 ∝ h∗1

w2 ∝ h∗2

Transmitter

Receiver

Feedback h1, h2

h1

h2

Figure 2.16 Closed loop transmit diversity based on MRC.

to different transmit antenna is available at the transmitter side, then we eas-ily apply the conventional combining schemes discussed in the previous subsec-tion. For example, if the complete complex gain aie

jθi for the channel from theith transmit antenna to the receive antenna is available at the transmitter, wecan implement the so-called transmit MRC by multiplying the transmit signalby a weight wi , which is proportional to aie

−jθi , before transmitting it on theith antenna. To satisfy the total transmit energy constraint, we can chose theamplitude of wi as

|wi | =ai√∑Lj=1 a2

j

, (2.86)

which leads to∑L

i=1 |wi |2 = 1. After propagating through the fading channels,the signal transmitted from different antennas will add up coherently at thereceiver and each is weighted proportional to the amplitude of channel gain. Assuch, the SNR of the received signal is the same as the combiner output SNRwith received MRC, i.e. γc =

∑Li=1 γi . Figure 2.16 illustrates the structure of the

MRC-based transmit diversity system with perfect CSI. Note that the require-ment of complete CSI at the transmitter side can be challenging to satisfy inpractice systems, considering the fact that they usually entail the channel esti-mation and feedback from the receiver. From this perspective, the low-complexitycombining schemes including SC and threshold combining are more attractive,as much less channel information is required for their implementation. For exam-ple, the receiver with transmit SC only needs to inform the transmitter whichantenna leads to the highest receive SNR and therefore should be used, whichresults in log2 L bits of feedback load. With transmitter SSC, the receiver onlyneeds to feed back one bit of information to indicate whether the transmittershould switch antenna or not.

When no CSI can be made available at the transmitter side, we can still explorethe diversity benefit inherent in the multiple transmit antennas through a class oflinear coding schemes, termed space-time block codes [19]. While the code design

Summary 37

for more than two transmit antennas scenario is available, the particular designfor the dual transmit antenna case, widely recognized as Alamouti’s scheme [20],is of great practical importance. Unlike other designs, Alamouti’s scheme achievesfull diversity gain over fading channels without incurring any rate loss. Thescheme transmits two complex data symbols (s1 , s2) over two symbol periodsusing two antennas as follows. In the first symbol period, s1 from antenna 1 ands2 from antenna 2. In the second symbol period, −s∗2 from antenna 1 and s∗1from antenna 2. Assuming that channel gains corresponding to the i transmitantenna hi = aie

jθi , i = 1, 2 remain constant during the two symbol periods, thereceived symbols over two symbol periods are

y1 = h1s1 + h2s2 + n1

y2 = h1(−s∗2) + h2s∗1 + n2 , (2.87)

where n1 and n2 are additive Gaussian noise collected by the receiver. Aftertaking the complex conjugate of y2 , we can rewrite the two equations into matrixform as [

y1

y∗2

]=[h1 h2

h∗2 −h∗

1

] [s1

s2

]+[n1

n∗2

]. (2.88)

With the estimated channel gains, the receiver can perform detection on[z1

z2

]=[h1 h2

h∗2 −h∗

1

]H [y1

y∗2

](2.89)

where [·]H denotes the Hermitian transpose. Due to the special structure of thematrix, we have [

z1

z2

]= (a2

1 + a22)[s1

s2

]+[n1

n2

], (2.90)

where ni can be shown to be zero-mean Gaussian with variance (a21 + a2

2)N0 . Assuch, we can show that the effective SNR of zi for the detection of si is

γi = (a21 + a2

2)Es

2N0=

12(γ1 + γ2). (2.91)

Therefore, Alamouti’s scheme achieves the same diversity gain as received MRCexcept for a 3 dB power loss due to the fact that each symbol is transmittedtwice. The key advantage of Alamouti’s scheme is that absolutely no CSI isrequired at the transmitter side. As a result, this scheme has been included inseveral wireless standards.

2.5 Summary

In this chapter, we reviewed the basics of digital wireless communications, includ-ing channel modeling, digital bandpass modulation, and performance analysis.


Several basic performance improvement techniques, such as adaptive modulation,diversity combining, and transmit diversity were also discussed. While most ofthe materials in this chapter are intended as the background for later chapters,we also present some trade-off analysis of different variants of threshold com-bining. Specifically, we introduce the general performance analysis procedureand sample complexity measures, which will apply to the analysis of advancedwireless technologies in later chapters.

2.6 Bibliography notes

For a more detailed discussion on most of the subjects in this chapter, the readercan refer to the popular textbook of A. Goldsmith [2]. Simon and Alouini’s book[1] provides a thorough and in-depth coverage of digital transmission schemesfor fading channel and their performance evaluation. Ref. [21] gives a completeoverview of the OFDM technology and its application in wireless systems. Thereader can refer to [19] for more general space-time code designs. The Markovchain-based analysis of the different implementation option of the switch and staycombining scheme and their analysis over a general fading channel is availablein [16].

References


Channels, 2nd ed. New York, NY: John Wiley & Sons, 2004.[2] A. J. Goldsmith, Wireless Communications. New York, NY: Cambridge University

Press, 2005.[3] G. L. Stuber, Principles of Mobile Communications, 2nd ed. Norwell, MA: Kluwer

Academic Publishers, 2000.[4] W. C. Jakes, Microwave Mobile Communication, 2nd ed. Piscataway, NJ: IEEE Press,

1994.[5] A. J. Goldsmith and S.-G. Chua, “Adaptive coded modulation for fading channels,”

IEEE Trans. Commun., vol. COM-46, no. 5, pp. 595–602, May 1998.[6] M.-S. Alouini and A. J. Goldsmith, “Adaptive modulation over Nakagami fading chan-

nels,” Kluwer J. Wireless Commun., vol. 13, nos. 1–2, pp. 119–143, 2000.[7] K. J. Hole, H. Holm, and G. E. Oien, “Adaptive multidimensional coded modulation

over flat fading channels,” IEEE J. Select. Areas Commun., vol. SAC-18, no. 7, pp.1153–1158, July 2000.

[8] K. Cho and D. Yoon, “On the general BER expression of one- and two-dimensionalamplitude modulation,” IEEE Trans. Commun., vol. 50, no. 7, pp. 1074–1080, July2002.

[9] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed.San Diego, CA: Academic Press, 1994.

[10] A. H. Nuttall, “Some integrals involving the QM function,” IEEE Trans. on Informa-

tion Theory, vol. 21, no. 1, pp. 95–96, January 1975.

References 39

[11] M. A. Blanco and K. J. Zdunek, “Performance and optimization of switched diversitysystems for the detection of signals with Rayleigh fading,” IEEE Trans. Commun., vol.COM-27, no. 12, pp. 1887–1895, December 1979.

[12] A. A. Abu-Dayya and N. C. Beaulieu, “Analysis of switched diversity systems ongeneralized-fading channels,” IEEE Trans. Commun., vol. COM-42, no. 11, pp. 2959–2966, November 1994.

[13] ——, “Switched diversity on microcellular Ricean channels,” IEEE Trans. Veh. Tech-

nol., vol. VT-43, no. 4, pp. 970–976, November 1994.[14] Y.-C. Ko, M.-S. Alouini, and M. K. Simon, “Analysis and optimization of switched

diversity systems,” IEEE Trans. Veh. Technol., vol. VT-49, no. 5, pp. 1569–1574,September 2000, see also Y.-C. Ko, M.-S. Alouini, and M. K. Simon, “Correction toanalysis and optimization of switched diversity systems,” IEEE Trans. Veh. Technol.,vol. VT-51, pp. 216, January 2002.

[15] C. Tellambura, A. Annamalai and V. K. Bhargava, “Unified analysis of switched diver-sity systems in independent and correlated fading channel,” IEEE Trans. Commun.,vol. COM-49, no. 11, pp. 1955–1965, November 2001.

[16] H.-C. Yang and M.-S. Alouini, “Markov chain and performance comparison of switcheddiversity systems,” IEEE Trans. Commu., vol. COM-52, no. 7, pp. 1113–1125, July2004.

[17] H.-C. Yang and M.-S. Alouini, “Performance analysis of multibranch switched diversitysystems,” IEEE Trans. Commun., vol. COM-51, no. 5, pp. 782–794, May 2003.

[18] H.-C. Yang and M.-S. Alouini, “Improving the performance of switched diversity withpost-examining selection,” IEEE Trans. Wireless Commun., vol. TWC-5, no. 1, pp. 67–71, January 2006.

[19] A. Paulraj, R. Nabar and D. Gore, Introduction to Space-Time Wireless Communica-

tions. Cambridge: Cambridge University Press, 2003.[20] S. M. Alamouti, “A simple transmitter diversity scheme for wireless communications,”

IEEE J. Select. Areas Commun., vol. 16, no. 8, pp. 1451–458, October 1998.[21] R. Prasad, OFDM for Wireless Communication Systems. Boston, MA: Artech House

Publishers, 2004.

3 Distributions of order statistics

3.1 Introduction

The previous chapter shows that the performance analysis of wireless commu-nication systems requires the statistics of the signal-to-noise ratio (SNR) at thereceiver. In the analysis of many advanced wireless communication techniques inlater chapters, we will make use of some order statistical results. This chaptersummarizes these results and their derivations for easy reference. Specifically, wefirst review the basic distribution functions of ordered random variables. Afterthat, we derive some new order statistics results, including the joint distribu-tion functions of partial sums of ordered random variables, for which we alsopresent a novel analytical framework based on the moment generating function(MGF). The chapter is concluded with a discussion on the limiting distributionsof extremes. Whenever appropriate, we use the exponential random variable spe-cial case as an illustrative example. Note that we focus on those order statisticsresults that will be employed in later chapters in the performance and complex-ity analysis of different wireless technologies. For a more thorough treatment oforder statistics, the readers are referred to [1, 2].

3.2 Basic distribution functions

Order statistics deals with the distributions and statistical properties of the newrandom variables obtained after ordering the realizations of some random vari-ables. Let γj ’s, j = 1, 2, · · · , L denote L independent and identically distributed(i.i.d.) nonnegative random variables with common PDF pγ (·) and CDF Fγ (·).Let γl:L denote the random variable corresponding to the lth largest observationof the L original random variables, such that γ1:L ≥ γ2:L ≥ · · · ≥ γL :L . γl:L isalso called lth order statistics. The ordering process is illustrated in Fig. 3.1.

3.2.1 Marginal and joint distributions

As an immediate result of the ordering process, these order statistics are nolonger identically distributed [1]. The PDF of γl:L , l = 1, 2, · · · , L, can be shown

Basic distribution functions 41

Rankingγ1, γ2, · · · , γL γ1:L ≥ γ2:L ≥ · · · ≥ γL:L

Figure 3.1 Ordering random variables.

to be given by

pγl :L (x) =L!

(L − l)!(l − 1)![Fγ (x)]L−l [1 − Fγ (x)]l−1pγ (x). (3.1)

The PDF of the largest random variable, γ1:L , and the smallest random variable,γL :L , can be obtained as

pγ1 :L (x) = L[Fγ (x)]L−1pγ (x), (3.2)

and

pγL :L (x) = L[1 − Fγ (x)]L−1pγ (x), (3.3)

respectively.Another consequence of the ordering operation is that the ordered random

variables are dependent of one another. Specifically, the joint PDF of two arbi-trary order statistics, γl:L and γk :L , l < k, can be shown to be given by [1]

pγl :L ,γk :L (x, y) =L!

(l − 1)!(k − l − 1)!(L − k)!(1 − Fγ (x))l−1 pγ (x) (3.4)

× (Fγ (x) − Fγ (y))k−l−1 pγ (y) (Fγ (y))L−k ,

which is clearly not equal to the product of the marginal PDFs, pγl :L (x) andpγk :L (x). Furthermore, while the joint PDF of the L-original random variablescan be written as the product of individual PDFs, because of the independenceproperty, as

pγ1 ,γ2 ,··· ,γL(x1 , x2 , · · · , xL ) =

L∏i=1

pγ (xi), (3.5)

the joint PDF of the L-ordered random variables becomes

pγ1 :L ,γ2 :L ,··· ,γL :L (x1 , x2 , · · · , xL ) = L!L∏

i=1

pγ (xi), x1 ≥ x2 ≥ · · · ≥ xL . (3.6)

3.2.2 Conditional distributions

Now let us consider the distribution of an ordered random variable given therealization of another ordered random variable [1]. These results are summarizedin the following two theorems.

Theorem 3.1. The conditional distribution of the jth order statistics, γj :L , giventhat the lth order statistics γl:L is equal to y, where j > l, is the distribution of

42 Chapter 3. Distributions of order statistics

the L − jth order statistics of L − l i.i.d. random variables γ−j , whose PDF is

the truncated PDF of the original random variable γ on the right at y, i.e.

pγj :L |γl :L =y (x) = pγ−L −j :L −l

(x), (3.7)

where the PDF of γ−j is given by

pγ−j(x) =

pγ (x)Fγ (y)

, x ≤ y. (3.8)

Proof. By definition, the conditional PDF of γj :L given γl:L = y is

pγj :L |γl :L =y (x) =pγj :L ,γl :L (x, y)

pγl :L (y). (3.9)

After substituting (3.4) and (3.1) into (3.9) and some manipulations, we canshow that

pγj :L |γl :L =y (x) =(L − l)!

(j − l − 1)!(L − j)!

(1 − Fγ (x)

Fγ (y)

)j−l−1pγ (x)Fγ (y)

(Fγ (x)Fγ (y)

)L−j

,

(3.10)which is the PDF of the L − jth order statistics of L − l i.i.d. random variableswith distribution function pγ (x)/Fγ (y) (x ≤ y).

Theorem 3.2. The conditional distribution of the jth order statistics, γj :L ,given that the lth order statistics γl:L is equal to y, where j < l, is the same asthe distribution of the jth order statistics of l − 1 i.i.d. random variables γ+

j ,whose PDF is the truncated PDF of the original random variable γ on the leftat y, i.e.

pγj :L |γl :L =y (x) = pγ +j : l−1

(x), (3.11)

where the PDF of γ+j is given by

pγ +j(x) =

pγ (x)1 − Fγ (y)

, x ≥ y. (3.12)

The theorem can be similarly proved as Theorem 3.1 [3]. These two theoremsestablish that order statistics of samples from a continuous distribution form areversible Markov chain. Specifically, the distribution of γj :L given that γl:L = y

with j < l is independent of γl+1:L , γl+2:L , · · · , γL :L and the distribution of γj :L

given that γl:L = y with j > l is independent of γl−1:L , γl−2:L , · · · , γ1:L . Theseresults become very useful in the derivation of the following sections.

3.3 Distribution of the partial sum of largest order statistics

In the analysis of certain advance diversity combining schemes, we require thestatistics of the sum of the largest order statistics, i.e.

∑Ls

i=1 γi:L , where Ls < L

Distribution of the partial sum of largest order statistics 43

[4–7]. The direct calculation of such a distribution function can be very tedious[8], as the largest Ls order statistics γi:L , i = 1, 2, · · · , Ls are correlated randomvariables with distribution function

pγ1 :L ,γ2 :L ,··· ,γL s :L (x1 , x2 , · · · , xLs) =

L!(L − Ls)!

[Fγ (xLs)]L−Ls

Ls∏i=1

pγ (xi),

x1 ≥ x2 ≥ · · · ≥ xLs. (3.13)

3.3.1 Exponential special case

When γi are i.i.d. exponential random variables, i.e. the PDF of γi is commonlygiven by

pγ (x) =1γ

e−xγ , x ≥ 0, (3.14)

where γ is the common mean, we can invoke the classical result by Sukhatmeto convert the sum of correlated random variables to the sum of independentrandom variables [4, 9].

Theorem 3.3. The spacings between ordered exponential random variablesxl = γl:L − γl+1:L , l = 1, 2, · · · , L, are independently exponential random vari-ables with distribution functions given by

pxl(x) =

l

γe−lx/γ , x ≥ 0. (3.15)

Proof. Omitted.

The sum of the largest order statistics can be rewritten as

Ls∑i=1

γi:L =Ls∑i=1

L∑l=i

xl (3.16)

= x1 + 2x2 + · · · + LsxLs+ LsxLs +1 + · · · + LsxL ,

which becomes the sum of independent random variables. We can then followthe moment generating function (MGF) approach to derive the statistics of thesum. Specifically, the MGF of lxk can be shown to be given by

Mlxk(s) =

∫ ∞

0plxk

(x)esxdx =(

1 − sγl

k

)−1

. (3.17)

Consequently, the MGF of the partial sum∑Ls

i=1 γi:L can be obtained as theproduct of individual MGF as

M∑L si = 1 γi :L

(s) = (1 − sγ)−Ls

L∏l=Ls +1

(1 − sγLs

l

)−1

. (3.18)


Finally, the PDF and CDF of∑Ls

i=1 γi:L can be routinely obtained after applyingproper inverse Laplace transform, for which closed-form results can be obtained.For example, the PDF of

∑Ls

i=1 γi:L is given by

p∑L si = 1 γi :L

(x) =L!

(L − Ls)!Ls !e−

xγ

[xLs −1

γLs (Ls − 1)!

+1γ

L−Ls∑l=1

(−1)Ls + l−1 (L − Ls)!(L − Ls − l)!l!

(Ls

l

)Ls −1

×(

e−l x

L s γ −Ls −2∑m=0

1m!

(− lx

Lsγ

)m)]

. (3.19)

3.3.2 General case

The statistics of∑Ls

i=1 γi:L becomes more challenging to obtain when the γisare not exponentially distributed. Existing solutions usually involve several foldsof integration, which does not lead to convenient mathematical evaluation [5].Based on the results in the previous section, we propose an alternative approachto obtain the statistics of

∑Ls

i=1 γi:L for the general case. The basic idea is totreat the partial sum as the sum of two correlated random variables as

Ls∑i=1

γi:L =Ls −1∑i=1

γi:L + γLs :L . (3.20)

Then, the PDF of∑Ls

i=1 γi:L can be obtained from the joint PDF of γLs :L and∑Ls −1i=1 γi:L ≡ yLs

, denoted by pγL s :L ,yL s(·, ·) as

p∑L si = 1 γi :L

(x) =∫ ∞

0pγL s :L ,yL s

(x − y, y)dy. (3.21)

We now derive the general result of the joint PDF of the lth order statis-tics, γl:L , and the partial sum of the first l − 1 order statistics, yl =

∑l−1j=1 γj :L ,

pγl :L ,yl(·, ·). Note that while the γj s are independent random variables, yl and

γl:L are correlated random variables. Applying the Bayesian formula, we canwrite pγl :L ,yl

(·, ·) as

pγl :L ,yl(x, y) = pγl :L (x) × pyl |γl :L =x(y), (3.22)

where pγl :L (·) is the PDF of the lth order statistics, given in (3.1), and pyl |γl :L =x(·)is the conditional PDF of the sum of the first l − 1 order statistics given thatγl:L is equal to x. The conditional PDF pyl |γl :L =x(·) can be determined with thehelp of the following corollary of Theorem 3.2 [10].

Corollary 3.1. The conditional PDF of the sum of the first l − 1 order statisticsof L i.i.d. random variables (l ≤ L), given that the lth order statistics is equalto y, is the same as the distribution of the sum of l − 1 different i.i.d. randomvariables whose PDF is the truncated PDF of the original random variable on

Distribution of the partial sum of largest order statistics 45

the left at y, i.e.

pyl |γl :L =y (x) = p∑L = l−1j = 1 γ +

j(x), (3.23)

where the PDF of γ+j was given in (3.12).

Proof. Theorem 3.2 states that given that the lth order statistics γl:L = y, theconditional distribution of the jth order statistics with j < l is the same as thedistribution of the jth order statistics of l − 1 different i.i.d. random variableswhose PDF is the truncated PDF of the original random variable on the left aty, as shown in (3.12). Therefore, the distribution of the sum of the first l − 1order statistics of L i.i.d. random variables given that the lth order statisticsγl:L = y is the same as the distribution of the sum of the l − 1 order statistics ofl − 1 i.i.d. random variables with the truncated distribution. The proof of thiscorollary is completed by noting that the ordering operation does not affect thestatistics of the sum of the random variables.

Based on the above corollary, the conditional PDF in (3.22) can be obtainedas the PDF of the sum of l − 1 i.i.d. random variables. The MGF approach canalso apply, which is demonstrated for the exponential r. v. case in the followingexample.

Example 3.1: When γi are i.i.d. exponentially distributed with CDF given by

Fγ (x) = 1 − e−xγ , γ ≥ 0, (3.24)

the truncated PDF in (3.12) specializes to

pγ +j(y) =

1γ

e−x −y

γ , x ≥ y. (3.25)

The corresponding MGF of γ+j can be shown to be given by

Mγ +j(t) =

∫ +∞

0pγ +

j(y)esxdx =

11 − sγ

esy . (3.26)

Consequently, the MGF of the sum of l − 1 i.i.d. random variable γ+j can be

shown to be given by

M∑ l−1j = 1 γ +

j(t) = [Mγ +

j(t)]l−1 =

1(1 − sγ)l−1 e(l−1)sx , (3.27)

which is also the MGF of the conditional random variable. After carrying outa proper inverse Laplace transform, we can obtain the conditional PDF of thesum of the first l − 1 order statistics pyl |γl :L =y (·) as

pyl |γl :L =y (x) = p∑ l−1j = 1 γ +

j(x)

=1

(l − 2)!γl−1 [x − (l − 1)y](l−2)e−x −( l−1 ) y

γ , x ≥ (l − 1)y. (3.28)


Finally, we can obtain a generic expression of the joint PDF pγl :L ,yl(·, ·) after

appropriate substitution into (3.22) as

pγl :L ,yl(x, y) =

L!(L − l)!(l − 1)!

[Fγ (x)]L−l [1 − Fγ (x)]l−1pγ (x) p∑ l−1j = 1 γ +

j(y).

(3.29)

Note that since the random variable γj s are nonnegative and γj :L ≥ γl:L for allj = 1, 2, · · · , l − 1, we have yl =

∑i−1j=1 γj :L ≥ (l − 1)γl:L . Therefore, the support

of the joint PDF pγl :L ,yl(x, y) is the region x > 0, y > (l − 1)x.

For the exponential r. v. special case, noting that the PDF of the lth orderstatistics of L i.i.d. exponential random variables specializes to

pγl :L (x) =L!

γ(l − 1)!

L−l∑j=0

(−1)j

(L − l − j)!j!e−

( l + j )x

γ , (3.30)

we obtain the joint PDF of γl:L and yl , by substituting (3.30) and (3.28) into(3.22), as

pγl :L ,yl(x, y) =

L−l∑j=0

(−1)jL!(L − l − j)!(l − 1)!(l − 2)!j!γl

[y − (l − 1)x](l−2)e−y + ( j + 1 )x

γ ,

x ≥ 0, y ≥ (l − 1)x. (3.31)

As an illustration, the joint PDF of γl:L and yl for exponential special case, asgiven in (3.31), is plotted in Fig. 3.2. Note that since l = 3 here, the joint PDFis equal to zero when yl < 2γl:L .

3.4 Joint distributions of partial sums

In this section, we generalize the result in previous section by deriving the jointdistributions of partial sums of order statistics [3]. We consider two scenariosdepending on whether all ordered random variables are involved or not.

3.4.1 Cases involving all random variables

We first investigate the three-dimensional joint PDF pyl ,γl :L ,zl(y, γ, z), where yl

is the sum of the first l − 1 order statistics, i.e. yl =∑l−1

j=1 γj :L , and zl is the sumof the last L − l order statistics, i.e. zl =

∑Lj= l+1 γj :L . The definition of yl and

zl is illustrated below.

yl =

l−1∑i=1

γi:L

︷︸︸︷γ1:L , · · · , γl−1:L , γl:L ,

zl =

k−1∑i= l+1

γi:L

︷︸︸︷γl+1:L , · · · , γL :L . (3.32)

Joint distributions of partial sums 47

0

5

10

15

20

0

5

10

15

20−0.01

0

0.01

0.02

0.03

0.04

Γi

γ(i)

Join

t PD

F

Figure 3.2 Joint PDF of Γi−1 and γ(i) for the exponential random variable case(L = 6, i = 3, and γ = 6 dB) [3]. c© 2006 IEEE.

Applying twice the Bayesian rule, we can write this joint PDF pyl ,γl :L ,zl(y, γ, z)

as the product of the PDF of γl:L and two conditional PDFs as follows

pyl ,γl :L ,zl(y, γ, z) = pγl :L (γ) × pzl |γl :L =γ (z) × pyl |γl :L =γ ,zl =z (y). (3.33)

The generic expression of pγl :L (·) was given in (3.1) in terms of the common PDFand CDF of γi , pγ (γ) and Pγ (γ). As noted earlier, the order statistics of samplesfrom a continuous distribution form a Markov chain, the conditional distributionof the jth order statistics of L i.i.d. random samples, given that the lth orderstatistics is equal to γ (j < l) and that the sum of the last L − l order statistics isequal to z, is the same as the conditional distribution of the jth order statisticsgiven only that the lth order statistics is equal to γ, i.e.

pγj :L |γl :L =γ ,∑L

k = l + 1 γk :L =z (y) = pγj :L |γl :L =γ (y). (3.34)

With the application of Corollary 1, we have

pyl |γl :L =γ ,zl =z (y) = p∑ l−1j = 1 γ +

j(y), y > (l − 1)γ, (3.35)

where the PDF of γ+j was given in (3.12)

The conditional PDF pzl |γl :L =γ (z) can be determined with the help of thefollowing corollary [3].


Corollary 3.2. The conditional distribution of the sum of the last L − l orderstatistics of L i.i.d. random samples (l ≤ L), given that the lth order statisticsis equal to γ, is the same as the distribution of the sum of L − l different i.i.d.random variables whose PDF is the PDF of the original (unordered) randomvariable truncated on the right of γ, i.e.

pzl |γl :L =γ (z) = p∑L −lj = 1 γ−

j(z), z < (L − l)γ, (3.36)

where the PDF of γ−j is given by

pγ−j(x) =

pγ (x)Fγ (γ)

, 0 < x < γ. (3.37)

Proof. Based on Theorem 3.1, the distribution of the sum of the last L − l orderstatistics of L i.i.d. random variables given that the lth order statistics γl:L = γ

is the same as the distribution of the sum of L − l order statistics of L − l i.i.d.random variables with truncated distribution. The proof is completed by notingthat the ordering operation does not affect the statistics of the sum of randomvariables.

Finally, we obtain the joint PDF pyl ,γl :L ,zl(y, γ, z) as

pyl ,γl :L ,zl(y, γ, z) = pγl :L (γ)p∑ L −l

j = 1 γ−j(z)p∑ l−1

j = 1 γ +j(y), (3.38)

γ > 0, y > (l − 1)γ, z < (L − l)γ.

Example 3.2: When γi are i.i.d. exponentially distributed, pγ−j(x) defined in

(3.37) specializes to

pγ−j(x) =

1γ e−

xγ

1 − e−γγ

, x < γ, (3.39)

Following again the MGF approach, we can obtain the closed-form expressionfor pzl |γl :L =γ (·) as

p∑L −lj = 1 γ−

j(z) =

L−l∑i=0

(L − l

i

)e−z/γ

(1 − e−γ/γ )L−l

(−1)i(z − iγ)L−l−1

γL−l(L − l − 1)!U(z − iγ),

(3.40)where U(·) is the unit step function.

It follows, after properly substituting (3.14), (3.24), (3.28), and (3.36) into(3.38), we can obtain a closed-form expression for the joint PDF pyl ,γl :L ,zl

(y, γ, z)


as

pyl ,γl :L ,zl(y, γ, z) =

L!(L − l)!(l − 1)!γL

[y − (l − 1)γ]l−2

(l − 2)!(L − l − 1)!e−

y + γ + zγ U(y − (l − 1)γ)

×L−l∑i=0

(L − l

i

)(−1)i(z − iγ)L−l−1U(z − iγ),

γ > 0, y > (l − 1)γ, z < (L − l)γ. (3.41)

We can apply the three-dimensional joint PDF of partial sums derived aboveto obtain the joint PDF of the lth order statistics and the sum of the remainingL − 1 order statistics. Specifically, setting l = 1, (3.38) reduces to the joint PDFof the largest random variable and the sum of the remaining ones, denoted bypγ1 :L ,z1 (γ, z), which is given by

pγ1 :L ,z1 (γ, z) = pγ1 :L (γ)p∑ L −1j = 1 γ−

j(z), γ > 0, z < (L − l)γ. (3.42)

For the general case where 1 < l < L, we can obtain the joint PDF of γl:L and∑i = l γi:L , while noting that the latter is equal to yl + zl , as

pγl :L ,yl +zl(γ,w) =

∫ (L−l)γ

0pyl ,γl :L ,zl

(w − z, γ, z)dz, y > (l − 1)γ. (3.43)

These joint PDFs are useful in the performance analysis of different wirelesstransmission technologies in later chapters.

3.4.2 Cases only involving the largest random variables

We now consider the joint PDFs involving only the first k-ordered random vari-ables, i.e. γl:L , l = 1, 2, · · · , k. More specifically, we are interested in obtainingthe joint PDF of the lth largest random variable (l < k) and the sum of theremaining k − 1 ones, which is denoted by

∑kj=1,j = l γj :L [11]. Note that the case

of l = k has been addressed in the previous section.We first consider the general case where 1 < l < k − 1. In this case, we start

with the joint PDF of the following four random variables, yl , γl:L , zkl , γk :L , where

yl and zkl are partial sums defined based on the first k order statistics, as

yl =

l−1∑i=1

γi:L

︷︸︸︷γ1:L , · · · , γl−1:L , γl:L ,

zkl =

k−1∑i= l+1

γi:L

︷︸︸︷γl+1:L , · · · , γk−1:L , γk :L , γk+1:L , · · · , γL :L . (3.44)

Applying the Bayesian rule twice, the 4D joint PDF under consideration,pyl ,γl :L ,z k

l ,γk :L(y, γ, z, β), can be written as

pyl ,γl :L ,z kl ,γk :L

(y, γ, z, β) = pγl :L ,γk :L (γ, β) · pyl |γl :L =γ ,γk :L =β (y)

× pzkl |γl :L =γ ,γk :L =β ,yl =y (z), (3.45)


where pγl :L ,γk :L (·, ·) is the joint PDF of γl:L and γk :L , the generic expression ofwhich was given in (3.4). As shown earlier, ordered random variables satisfy theMarkovian property. Therefore, the conditional PDFs pyl |γl :L =γ ,γk :L =β (y), andpzk

l |γl :L =γ ,γk :L =β ,yl =y (z) are equivalent to pyl |γl :L =γ (y), and pzkl |γl :L =γ ,γk :L =β (z),

respectively. With the application of Corollary 1, we obtain the conditional PDFpyl |γl :L =γ ,γk :L =β (y) as

pyl |γl :L =γ ,γk :L =β (y) = pyl |γl :L =γ (y)

= p∑ l−1i = 1 γ +

i(y), 0 < (l − 1)γ < y, (3.46)

where γ+i is the random variable with truncated distribution given in (3.12).

The conditional PDF pzkl |γl :L =γ ,γk :L =β ,yl =y (z), or equivalently

pzkl |γl :L =γ ,γk :L =β (z), can be determined with the help of the following corol-

lary [11].

Corollary 3.3. The conditional distribution of the sum of k − l + 1 order statis-tics, i.e.

∑k−1i= l+1 γi:L , given that the lth order statistics is equal to γ and the kth

order statistics is equal to β, is the same as the distribution of the sum of k − l + 1different i.i.d. random variables whose PDF is the truncated PDF of the originalrandom variable on the right at γ and on the left at β, i.e.

pzkl |γl :L =γ ,γk :L =β (z) = p∑ k −l−1

i = 1 γ±i(z),

0 < β <z

k − l − 1< γ, (3.47)

where γ±i denotes the random variable with PDF

pγ±i(x) =

pγ (x)Fγ (γ) − Fγ (β)

, β ≤ x ≤ γ.

The results can be proved easily after sequentially applying Corollary 1 andCorollary 2.

Finally, after substituting (3.4), (3.46), and (3.47) into (3.45), we obtain thegeneric expression of the joint PDF pyl ,γl :L ,z k

l ,γk :L(y, γ, z, β) as


(y, γ, z, β) = pγl :L ,γk :L (γ, β) · p∑ l−1i = 1 γ +

i(y) · p∑ k −l−1

i = 1 γ±i(z), (3.48)

0 < β < γ, y > (l − 1)γ, β <z

k − l − 1< γ.

In the case of k = 1, it is sufficient to consider the 3D joint PDF ofpγ1 :L ,z k

1 ,γk :L(γ, z, β), which can be written as

pγ1 :L ,z k1 ,γk :L

(γ, z, β) = pγ1 :L ,γk :L (γ, β) · pzk1 |γ1 :L =γ ,γk :L =β (z). (3.49)

Applying Corollary 3, the 3D joint PDF can be obtained as

pγ1 :L ,z k1 ,γk :L

(γ, z, β) = pγ1 :L ,γk :L (γ, β) · p∑ k −2i = 1 γ±

i(z), (3.50)

0 < β < γ, (k − 2)β < z < (k − 2)γ.


When l = k − 1, the 3D joint PDF of interest becomes pyk −1 ,γk −1 :L ,γk :L (y, γ, β),which can be written, after applying the Bayesian rule, as

pyk −1 ,γk −1 :L ,γk :L (y, γ, β) = pγk −1 :L ,γk :L (γ, β) · pyk −1 |γk −1 :L =γ ,γk :L =β (y). (3.51)

It follows, after proper substitution, pyk −1 ,γk −1 :L ,γk :L (y, γ, β) can be shown to begiven by

pyk −1 ,γk −1 :L γk :L (y, γ, β) = pγk −1 :L ,γk :L (γ, β) · p∑ k −2i = 1 γ +

i(y), (3.52)

0 < β < γ, y > (k − 2)γ.

The 3D joint PDFs pγ1 :L ,z k1 ,γk :L

(γ, z, β) and pyk −1 ,γk −1 :L γk :L (y, γ, β) for the expo-nential random variable special case can be similarly obtained after proper sub-missions into (3.50) and (3.52).

Example 3.3: When γi are i.i.d. exponential random variables, it can be alsoshown that pγ±

i(x) specializes to

pγ±i(x) =

e−x/γ

γ(e−β/γ − e−γ/γ

) , β ≤ x ≤ γ, (3.53)

The corresponding MGF is given by

Mγ±i(s) =

∫ γ

β

pγ±i(x)esxdx =

esβ−β/γ − esγ−γ/γ

(e−β/γ − e−γ/γ )(1 − sγ). (3.54)

Noting that the MGF of∑k−l−1

i=1 γ±i is equal to [Mγ±

i(s)]k−l−1 , we can obtain the

closed-form expression for (3.47) by taking the inverse Laplace transform as

pzkl |γl :L =γ ,γk :L =β ,yl =y (z) = L−1[Mγ±

i(s)]k−l−1 (3.55)

=k−l−1∑j=0

(k − l − 1

j

)(−1)j e−z/γ [z − β(k − l − j − 1) − γj]k−l−2

[(e−β/γ − e−γ/γ )γ]k−l−1(k − l − 2)!

×U(z − β(k − l − j − 1) − γj),

(k − l − 1)β < z < (k − l − 1)γ.

Meanwhile, the joint PDF of γl:L and γk :L given in (3.4) becomes

pγl :L ,γk :L (x, y) =L!

(l − 1)!(k − l − 1)!(L − k)!γ2 e−l x + y

γ (3.56)

×(e−

yγ − e−

xγ

)k−l−1·(1 − e−

yγ

)L−k

,


After substituting (3.28),(3.55), and (3.56) into (3.48), we can obtain aclosed-form expression for the 4D joint PDF, pyl ,γl :L ,z k

l ,γk :L(y, γ, z, β), for the

exponential random variable case as


(y, γ, z, β) (3.57)

=L!e−(y+γ+z+β )/γ (1 − e−β/γ )L−k [y − (l − 1)γ]l−2

(L − k)!(k − l − 1)!(k − l − 2)!(l − 1)!(l − 2)!γk

×k−l−1∑j=0

(k − l − 1

j

)(−1)j [b − β(k − l − j − 1) − γj]k−l−2

×U(γ)U(γ − β)U(y − (l − 1)γ)U(z − β(k − l − j − 1) − γj),

(k − l − 1)β < z < (k − l − 1)γ.

Remark: We notice from the above three examples that the derivation of thejoint PDFs often involves the statistics of the sum of some truncated randomvariables. For convenience, we summarize the PDF and MGF of the truncatedrandom variables corresponding to the received SNR of the three most popularfading channel models in Table 3.1.

With these multiple dimensional joint PDFs available, we can readily derivethe joint PDF of any partial sums involving the largest k-ordered randomvariables. For example, we can obtain the joint PDF of Y = yl + γl:L andZ = zk

l + γk :L , pY ,Z (y, z)as

pY ,Z (y, z) =∫ z

k −l

0

∫ yl

zk −l


(y − γ, γ, z − β, β)dγdβ, y >l

k − lz.

(3.58)In addition, the joint PDF γl:L and W = yl + zk

l + γk :L , denoted bypγl :L ,W (γ,w), can be calculated as

pγl :L ,W (γ,w) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

∫ (k−2)γ

(k−2)w/(k−1)pγ1 :L ,z k

1 ,γk :L(γ, z, w − z)dz, l = 1;∫ γ

0

∫ (k−l−1)γ

(k−l−1)β


(w − z − β, γ, z, β)dzdβ, 1 < l < k − 1;∫ γ

0pyk −1 ,γk −1 :L γk :L (w − β, γ, β)dβ, l = k − 1.

(3.59)Note that only finite integrations of the joint PDFs are involved in these cal-culations. Therefore, even though a closed-form expression is tedious to obtain,this joint probability can be easily calculated with mathematical software, suchas Mathematica and Maple.

MGF-based unified analytical framework for joint distributions 53

Table 3.1 Truncated MGFs for the three fading models under consideration.

Mγ + (s) where γ+ has PDF pγ +j(x) = pγ (x)

1−Pγ (γT ) , x ≥ γT .

Rayleigh 11−sγ esγT

Rice 1+K1+K−sγ e

s γ K1 + K −s γ

Q 1

(√2 K ( 1 + K )1 + K −s γ

,√

2(1+K−sγ ) γ Tγ

)Q 1

(√2K ,

√2(1+K ) γ T

γ

)Nakagami-m

(1 − sγ

m

)−m Γ(

m,m γ T

γ−sγT

)Γ(

m,m γ T

γ

)Mγ−(s) where γ− has PDF pγ−

j(x) = pγ (x)

Pγ (γT ) , 0 < x < γT .

Rayleigh 11−sγ

1−es γ T − γ T

γ

1−e− γ T

γ

Rice 1+K1+K−sγ e

s γ K1 + K −s γ

1−Q 1

(√2 K ( 1 + K )1 + K −s γ

,√

2(1+K−sγ ) γ Tγ

)1−Q 1

(√2K ,

√2(1+K ) γ T

γ

)Nakagami-m

(1 − sγ

m

)−m 1−Γ(

m,m γ T

γ−sγT

)/Γ(m )

1−Γ(

m,m γ T

γ

)/Γ(m )

Mγ±(s) where γ± has pγ±i(x) = pγ (x)

Pγ (γT 1 )−Pγ (γT 2 ) , γT2 ≤ x ≤ γT1 .

Rayleigh 11−sγ

es γ T 2

−γ T 2

γ −es γ T 1

−γ T 1

γ

e−

γ T 2γ −e

−γ T 1

γ

Rice1 + K

1 + K −s γe

s γ K

1 + K −s γ Q 1

(√2 K ( 1 + K )1 + K −s γ

,√

2(1+K−sγ )γ T 2

γ

)Q 1

(√2K ,

√2(1+K )

γ T 2γ

)−Q 1

(√2K ,

√2(1+K )

γ T 1γ

)

−1 + K

1 + K −s γe

s γ K

1 + K −s γ Q 1

(√2 K ( 1 + K )1 + K −s γ

,√

2(1+K−sγ )γ T 1

γ

)Q 1

(√2K ,

√2(1+K )

γ T 2γ

)−Q 1

(√2K ,

√2(1+K )

γ T 1γ

)

Nakagami-m(1 − sγ

m

)−m Γ(

m,m γ T 2

γ−sγT 2

)−Γ

(m,

m γ T 1γ

−sγT 1

)Γ(

m,m γ T 2

γ

)−Γ

(m,

m γ T 1γ

)

3.5 MGF-based unified analytical frameworkfor joint distributions

In this section, we present a unified analytical framework to determine the jointstatistics of partial sums of ordered RVs using an MGF-based approach [15]. Morespecifically, we extend the result in [12–14], which only derives the joint MGF ofthe selected individual order statistics and the sum of the remaining ones, andsystematically solves for the joint statistics of arbitrary partial sums of orderedRVs. The main advantage of the proposed MGF-based unified framework is thatit applies not only to the cases when all the K-ordered RVs are considered,but also to those cases when only the Ks (Ks < K) best RVs are involved. Foreach scenario, we present some selected examples to illustrate the derivationprocedure.


3.5.1 General steps

The proposed analytical framework adopts a general two-step approach:

(i) obtain the analytical expressions of the joint MGF of partial sums (notnecessarily the partial sums of interest as will be seen later);

(ii) apply the proper inverse Laplace transform to derive the joint PDF of partialsums (additional integration may be required to obtain the desired jointPDF).

To facilitate the inverse Laplace transform calculation, the joint MGF from step(i) should be made as compact as possible. An observation made in [13–15]involving the interchange of multiple integrals of ordered RVs becomes useful inthe following analysis. Suppose, for example, that we need to evaluate a multipleintegral over the range γa ≥ γ1 ≥ γ2 ≥ γ3 ≥ γ4 ≥ γb . More specifically, let

I =∫ γa

γb

dγ1

∫ γ1

γb

dγ2

∫ γ2

γb

dγ3

∫ γ3

γb

dγ4 p (γ1 , γ2 , γ3 , γ4). (3.60)

It can be shown that by interchanging the order of integration, while ensuringeach pair of integration limits is chosen to be as tight as possible, the multipleintegral in (3.60) can be rewritten into the following equivalent representations,

I =∫ γa

γb

dγ4

∫ γa

γ4

dγ3

∫ γa

γ3

dγ2

∫ γa

γ2

dγ1 p (γ1 , γ2 , γ3 , γ4)

=∫ γa

γb

dγ2

∫ γ2

γb

dγ3

∫ γ3

γb

dγ4

∫ γa

γ2

dγ1 p (γ1 , γ2 , γ3 , γ4)

=∫ γa

γb

dγ3

∫ γa

γ3

dγ1

∫ γ3

γb

dγ4

∫ γ1

γ3

dγ2 p (γ1 , γ2 , γ3 , γ4)

=∫ γa

γb

dγ1

∫ γ1

γb

dγ4

∫ γ1

γ4

dγ3

∫ γ1

γ3

dγ2 p (γ1 , γ2 , γ3 , γ4)

=∫ γa

γb

dγ4

∫ γa

γ4

dγ1

∫ γ1

γ4

dγ3

∫ γ1

γ3

dγ2 p (γ1 , γ2 , γ3 , γ4). (3.61)

The general rule is that the integration limits should be selected as tight aspossible using the remaining variables. For example, in the first equation of(3.61), the variables are integrated in the order of γ1 , γ2 , γ3 , and γ4 . Based onthe given inequality condition γa ≥ γ1 ≥ γ2 ≥ γ3 ≥ γ4 ≥ γb , the integration limitof γ1 should be from γ2 to γa , because γ2 is the tightest among the remainingRVs. Similarly, the integration limit γ3 is from γ4 to γa , because γ1 and γ2 werealready integrated out.

After obtaining the joint MGF in a compact form, we can derive the jointPDF of a selected partial sum through an inverse Laplace transform. For mostof our cases of interest, the joint MGF involves basic functions, for which theinverse Laplace transform can be calculated analytically. In the worst case, wemay rely on the Bromwich contour integral. The final joint PDF involves at


most one single one-dimensional contour integration, which can be easily andaccurately evaluated numerically with the help of the integral tables [16, 17] orusing standard mathematical packages such as Mathematica and Matlab.

The general steps can be directly applied when all K-ordered RVs are consid-ered and the RVs in the partial sums are continuous. When these conditions donot hold, we need to apply some extra steps in the analysis in order to obtain avalid joint MGF. Specifically, when only the best Ks (Ks < K)-ordered RVs areinvolved in the partial sums, we should consider the Ksth order statistics γKs :K

separately. When the RVs involved in a partial sum are not continuous, i.e. areseparated by the other RVs, we need to divide this partial sum into smallersums. Without such separation, we cannot find the valid integration limit whencalculating the joint MGF. In both cases, when the joint PDF of the new partialsums are derived, we need to perform another finite integration to obtain thedesired joint PDF.

3.5.2 Illustrative examples

In the following, we present several examples to illustrate the proposed analyticalframework. Our focus is on how to obtain a compact expression of the jointMGFs, which can be greatly simplified with the application of the followingfunctions and relations.

Common functions

(i) A mixture of a CDF and an MGF c (γ, λ):

c (γ, λ) =∫ γ

0dx p (x) exp (λx), (3.62)

where p (x) denotes the PDF of the RV of interest. Note that c (γ, 0) = c (γ)is the CDF and c (∞, λ) leads to the MGF. Here, the variable γ is real,while λ can be complex.

(ii) A mixture of an exceedance distribution function (EDF) and an MGF,e (γ, λ):

e (γ, λ) =∫ ∞

γ

dx p (x) exp (λx). (3.63)

Note that e (γ, 0) = e (γ) is the EDF while e (0, λ) gives the MGF.(iii) An interval MGF µ (γa , γb , λ):

µ (γa , γb , λ) =∫ γb

γa

dx p (x) exp (λx). (3.64)

Note that µ (0,∞, λ) gives the MGF.


The functions defined in (3.62), (3.63) and (3.64) are related as follows

c (γ, λ) = e (0, λ) − e (γ, λ) (3.65)

= c (∞, λ) − e (γ, λ)

e (γ, λ) = c (∞, λ) − c (γ, λ) (3.66)

= e (0, λ) − c (γ, λ)

µ (γa , γb , λ) = c (γb, λ) − c (γa , λ) (3.67)

= e (γa , λ) − e (γb, λ) .

Simplifying relationship

(i) Integral Im defined as

Im =∫ γm −1 :K

0dγm :K p (γm :K ) exp (λγm :K )

×∫ γm :K

0dγm+1:K p (γm+1:K ) exp (λγm+1:K )

×∫ γm + 1 :K

0dγm+2:K p (γm+2:K ) exp (λγm+2:K )

· · ·∫ γK −1 :K

0dγK :K p (γK :K ) exp (λγK :K ), (3.68)

can be expressed in terms of the function c (γ, λ) as

Im =1

(K − m + 1)![c (γm−1:K , λ)](K−m+1) . (3.69)

(ii) Integral I ′m defined as

I ′m =∫ ∞

γm + 1 :K

dγm :K p (γm :K ) exp (λγm :K )∫ ∞

γm :K

dγm−1:K p (γm−1:K ) exp (λγm−1:K )

×∫ ∞

γm −1 :K

dγm−2:K p (γm−2:K ) exp (λγm−2:K )

· · ·∫ ∞

γ2 :K

dγ1:K p (γ1:K ) exp (λγ1:K ), (3.70)

can be expressed in terms of the function e (γ, λ) as

I ′m =1m!

[e (γm+1:K , λ)]m . (3.71)


(iii) Integral I ′′a,b defined as

I ′′a,b =∫ γa :K

γb :K

dγb−1:K p (γb−1:K ) exp (λγb−1:K )∫ γa :K

γb−1 :K

dγb−2:K p (γb−2:K ) exp (λγb−2:K )

×∫ γa :K

γb−2 :K

dγb−3:K p (γb−3:K ) exp (λγb−3:K )

· · ·∫ γa :K

γa + 2 :K

dγa+1:K p (γa+1:K ) exp (λγa+1:K ), (3.72)

can be expressed in terms of the function µ (·, ·) as

I ′′a,b =1

(b − a − 1)![µ (γb:K , γa :K , λ)](b−a−1) for a < b. (3.73)

The proof of this simplifying relationship can be found in the Appendix of thischapter.

Let us first consider the joint PDF ofm∑

n=1γn :K and

K∑n=m+1

γn :K , which involves

all K-ordered random variables. Let Z1 =m∑

n=1γn :K and Z2 =

K∑n=m+1

γn :K for

convenience. The following theorem presents the joint PDF of Z1 and Z2 .

Theorem 3.4. The two-dimensional joint PDF of Z1 and Z2 can be calculatedas

pZ1 ,Z2 (z1 , z2) = L−1S1 ,S2

MGFZ1 ,Z2 (−S1 ,−S2)

=K!

(K − m)! (m − 1)!

∞∫0

dγm :K

[p (γm :K )

×L−1S1

exp (−S1γm :K ) [e (γm :K ,−S1)]

(m−1)

×L−1S2

[c (γm :K ,−S2)]

(K−m )]

for z1 ≥ m

K − mz2 . (3.74)

Proof. The second-order MGF of Z1 and Z2 is given by the expectation

MGFZ1 ,Z2 (λ1 , λ2) = E exp (λ1Z1 + λ2Z2)

= K!

∞∫0

dγ1:K p (γ1:K ) exp (λ1γ1:K ) · · ·γm −1 :K∫

0

dγm :K p (γm :K ) exp (λ1γm :K ) (3.75)

×γm :K∫0

dγm+1:K p (γm+1:K ) exp (λ2γm+1:K )· · ·γK −1 :K∫

0

dγK :K p (γK :K ) exp (λ2γK :K ).


After applying (3.69), the MGF can be rewritten into the following form.

MGFZ1 ,Z2 (λ1 , λ2) = K!

∞∫0

dγ1:K p (γ1:K ) exp (λ1γ1:K )

× · · ·γm −1 :K∫

0

dγm :K p (γm :K ) exp (λ1γm :K )1

(K − m)![c (γm :K , λ2)]

(K−m ) .

(3.76)

Changing the order of integration based on the principle of (3.61), we can rewrite(3.76) as

MGFZ1 ,Z2 (λ1 , λ2) =K!

(K − m)!

∞∫0

dγm :K p (γm :K ) exp (λ1γm :K )

× [c (γm :K , λ2)](K−m )

∞∫γm :K

dγm−1:K p (γm−1:K ) exp (λ1γm−1:K )

× · · ·∞∫

γ2 :K

dγ1:K p (γ1:K ) exp (λ1γ1:K ). (3.77)

Finally, applying (3.71), we can obtain the second-order MGF of Z1 and Z2 as

MGFZ1 ,Z2 (λ1 , λ2) =K!

(K − m)! (m − 1)!∞∫

0

dγm :K p (γm :K ) exp (λ1γm :K ) [c (γm :K , λ2)](K−m ) [e (γm :K , λ1)]

(m−1) .

(3.78)

The desired two-dimensional joint PDF of Z1 =m∑

n=1γn :K and Z2 =

K∑n=m+1

γn :K

can be obtained after proper inverse Laplace transform.

For the exponential random variable special case, with the help of the inverseLaplace transform pair [16]

Ls−1(

1s + a

)n=

1(n − 1)!

tn−1e−at , t ≥ 0, n = 1, 2, 3, . . . , (3.79)

and the Laplace transform property [16]

Ls−1 e−asF (s)

= f (t − a) U (t − a) , a > 0, (3.80)


we can obtain the following compact expression for the joint PDF of Z1 and Z2 ,as

pZ1 ,Z2 (z1 , z2) =⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

K !(K−m )!(K−m−1)!(m−1)!(m−2)!γ K exp

(− z1 +z2

γ

)×

∞∫0

dγm :K

[[z1 − mγm :K ]m−2 U (z1 − mγm :K )

×K−m∑j=0

(−1)j (K−mj

)[z2 − jγm :K ]K−m−1 U (z2 − jγm :K )

], m ≥ 2;

K !(K−1)!(K−2)!γ K exp

(− z1 +z2

γ

)×

K−1∑j=0

(−1)j (K−1j

)[z2 − jz1 ]

K−2 U (z2 − jz1), m = 1.

(3.81)

We now consider the joint PDF of γm :K andKs∑

n=1n =m

γn :K , which involves the Ks

largest ordered random variables among a total K ones. Note also that dependingon the value of m, the ordered random variables involved in the summation maybe discontinuous. In these cases, we may need to first consider the joint PDF ofsome new partial sums. The results are summarized in the following theorem.

Theorem 3.5. The joint PDF of γm :K andKs∑

n=1n =m

γn :K can be obtained as

pγm :K ,

K s∑n=1n =m

γn :K

(x, y) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

∫ (Ks −2)x

(K s −2K s −1 )y

pγ1 :K ,

K s −1∑n = 2

γn :K ,γK s :K

(x, z2 , y − z2) dz2 , m = 1,

∫ x

0

∫ y−(Ks −m )z4

(m−1)x

pm −1∑n = 1

γn :K ,γm :K ,K s −1∑

n = m + 1γn :K ,γK s :K

(z1 , x, y − z1 − z4 , z4) dz1dz4 , 1 < m < Ks − 1,

∫ y

(Ks −2)x pK s −2∑n = 1

γn :K ,γK s −1 :K ,γK s :K

(z1 , x, y − z1) dz1 , m = Ks − 1,

pγK s :K ,

K s −1∑n = 1

γn :K

(x, y) , m = Ks,

(3.82)

where the joint PDFs involved are given by, for m = 1,

pγ1 :K ,

K s −1∑n = 2

γn :K ,γK s :K

(z1 , z2 , z3)

=K!

(Ks − 2)!p (z1) p (z3) [c (z3)]

(K−Ks ) U (z1 − z3)L−1S2

[µ (z3 , z1 ,−S2)]

(Ks −2)

,

z3 < z1 , (Ks − 2) z3 < z2 < (Ks − 2) z1 , (3.83)


for 1 < m < Ks − 1,

pm −1∑n = 1

γn :K ,γm :K ,K s −1∑

n = m + 1γn :K ,γK s :K

(z1 , z2 , z3 , z4)

=K!

(Ks − m − 1)! (m − 1)!p (z2) p (z4) [c (z4)]

(K−Ks ) U (z2 − z4)

×L−1S1

[e (z2 ,−S1)]

(m−1)L−1

S3

[µ (z4 , z2 ,−S3)]

(Ks −m−1)

,

z4 < z2 , (m − 1)z2 < z1 , (Ks − m − 1) z4 < z3 < (Ks − m − 1) z2 , (3.84)

for m = Ks − 1,

pK s −2∑n = 1


(z1 , z2 , z3)

=K!

(Ks − 2)!p (z2) p (z3) [c (z3)]

(K−Ks ) U (z2 − z3)L−1S1

[e (z2 ,−S1)]

(Ks −2)

,

z3 < z2 , z1 > (Ks − 2) z2 , (3.85)

for m = Ks ,

pγK s :K ,

K s −1∑n = 1

γn :K

(z1 , z2)

=K!

(Ks − 1)!p (z1) [c (z1)]

(K−Ks ) L−1S2

[e (z1 ,−S2)]

(Ks −1)

,

z2 ≥ (Ks − 1) z1 , (3.86)

where U (·) is the unit step function.

Proof. The proof is presented in the Appendix of this chapter.

Note that (3.82) involves only finite integrations of higher-dimensional jointPDFs, for which the closed-form expression can usually be obtained. For example,for the exponential random variable special case, the joint PDFs involved in(3.82) can be shown to be given by, for m = 1,

pγ1 :K ,

K s −1∑n = 2

γn :K ,γK s :K

(z1 , z2 , z3) (3.87)

=K!

(K − Ks)! (Ks − 2)! (Ks − 3)!γKsexp

(−z1 + z2 + z3

γ

)

×[1 − exp

(−z3

γ

)]K−Ks

U (z1 − z3)

×Ks −2∑j=0

[(−1)j

(Ks − 2

j

)[z2 − (Ks − 2 − j) z3 − jz1 ]

Ks −3

×U (z2 − (Ks − 2 − j) z3 − jz1)

], (3.88)

Limiting distributions of extreme order statistics 61

for 1 < m < Ks − 1,

pm −1∑n = 1

γn :K ,γm :K ,K s −1∑

n = m + 1γn :K ,γK s :K

(z1 , z2 , z3 , z4)

=K!

(K − Ks)! (Ks − m − 1)! (Ks − m − 2)! (m − 1)! (m − 2)!γKs

× exp(−z1 + z2 + z3 + z4

γ

)[1 − exp

(−z4

γ

)]K−Ks

[z1 − (m − 1) z2 ]m−2

×Ks −m−1∑

j=0

[(−1)j

(Ks − m − 1

j

)[z3 − (Ks − m − 1 − j) z4 − jz2 ]

Ks −m−2

×U (z2 − z4) U (z1 − (m − 1) z2) U (z3 − (Ks − m − 1 − j) z4 − jz2)

]. (3.89)

for m = Ks − 1,

pK s −2∑n = 1


(z1 , z2 , z3)

=K!

(K − Ks)! (Ks − 2)! (Ks − 3)!γKsexp

(−z1 + z2 + z3

γ

)

×[1 − exp

(−z3

γ

)]K−Ks

[z1 − (Ks − 2) z2 ]Ks −3 U (z2 − z3)

×U (z1 − (Ks − 2) z2) . (3.90)

for m = Ks ,

pγK s :K ,

K s −1∑n = 1

γn :K

(z1 , z2)

=K!

(K − Ks)! (Ks − 1)! (Ks − 2)!γKsexp

(−z1 + z2

γ

)

×[1 − exp

(−z1

γ

)]K−Ks

[z2 − (Ks − 1) z1 ]Ks −2 U (z2 − (Ks − 1) z1) . (3.91)

Therefore, the desired joint PDF can be evaluated easily numerically with thehelp of integral tables [16, 17] or using standard mathematical packages such asMathematica or Matlab, etc.

3.6 Limiting distributions of extreme order statistics

The limiting distribution of the largest order statistics from L i.i.d. samplescan be useful in the asymptotic analysis of multiuser wireless systems. It has


been established that the limiting distribution of γ1:L , i.e. limL→+∞ Fγ1 :L (x), ifit exists, must be one of the following three types [1, 18]:

Frechet distribution with CDF

F (1)(x) = exp(−x−α ), x > 0, α > 0; (3.92)

Weibull distribution with CDF

F (2)(x) =

exp[−(−x)α ], x ≤ 0, α > 0,

1, x > 0;(3.93)

Gumbel distribution with CDF

F (3)(x) = exp(−e−x). (3.94)

Specifically, there exist constants aL > 0 and bL such that

limL→+∞

Pγ1 :L (aLx + bL ) = F (i)(x), i = 1, 2, or 3. (3.95)

Note that these three distributions are members of the family of generalizedextreme-value (GEV) distribution with CDF given by

FGEV (x) = exp

−[1 + ξ

(x − µ

σ

)]−1/ξ

, (3.96)

where ξ, µ, and σ are constant parameters. Specifically, when ξ = 0, the CDF ofthe GEV distribution simplifies to

FGEV (x) = exp(−e−

x −µσ

), (3.97)

which is of the Gumbel type. Furthermore, setting ξ > 0 and ξ < 0 will lead toFrechet and Weibull types, respectively.

The type of the limiting distribution depends on the properties of the distri-bution functions of the original unordered random variables. It can be shownthat if the distribution functions of γi satisfy

limx→+∞

xpγ (x)1 − Fγ (x)

= α, (3.98)

for some constant α > 0, then the limiting distribution of γ1:L will be of theFrechet type. This indicates that for some constant aL > 0, we have

limL→+∞

Fγ1 :L (aLx) = F (1)(x), (3.99)

Bibliography notes 63

where aL can be computed by solving 1 − Pγ (aL ) = 1/L based on thecharacteristic of extremes. If, instead, the following condition is satisfied by thedistribution of γi

limx→+∞

1 − Fγ (x)pγ (x)

= α, (3.100)

where the constant α > 0, then the limiting distribution of γ1:L will be of theGumbel type. More specifically, we have

limL→+∞

Fγ1 :L (x) = F (3)(

x − bL

aL

)= exp

(−e

− x −b La L

), (3.101)

where the constant bL and aL can be computed by sequentially solving equations1 − Fγ (bL ) = 1/L and 1 − Fγ (aL + bL ) = 1/(eL).

Example 3.4: When γis follow i.i.d. exponential distribution with PDF pγ (x) =e−x , we can easily verify that the condition in (3.100) holds and, as such, the lim-iting distribution of γ1:L is of the Gumbel type. Applying the results in (3.101),we can show that aL = 1 and bL = log L. Finally, we have

limL→+∞

Fγ1 :L −log L (x) = limL→+∞

Pr[γ1:L < log L + x] = exp(−e−x

). (3.102)

3.7 Summary

In this chapter, the relevant order statistics results were presented in a sys-tematic fashion. Starting from the basic distribution functions of order randomvariables, we derived the distribution functions of the partial sum and the jointdistribution function of several partial sums, first using the successive condition-ing approach then with the MGF-based analytical framework. The results wereapplied to the exponential random variable special case for illustration wheneverfeasible.


For more thorough coverage of order statistics, the reader may refer to the bookby David [1]. Section 9.11.2 of [19] addresses the distribution function of thesum of the largest order statistics for non-identically distributed original ran-dom variables. Ref. [12] presents more illustrative examples for the MGF-based


analytical framework for the joint distribution function of several partialsums.

Appendix to chapter 3. Order statistics

Proof of the simplifying relationship involving Im

In this subsection, we prove the following relationship by induction.

Im =1

(K − m + 1)![c (γm−1:K , λ)](K−m+1) . (3.103)

Let us first consider the case m = K. Noting that p (γK :K ) exp (λγK :K ) =c′ (γK :K , λ), it is easy to show that

γK −1 :K∫0

dγK :K p (γK :K ) exp (λγK :K ) =

γK −1 :K∫0

dγK :K c′ (γK :K , λ) (3.104)

= c (γK :K , λ)|γK −1 :K0

= c (γK−1:K , λ) .

Now let us assume the relationship in (3.103) holds for K ≥ m ≥ k, and con-sider the case m = k − 1. Starting from the definition of Im , we have

Ik−1 =

γk −2 :K∫0

dγk−1:K p (γk−1:K ) exp (λγk−1:K )

γk −1 :K∫0

dγk :K p (γk :K ) exp (λγk :K )

· · ·γK −1 :K∫

0

dγK :K p (γK :K ) exp (λγK :K )

=

γk −2 :K∫0

dγk−1:K c′ (γk−1:K , λ)1

(K − k + 1)![c (γk−1:K , λ)](K−k+1) .

(3.105)

Applying integration by part, we have

Ik−1 =1

(K − k + 1)![c (γk−1:K , λ)](K−k+2)

∣∣∣∣γk −2 :K

0

−γk −2 :K∫

0

dγk−1:K1

(K − k)![c (γk−1:K , λ)](K−k+1) c′ (γk−1:K , λ).

(3.106)

Appendix 65

It follows after some rearrangement of terms and manipulation that

γk −2 :K∫0

dγk−1:K [c (γk−1:K , λ)](K−k+1) c′ (γk−1:K , λ) (3.107)

=1

(K − k + 2)[c (γk−2:K , λ)](K−k+2)

Substituting (3.107) into (3.106), we can finally show

Ik−1 =1

(K − k + 1)![c (γk−2:K , λ)](K−k+2)

− 1(K − k)!

1(K − k + 2)

[c (γk−2:K , λ)](K−k+2)

=1

(K − k + 2)![c (γk−2:K , λ)](K−k+2) . (3.108)

We can now conclude that the relationship in (3.103) holds for m.The simplifying relationship involving I ′m and I ′′a,b , given in (3.71) and (3.73),

respectively, can be proved similarly.

Derivation of the joint PDF of γm:K andKs∑

n=1n =m

γn:K among K-ordered RVs

In this Appendix, we derive the joint PDF of γm :K andKs∑

n=1n =m

γn :K among K-

ordered RVs by considering four cases: (i) m = 1, (ii) 1 < m < Ks − 1, (iii) m =Ks − 1, and (iv) m = Ks separately.

(i) m = 1In this case, we need to consider the random variable γKs :K separately. Assuch, we start by calculating the three-dimensional joint MGF of Z1 = γ1:K ,

Z2 =Ks −1∑n=2

γn :K , and Z3 = γKs :K as

MGF (λ1 , λ2 , λ3) = E exp (λ1Z1 + λ2Z2 + λ3Z3)

= K!

∞∫0


γ1 :K∫0


× · · ·γK s −2 :K∫

0

dγKs −1:K p (γKs −1:K ) exp (λ2γKs −1:K )

×γK s −1 :K∫

0

dγKs :K p (γKs :K ) exp (λ3γKs :K ) [c (γKs :K )](K−Ks ) . (3.109)


With the help of (3.61) and (3.73), we can simplify the three-dimensionalMGF to

MGF (λ1 , λ2 , λ3) = K!

∞∫0

dγKs :K p (γKs :K ) exp (λ3γKs :K ) [c (γKs :K )](K−Ks )

×∞∫

γK s :K

dγ1:K p (γ1:K ) exp (λ1γ1:K )1

(Ks − 2)![µ (γKs :K , γ1:K , λ2)]

(Ks −2) .

(3.110)

The joint PDF of Z1 = γ1:K , Z2 =Ks −1∑n=2

γn :K and Z3 = γKs :K can then

be calculated after applying proper inverse Laplace transform to (3.110)as

pγ1 :K ,

K s −1∑n = 2

γn :K ,γK s :K

(z1 , z2 , z3) = L−1S1 ,S2 ,S3

MGFZ (−S1 ,−S2 ,−S3)

=K!

(Ks − 2)!

∞∫0

dγKs :K p (γKs :K )L−1S3

exp (−S3γKs :K ) [c (γKs :K )](K−Ks )

×∞∫

γK s :K

dγ1:K p (γ1:K )L−1S1

exp (−S1γ1:K )

×L−1S2

[µ (γKs :K , γ1:K ,−S2)]

(Ks −2)

=K!

(Ks − 2)!p (z1) p (z3) [c (z3)]

(K−Ks ) U (z1 − z3)

×L−1S2

[µ (z3 , z1 ,−S2)]

(Ks −2)

. (3.111)

Finally, the desired two-dimensional joint PDF of γm :K andKs∑

n=1n =m

γn :K can becalculated as

pγm :K ,

K s∑n=1n =m

γn :K

(x, y) =∫ (Ks −2)x

(K s −2K s −1 )y

pγ1 :K ,

K s −1∑n = 2

γn :K ,γK s :K

(x, z2 , y − z2) dz2 .

(3.112)

(ii) 1 < m < Ks − 1

In this case, we divide the partial sumKs∑

n=1n =m

γn :K into three parts by con-

sidering γKs :K and discontinuous ordered random variables separately. In

Appendix 67

particular, we first calculate the four-dimensional MGF of Z1 =m−1∑n=1

γn :K ,

Z2 = γm :K , Z3 =Ks −1∑

n=m+1γn :K and Z4 = γKs :K as

MGF (λ1 , λ2 , λ3 , λ4) = K!

∞∫0


· · ·γm −2 :K∫

0

dγm−1:K p (γm−1:K ) exp (λ1γm−1:K )

×γm −1 :K∫

0

dγm :K p (γm :K ) exp (λ2γm :K )

×γm :K∫0

dγm+1:K p (γm+1:K ) exp (λ3γm+1:K )

· · ·γK s −2 :K∫

0


×γK s −1 :K∫

0

dγKs :K p (γKs :K ) exp (λ4γKs :K ) [c (γKs :K )](K−Ks ) .

(3.113)

With the help of (3.61), (3.103), (3.71), and (3.73), we can arrive at the followcompact expression for the MGF.

MGF (λ1 , λ2 , λ3 , λ4) =K!

(Ks − m − 1)! (m − 1)!(3.114)

×∞∫

0


×∞∫

γK s :K

dγm :K p (γm :K ) exp (λ2γm :K ) [e (γm :K , λ1)](m−1)

× [µ (γKs :K , γm :K , λ3)](Ks −m−1) .

Similar to the previous case, we can calculate the four-dimensional joint PDF

of Z1 =m−1∑n=1

γn :K , Z2 = γm :K , Z3 =Ks −1∑

n=m+1γn :K and Z4 = γKs :K by applying


an inverse Laplace transform, yielding

pm −1∑n = 1

γn :K ,γm :K ,K s −1∑

n = m + 1γn :K ,γK s :K

(z1 , z2 , z3 , z4) =

K!(Ks − m − 1)! (m − 1)!

∞∫0

dγKs :K p (γKs :K )L−1S4

exp (−S4γKs :K ))

× [c (γKs :K )](K−Ks )

∞∫γK s :K

dγm :K

[p (γm :K )L−1

S2exp (−S2γm :K )

×L−1S1

[e (γm :K ,−S1)]

(m−1)L−1

S3

[µ (γKs :K , γm :K ,−S3)]

(Ks −m−1)]

=F

(Ks − m − 1)! (m − 1)!p (z2) p (z4) [c (z4)]

(K−Ks ) U (z2 − z4)

×L−1S1

[e (z2 ,−S1)]

(m−1)L−1

S3

[µ (z4 , z2 ,−S3)]

(Ks −m−1)

. (3.115)


n=1n =m

γn :K can becalculated as

pγm :K ,

K s∑n=1n =m

γn :K

(x, y) (3.116)

=∫ x

0

∫ y−(Ks −m )z4

(m−1)xpm −1∑

n = 1γn :K ,γm :K ,

K s −1∑n = m + 1

γn :K ,γK s :K

(z1 , x, y − z4 , z4) dz1dz4 .

(iii) m = Ks − 1

In this case, we can start with the three-dimensional MGF of Z1 =Ks −2∑n=1

γn :K ,

Z2 = γKs −1:K , and Z3 = γKs :K , which is given by

MGF (λ1 , λ2 , λ3) = K!

∞∫0


×γK s −2 :K∫

0


×γK s −1 :K∫

0


Appendix 69

The three-dimensional MGF can be simplified, with the help of (3.61) and(3.71), to

MGF (λ1 , λ2 , λ3) =K!

(Ks − 2)!(3.118)

×∞∫

0


×∞∫

γK s :K

dγKs −1:K p (γKs −1:K ) exp (λ2γKs −1:K ) [e (γKs −1:K , λ1)](Ks −2) .

The corresponding three-dimensional joint PDF ofKs −2∑n=1

γn :K , γKs −1:K and

γKs :K is then given by

pK s −2∑n = 1


(z1 , z2 , z3) =K!

(Ks − 2)!(3.119)

p (z2) p (z3) [c (z3)](K−Ks ) U (z2 − z3)L−1

S1

[e (z2 ,−S1)]

(Ks −2)

.


n=1n =m

γn :K can be

obtained as

pγm :K ,

K s∑n=1n =m

γn :K

(x, y) =∫ x

0pK s −2∑

n = 1γn :K ,γK s −1 :K ,γK s :K

(y − z3 , x, z3) dz3 .

(3.120)(iv) m = Ks

For this case, we can consider directly the two-dimensional MGF of γm :K andKs∑

n=1n =m

γn :K , which is given by

MGF (λ1 , λ2) = K!

∞∫0


· · ·γK s −2 :K∫

0


×γK s −1 :K∫

0



With the help of (3.61) and (3.71), we can simplify the MGF to

MGF (λ1 , λ2) =K!

(Ks − 1)!

∞∫0

dγKs :K p (γKs :K ) exp (λ1γKs :K )

× [c (γKs :K )](K−Ks ) [e (γKs :K , λ2)](Ks −1) . (3.122)

The desired two-dimensional joint PDF of γKs :K andKs −1∑n=1

γn :K can be

obtained, by applying an inverse Laplace transform, as

pγm :K ,

K s∑n=1n =m

γn :K

(z1 , z2)

=K!

(Ks − 1)!p (z1) [c (z1)]

(K−Ks ) L−1S2

[e (z1 ,−S2)]

(Ks −1)

. (3.123)

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tions, 2nd ed. Amsterdam: North-Holland Elsevier, 1998.[3] Y.-C. Ko, H.-C. Yang, S.-S. Eom and M.-S. Alouini, “Adaptive modulation and diver-

sity combining based on output-threshold MRC,” IEEE Trans. Wireless Commun.,vol. 6, no. 10, pp. 3728–3737, October 2007.

[4] M.-S. Alouini and M. K. Simon, “An MGF-based performance analysis of generalizedselective combining over Rayleigh fading channels,” IEEE Trans. Commun., vol. 48,no. 3, pp. 401–415, March 2000.

[5] Y. Ma and C. C. Chai, “Unified error probability analysis for generalized selectioncombining in Nakagami fading channels,” IEEE J. Select. Areas Commun., vol. 18,no. 11, pp. 2198–2210, November 2000.

[6] M. Z. Win and J. H. Winters, “Virtual branch analysis of symbol error probability forhybrid selection/maximal-ratio combining Rayleigh fading,” IEEE Trans. Commun.,vol. 49, no. 11, pp. 1926–1934, 2001.

[7] A. Annamalai and C. Tellambura, “Analysis of hybrid selection/maximal-ratio diver-sity combiners with Gaussian errors,” IEEE Trans. Wireless Commun., vol. 1, no. 3,pp. 498–511, July 2002.

[8] Y. Roy, J.-Y. Chouinard and S. A. Mahmoud, “Selection diversity combining with mul-tiple antennas for MM-wave indoor wireless channels,” IEEE J. Select. Areas Commun.,vol. SAC-14, no. 4, pp. 674–682, May 1998.

[9] P. V. Sukhatme, “Tests of significance for samples of the population with two degreesof freedom,” Ann. Eugenics, vol. 8, p. 5256, 1937.

[10] H.-C. Yang, “New results on ordered statistics and analysis of minimum-selection gen-eralized selection combining (GSC),” IEEE Trans. Wireless Commun., vol. 5, no. 7,pp. 1876–1885, 2006.

References 71

[11] S. Choi, M.-S. Alouini, K. A. Qaraqe and H.-C. Yang, “Finger replacement methodfor RAKE receivers in the soft handover region,” IEEE Trans. Wireless Commun.,vol. TWC-7, no. 4, pp. 1152–1156, 2008.

[12] A. H. Nuttall, “An integral solution for the joint PDF of order statistics and residualsum,” NUWC-NPT, Technical Report, October 2001.

[13] ——, “Joint probability density function of selected order statistics and the sum of theremaining random variables,” NUWC-NPT, Technical Report, January 2002.

[14] A. H. Nuttall and P. M. Baggenstoss, “Joint distributions for two usefulclasses of statistics, with applications to classification and hypothesis testing,”IEEE Trans. Signal Processing, submitted for publication. [Online]. Available:http://www.npt.nuwc.navy.mil/Csf/papers/order.pdf

[15] S. Nam, M.-S. Alouini and H.-C. Yang, “An MGF-based unified framework to deter-mine the joint statistics of partial sums of ordered random variables,” IEEE Trans. on

Inform. Theory, vol. IT-56, no. 11, pp. 5655–5672, November 2010.[16] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. New York,

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of a sample,” in Studies in Mathematics and Mechanics Presented to Richard von Mises.New York, NY: Academic, 1954, pp. 346–353.


Channels, 2nd ed. New York, NY: John Wiley & Sons, 2004.

4 Advanced diversity techniques

4.1 Introduction

Diversity combining can considerably improve the performance of wireless systemoperating over fading channels [1, 2]. In general, the performance gain increasesas the number of diversity branches increases, although with a diminishing gain.These results have motivated the recent development of several wideband wire-less systems, where a large number of diversity paths exist. Examples includewideband code division multiple access (WCDMA) systems [3], ultra wideband(UWB) systems [4], and millimeter (MM)-wave systems [5]. While huge diver-sity benefits can be explored in the resulting diversity-rich environment, it iscritical to exploit those benefits efficiently [6]. In particular, MRC is well-knownto be the optimal combining scheme in a noise-limited environment. However,applying MRC in such a diversity-rich environment will entail very high systemcomplexity. Note that we need to implement a receiver chain for each resolvablediversity path. In addition, processing all the diversity paths will lead to veryhigh processing power consumption.

In this chapter, we present several advanced combining schemes for diversity-rich environments. The general principle of these schemes is to select a propersubset of good diversity paths among those available and then combine themin an optimal MRC fashion. Specifically, these schemes will determine the pathsubset using the best-selection process or thresholding, or a combination of both.As expected intuitively, different schemes will lead to different performance ver-sus complexity trade-offs. Based on accurate mathematical analysis, sometimeswith the help of the order statistics results from the previous chapter, we are ableto obtain the exact statistics of the combiner output SNR with these schemes,which are then utilized to quantify the trade-off study between performance andcomplexity between them. For analytical tractability and clarity, we adopt anindependent flat-fading channel model for the available diversity paths.

4.2 Generalized selection combining (GSC)

GSC is one of the most widely studied low-complexity schemes for diversity-rich environments, which is also known as hybrid selection and maximum ratio

Generalized selection combining 73

Estimate, rank,

and select

best Lc paths

1

2

L

1

Lc

Lc branch

MRC

Figure 4.1 Structure of a GSC-based diversity combiner.

combining scheme (see for example [3–5,7–15]). The basic idea is to select a subsetof the best paths and then combine them in the MRC fashion. The rationale isthat applying MRC to the weak paths will bring little additional performancebenefit, while entailing extra hardware complexity of additional RF chains. Inaddition, the channel estimation of weak paths can be unreliable, which furtherlimits the benefit of applying MRC to them, as imperfect channel estimation willconsiderably degrade the performance of MRC [16, 17]. The structure of a GSCcombiner is shown in Fig. 4.1, which consists of the concardination of a multi-branch selection combiner and a conventional MRC combiner. The immediatebenefit with pre-selection is that only an Lc -branch MRC is needed with GSC.

The mode of operation of a GSC combiner can be summarized as follows:(i) Receiver first estimates the SNRs of all L available diversity paths, whichmay correspond to the resolved paths for UWB and WCDMA systems or theantenna branches in an MM-wave system; (ii) Receiver will then rank the SNRsand select the Lc strongest paths, i.e. those with the highest SNR; (iii) Receiverfinally determines the MRC weights for those selected paths, which entails thefull channel estimation (amplitude and phase) of the selected paths, and startsactual data reception. Note that unlike the conventional MRC combiner, wherethe receiver needs to estimate the complex channel gains for all diversity paths,the receiver with GSC only needs to estimate the complete channel states ofthose selected paths. The path selection process only requires the received signalpower to be estimated.

4.2.1 Statistics of output SNR

To evaluate the performance of the GSC combiner, we need the statistics of thecombined SNR γc , which is given by

γc =Lc∑i=1

γi:L , (4.1)

74 Chapter 4. Advanced diversity techniques

where γi:L is the ith largest one among total L SNRs. Based on the order statisticsresults in the previous chapter, we can obtain the closed-form expression of theMGF of the combined SNR with GSC for the i.i.d. Rayleigh fading scenario as

Mγc(s) = (1 − sγ)−Lc

L∏l=Lc +1

(1 − sγLc

l

)−1

. (4.2)

With the MGF of γc , we can directly apply the MGF-based approach to evaluatethe average error rate performance. In particular, the average error rate of M -aryphase-shift-keying (M -PSK) signals is given by [18]

Ps =1π

∫ (M −1 )πM

0Mγc

(− gPSK

sin2 φ

)dφ, (4.3)

where gPSK = sin2 (π/M). The PDF and CDF of γc with GSC can be derivedroutinely after taking the proper inverse Laplace transform. In particular, theCDF of γc with GSC is given in the following closed-form expression [10]

Fγc(x) =

L!(L − Lc)!Lc !

1 − e−

xγ

Lc −1∑k=0

1k!

(x

γ

)k

+L−Lc∑l=1

(−1)Lc + l−1 (L − Lc)!(L − Lc − l)!l!

(Lc

l

)Lc −1

×[(

1 +l

Lc

)−1 [1 − e−(1+ l

L c ) xγ

]

−Lc −2∑m=0

(− l

Lc

)m(

1 − e−xγ

m∑k=0

1k!

(x

γ

)k)]

, (4.4)

which can be directly applied to calculate the outage performance of GSC.For more general fading channel models, the statistical distribution function

of γc can still be obtained with the order statistics result in the previous chap-ter, although the closed-form expression may be not available. Specifically, thecombiner output SNR with GSC can be viewed as the sum of two correlatedrandom variables: the sum of the first Lc − 1 largest SNRs

∑Lc −1i=1 γi:L , denoted

by ΓLc −1 , and the Lcth largest path SNR γLc :L . It follows that the PDF of com-biner output SNR γc can be written in terms of the joint PDF of ΓLc −1 andγLc :L as

pγc(x) =

∫ ∞

0pΓL c −1 ,γL c :L (x − y, y)dy. (4.5)

The generic expression of the joint PDF pΓL c −1 ,γL c :L (·, ·) has been derived in theprevious chapter.

GSC with threshold test per branch 75

For the general independent and non-identical distributed (i.n.d.) fading chan-nel case, the MGF of the combined SNR with GSC can be obtained as

Mγc(s) =

∑n1 ,··· ,nL c −1

n1 <n2 < ···<nL c −1

∑nL c

∫ ∞

0e−sxpnL c

(x) (4.6)

×[

Lc −1∏l=1

Mnl(s, x)

][L∏

l ′=Lc +1

Fn ′l(x)

]dx,

where ni ∈ 1, 2, · · · , Lc, i = 1, · · · , Lc , are the index of the ith selectedbranches, pnL c

(x) is the PDF of the Lcth selected branch SNR, Fn ′l(x) is the

SNR CDF of the remain branches and Mnl(s, x) is the truncated MGF of the

lth selected branch SNR [15, eq. (4)]. Note that summation∑

n1 ,··· ,nL c −1n1 <n2 < ···<nL c −1

is

carrying over all possible index sets of the largest Lc − 1 branch SNRs out of thetotal L branches and

∑nL c

over the possible indexes of Lcth selected branch.

4.3 GSC with threshold test per branch (T-GSC)

The GSC scheme discussed in the previous section can be viewed as a naturalcombination of SC and MRC. While maintaining a fixed low-hardware complex-ity, the GSC combiner may discard diversity paths with good quality or includesome weak paths. To alleviate the above-mentioned shortcomings of the conven-tional GSC scheme, GSC with threshold test per branch (T-GSC) was proposedand studied in [19, 20]. With T-GSC, the combining decision on a particularpath is based on the comparison result of its SNR against a preseleted threshold.Specifically, if the path SNR is above the threshold, the path will be combined inan MRC fashion. Otherwise, it will be discarded. As such, T-GSC will combinea variable number of diversity paths over time. Note that as the T-GSC com-biner may need to combine all L diversity paths in the MRC fashion, T-GSChas the same hardware complexity as the conventional MRC scheme. However,T-GSC can save the receiver processing power by only combining those pathswith good-enough quality.

Depending on how the SNR threshold is chosen, there are two T-GSC schemes,i.e. absolute threshold GSC (AT-GSC) and normalized threshold GSC (NT-GSC) [20,21]. With AT-GSC, the ith diversity path is combined if γi ≥ γT , whereγT is a fixed threshold. With NT-GSC, however, the threshold γT is normalizedagainst the best path SNR, i.e.

γT = η maxl

γl,

where 0 < η < 1 is the normalized threshold [22,23]. Note that with AT-GSC, itmay happen in the worst case scenario that no path is combined, whereas withNT-GSC, such a problem is avoided as at least the best path will be selected.On the other hand, the NT-GSC scheme requires the additional complexity of


Table 4.1 MGF Mγ ′l(·) for the three fading models under consideration.

Model Mγ ′l(s)

Rayleigh 1 − e−γ Tγ + 1

1−tγ etγT − γ Tγ

Rice 1 − Q1

(√2K,

√2(1 + K) γT

γ

)+ 1+K

1+K−tγ et γ K

1 + K −t γ

×Q1

(√2K (1+K )1+K−tγ ,

√2(1 + K − tγ) γT

γ

)Nakagami-m 1 −

Γ(

m,m γ T

γ

)Γ(m ) +

(1 − tγ

m

)−m Γ(

m,m γ T

γ−tγT

)Γ(m )

determining the instantaneous best path, which involves the comparison of dif-ferent estimated path SNRs. In the following, we present the major steps inobtaining the statistics of the output SNR with both T-GSC schemes.


AT-GSCBased on the mode of operation of the AT-GSC scheme, the combiner can beviewed as an L-branch MRC combiner with input SNRs given by

γ′l =

γl , γl ≥ γth ;

0, 0 ≤ γl < γth ,(4.7)

where γl is the SNR of the lth diversity path. The combiner output SNR is equalto the sum of these L input SNRs, i.e. γc =

∑Ll=1 γ′

l . With the i.i.d. assumption,we can easily calculate the MGF of γc as the product of the MGFs of γ′

l . It canbe shown that the PDF of γ′

l can be given by

pγ ′l(γ) =

Fγl

(γT )δ(γ), γ = 0;

pγl(γ), γ ≥ γT ,

(4.8)

where Fγl(·) and pγl

(·) are the CDF and PDF of the lth path SNRs, respectively.It follows that the MGF of γ′

l can be obtained as

Mγ ′l(s) = Fγl

(γT ) +∫ ∞

γT

pγl(γ)esγ dγ. (4.9)

For most popular fading channel models, the closed-form expression of Mγ ′l(·) is

available. Table 4.1 summarizes these results for the three fading-channel modelsunder consideration. Finally, the MGF of the combined SNR with AT-GSC isgiven by

Mγc(s) =

L∏l=1

Mγ ′l(s) =

L∏l=1

Fγl(γT ) +

∫ ∞

γT

pγl(γ)esγ dγ. (4.10)

GSC with threshold test per branch 77

The PDF and CDF of the combined SNR can be obtained routinely after takingproper inverse Laplace transform and carrying out integration. With the statis-tics of the combined SNR, we can readily evaluate the performance of AT-GSCover fading channels.

NT-GSCTo derive the statistics of the combined SNR with NT-GSC, we consider L

mutually exclusive events depending on how many diversity paths are combined[22]. Let γ1:L ≥ γ2:L ≥ · · · ≥ γL :L denote the ordered path SNRs in descendingorder, where γ1:L corresponds to the largest among all L, i.e. γ1:L = maxlγl.It follows that Lc paths are combined with NT-GSC if and only if γ1:L ≥ γ2:L ≥· · · ≥ γLc :L ≥ ηγ1:L ≥ γLc +1:L ≥ · · · ≥ γL :L , which leads to the combiner outputSNR being

γc,Lc=

Lc∑i=1

γi:L , Lc = 1, 2, · · · , L. (4.11)

Applying the total probability theorem, the distribution function of the combinedSNR with NT-GSC can be obtained as the sum of the distribution functions foreach individual event. In particular, the MGF of γc can be written as

Mγc(s) =

L∑Lc =1

Mγc , L c(s), (4.12)

where Mγc , L c(s) is the MGF for the event that Lc paths are combined and is

given by

Mγc , L c(s) =

∫ ∞

0dγ1:L

∫ γ1 :L

ηγ1 :L

dγ2:L · · ·∫ γL c −1 :L

ηγ1 :L

dγLc :L

∫ ηγ1 :L

0dγLc +1:L (4.13)∫ γL c + 1 :L

0dγLc +2:L · · ·

∫ ηγL −1 :L

0es

∑L ci = 1 γi :L

pγ1 :L ,γ2 :L ,··· ,γL :L (γ1:L , γ2:L , · · · , γL :L )dγL :L ,

where pγ1 :L ,γ2 :L ,··· ,γL :L (·) is the joint PDF of the ordered path SNRs, given inthe previous chapter. After proper substitution and carrying out the integrationwith the help of the following definition of partial MGF

Mγ (s, x) =∫ ∞

x

pγ (γ)esγ dγ, (4.14)

we can obtain the MGF for the event that Lc paths are combined in the followingcompact form involving a single integral

Mγc , L c(s) = Lc

(L

Lc

)∫ ∞

0esγ pγ (γ)[Fγ (ηγ)]L−Lc [Mγ (s, γ) −Mγ (s, ηγ)]Lc −1dγ.

(4.15)Note that the partial MGF Mγ (s, x) is related to the MGF defined in (4.9) asMγ (s, γth) = Mγ ′

l(s) − Fγ (γth). We can easily obtain the closed-form expression


of the partial MGF for three popular fading channel models directly from Table4.1. The PDF and CDF of the combined SNR with NT-GSC can be obtainedroutinely.

4.3.2 Average number of combined paths

Both GSC and T-GSC will reduce the combiner complexity in terms of the num-ber of active MRC branches. With GSC, the number of active MRC branchesis fixed to Lc < L, whereas with T-GSC, the number of active branches is ran-domly varying. In particular, the probability that i branches will be active withAT-GSC is equal to

Pr[Nc = i] =(

L

i

)[1 − Fγ (γT )]i [Fγ (γT )]L−i . (4.16)

As such, the average number of combined paths with AT-GSC is given by

Nc =L∑

i=0

iPr[Nc = i] = L[1 − Fγ (γT )]. (4.17)

Based on the mode of operation of NT-GSC, the probability Pr[Nc = i] withNT-GSC can be calculated as

Pr[Nc = i] = Pr[γi:L ≥ ηγ1:L ≥ γi+1:L ], (4.18)

which can be calculated using the joint PDF of γ1:L , γi:L , and γi+1:L as

Pr[Nc = i] =∫ ∞

0

∫ x

ηx

∫ ηx

0pγ1 :L ,γi :L ,γi + 1 :L (x, y, z)dxdydz. (4.19)

After slightly generalized (3.4), the joint PDF pγ1 :L ,γi :L ,γi + 1 :L (·, ·, ·) can beobtained as

pγ1 :L ,γi :L ,γi + 1 :L (x, y, z) =L!

(i − 2)!(L − i − 1)!pγ (x) (4.20)

× (Fγ (x) − Fγ (y))i−2 pγ (y)pγ (z) (Fγ (z))L−i−1 .

4.4 Generalized switch and examine combining (GSEC)

Both GSC variants discussed in the previous section require the estimation ofall available diversity paths, regardless of whether they are eventually used ornot. The path estimation complexity can be further reduced with a switching-based mechanism during the path selection stage. Note that the main complexitysaving of threshold combining schemes over selection combining is fewer pathestimations. Generalized switch and examine combining (GSEC) is another low-complexity combining scheme that was proposed for a diversity-rich environmentin this context [6]. The basic idea of GSEC is to extend the notion of switch and

Generalized switch and examine combining 79

γ1

γ2

γ3

γ4

γ5

γ1 < γT

γ1

γ2

γ3

γ4

γ5

γ2 > γT

γ1

γ2

γ3

γ4

γ5

γ3 < γT

γ4 < γT

Examining

Connected

γ1

γ2

γ3

γ4

γ5

γ1

γ2

γ3

γ4

γ5

Figure 4.2 Sample operation of a GSEC-based diversity combiner.

examine combining studied in a previous chapter to multi-input-multi-outputscenario and use it for multiple path selection before cascading with a traditionalMRC combiner. As such, the GSEC scheme can also be viewed as hybrid switchand examine/maximum ratio combining.

Specifically, the receiver with GSEC tries to select Lc acceptable paths, i.e.those whose instantaneous SNR is above a pre-selected fixed threshold, out of atotal of L available paths for subsequent MRC combining. Figure 4.2 illustratesa sample path selection process of a GSEC combiner. Specifically, the receiverwith GSEC will sequentially estimate and compare the channel SNR of diversitypaths against sT until it finds Lc acceptable paths. After that, the receiver willstop path estimation. As such, the receiver with GSEC does not always need toestimate all L diversity paths, which manifest the major complexity saving ofGSEC over GSC and T-GSC. If there are not enough acceptable paths even afterexamining all paths, the receiver will randomly select some unacceptable pathsfor MRC combining. Therefore, similar to GSC, GSEC always combines a fixednumber of diversity branches. On the other hand, the path selection process withGSEC is much simpler than that of GSC, as the receiver does not need to rankall available diversity paths. Note that with GSC, the receiver needs to comparetwo estimated SNRs whereas with GSEC, only comparisons of an estimated SNRwith a fixed threshold are necessary.



To accurately quantify the new trade-off of performance versus complexity intro-duced by the GSEC scheme, we need the statistical characterization of its com-bined SNR. In this section, we derive the generic expression of the MGF of thecombined SNR with GSEC, which can be used to routinely obtain other distri-bution functions. Based on the mode of operation of GSEC, we note that thenumber of acceptable diversity paths that are eventually combined with GSEConly takes values from 0 to Lc . Since they are exclusive and disjoint events, wecan apply the total probability theorem and write the MGF of combiner outputγc in the following weighted sum form

Mγc(s) =

Lc∑i=0

πi M(i)γc

(s), (4.21)

where M(i)γc (·) is the conditional MGF of γc given that exactly i out of Lc com-

bined paths are acceptable and πi is the probability that there are exactly i

acceptable paths. With the i.i.d. fading assumption, it is not difficult to showthat the πis are mathematically given by

πi =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

(L

i

)[Fγ (γT )]L−i [1 − Fγ (γT )]i , i = 0, · · · , Lc − 1;

L∑j=Lc

(L

j

)[Fγ (γT )]L−j [1 − Fγ (γT )]j , i = Lc,

(4.22)

where Pγ (·) is the common CDF of the received SNR on each diversity path andgiven in Table 2.2 for the three fading models under consideration.

The conditional MGF M(i)γc (·) can be written as

M(i)γc

(s) = [Mγ + (s)]i [Mγ−(s)]Lc −i , (4.23)

where Mγ + (·) denotes the MGF of a single-path SNR given that it is greateror equal to γT and Mγ−(·) denotes the MGF of a single-path SNR given thatit is less than γT , both of which have been defined in the previous chapter.Their closed-form expressions for the three fading models under considerationare summarized in Table 3.1. Finally, after substituting (4.22) and (4.23) into(4.21), the generic expression for the MGF of the overall combiner output γc isgiven by

Mγc(s) =

Lc −1∑i=0

(L

i

)[Pγ (γT )]L−i [1 − Pγ (γT )]i [Mγ + (s)]i [Mγ−(s)]Lc −i

+L∑

j=Lc

(L

j

)[Pγ (γT )]L−j [1 − Pγ (γT )]j [Mγ + (s)]Lc . (4.24)


When Lc = L, it can be shown, with the help of the following relationship

[1 − Fγ (γT )]Mγ + (s) + Fγ (γT )Mγ−(s) = Mγ (s), (4.25)

where Mγ (·) is the common MGF of the received SNR, that (4.24) reduces to[Mγ (s)]L , i.e. the GSEC combiner is equivalent to an L branch MRC combiner,as expected. It can also be shown that when Lc = 1, (4.24) simplifies to the MGFof the output SNR of a traditional L branch SEC combiner [24, eq. (35)].

With the analytical expression of the MGF, we can readily derive the PDFand CDF of the combined SNR with GSEC through inverse Laplace transformand integration. As an example, the CDF γc over i.i.d. Rayleigh fading channelcan be obtained as

Fγc(x) =

Lc −1∑i=0

(L

i

)[1 − exp

(−γT

γ

)]L−Lcmin[Lc , x

γ T]−i∑

j=0

×(

Lc − i

j

)(−1)j exp

(− (i + j)γT

γ

)

×[1 − exp

(−x − (i + j) γT

γ

) Lc −1∑k=0

1k!

(x − (i + j) γT

γ

)k]

+ ILc γT(x)

L∑j=Lc

(L

j

)[1 − exp

(−γT

γ

)]L−j

exp(−jγT

γ

)

×[1 − exp

(−x − Lc γT

γ

) Lc −1∑k=0

1k!

(x − Lc γT

γ

)k]

, (4.26)

where ILc γT(x) is an indicator function, which is equal to 1 if x ≥ LcγT and zero

otherwise.

4.4.2 Average number of path estimations

We now quantify the complexity of GSEC in terms of the average number ofpath estimations. Because of the subsequent MRC combining, the receiver withGSEC needs always to estimate at least Lc diversity paths. In the case that thereare less than Lc acceptable paths among the total L ones, based on the modeof operation of GSEC, whenever the receiver encounters L − Lc unacceptablepaths, it will apply MRC to the remaining paths, which necessitates their fullestimations. As a result, in this case, the receiver needs to estimate all L availablediversity paths. It is not difficult to show that the probability that there are lessthan Lc acceptable paths is given by

PB =L∑

L−Lc +1

(L

k

)[Fγ (γT )]L−k [1 − Fγ (γT )]k . (4.27)


On the other hand, if there are at least Lc acceptable diversity paths, basedon the mode of operation of GSEC, the combiner will stop examining pathswhenever it has found Lc acceptable ones. In this case, the number of channelestimates during a guard period takes values from Lc to L. Note that the prob-ability that k channel estimates are performed in a guard period, denoted byP

(k)A , is equal to the probability that the Lcth acceptable path is the kth path

examined, or equivalently, exactly k − Lc ones of the first k − 1 examined paths,are unacceptable whereas the kth examined path is acceptable. It can be shown,with the assumption of i.i.d. diversity paths, that P

(k)A is mathematically given

by

P(k)A =

(k − 1k − Lc

)[1 − Fγ (γT )]Lc [Fγ (γT )]k−Lc . (4.28)

Finally, by combining these two mutually exclusive cases, we obtain the expres-sion for the overall average number of channel estimates needed by GSEC duringa guard period as

NE =L∑Lc

k P(k)A + L PB

=L∑

k=Lc

k [1 − Fγ (γT )]Lc

(k − 1k − Lc

)[Fγ (γT )]k−Lc

+LL∑

L−Lc +1

(L

k

)[Fγ (γT )]L−k [1 − Fγ (γT )]k . (4.29)

In the case of γT = 0, since Fγ (γT ) = 0, it can be shown that N = Lc , asexpected. On the other hand, when γT → ∞ and thus Fγ (γT ) = 1, it can besimilarly shown that N = L, also as expected.

4.4.3 Numerical examples

Figure 4.3 plots the average bit error probability of BPSK with GSEC as afunction of the switching threshold for L = 4 and different values of Lc . It canbe observed that there exists an optimal choice of the switching threshold inthe minimum average error probability sense except for the Lc = 4 case, whichcorresponds to a traditional 4-branch MRC. The performance advantage of theoptimal threshold diminishes and the value of the optimal threshold decreasesas Lc increases.

Figure 4.4 compares the error performance of GSEC and GSC in a diversity-rich environment with L = 12 available diversity paths. GSEC is not as goodas GSC performance-wise. We also note that as Lc increases, the performancegap between GSEC and GSC reduces significantly. This implies that GSEC cantake advantage of additional diversity paths with a relative lower complexitycompared to the competing GSC scheme. For instance, for a BER of 10−6 , there


−15 −10 −5 0 5 10 1510

−6

10−5

10−4

10−3

10−2

10−1

Switching Thresholds, dB

Ave

rage

Err

or P

roba

bilit

y

Lc=1

Lc=2

Lc=3

Lc=4

Figure 4.3 Average BER of BPSK with GSEC over L = 4 i.i.d. Rayleigh fading pathsas a function of the common switching threshold γT (γ = 10 dB) [6]. c© 2004 IEEE.

0 5 10 1510

−8

10−7

10−6

10−5

10−4

10−3

10−2

Average SNR, dB

Ave

rage

Err

or P

roba

bilit

y

Lc=2

Lc=4

Lc=6

Lc=8

GSEC

GSC

Figure 4.4 Error rate comparison of GSEC and GSC in diversity-rich environment,L = 12 [6]. c© 2004 IEEE.


−15 −10 −5 0 5 10 151

1.5

2

2.5

3

3.5

4

4.5

Switching Thresholds, dB

Ave

rage

Num

ber

of C

hann

el E

stim

ates

Lc=1

Lc=2

Lc=3

Lc=4

Figure 4.5 Average number of channel estimates of GSEC with L = 4 as function ofthe common switching threshold γT (γ = 10 dB) [6]. c© 2004 IEEE.

is only a 0.6 dB difference when Lc = 6. Notice that, in this scenario, the GSCcombiner needs to perform approximately Lc × L = 12 × 6 = 72 comparisons ineach guard period to select the 6 strongest paths, whereas GSEC requires atmost L = 12 comparisons.

Figure 4.5 plots the average number of path estimates of GSEC with L = 4as a function of the switching threshold for the Rayleigh fading case. As we cansee, as the switching threshold γT increases, the number of path estimates N

increases from Lc to L. Intuitively, because if the threshold is large, it becomesmore difficult for the receiver to find acceptable paths and as such more pathsneed to be estimated. We also marked the optimal operating points for GSECin terms of minimizing the average error rate, which were read from Fig. 4.3.Note that even for the best performance, GSEC does not need to estimate allthe diversity paths, whereas for GSC all L diversity paths need to be estimatedin each guard period.

4.5 GSEC with post-examining selection (GSECps)

The GSEC scheme works well when the channel condition is favorable and thereceiver can find enough acceptable paths. When there are not enough acceptable

GSEC with post-examining selection 85

paths, however, GSEC will combine all the acceptable paths and some ‘randomly’selected unacceptable paths. In this case, the selected unacceptable paths mayhave low instantaneous SNR, which degrades the performance of GSEC, espe-cially in a low SNR region. Noting that in this case all diversity paths have beenestimated anyway, a preferred alternative is to combine those unacceptable pathswith the highest instantaneous SNRs.

With this observation in mind, an improved version of GSEC, termed the gen-eralized switch and examine combining with post-examine selection (GSECps)was proposed [25]. In particular, the new scheme operates in exactly the same wayas GSEC when there are enough acceptable paths. When the number of accept-able paths is smaller than the number of paths required, instead of combiningsome ‘randomly’ selected unacceptable paths as with GSEC, GSECps combinesthe best unacceptable paths together with all acceptable paths. Intuitively, weexpect that GSECps can offer better performance than GSEC through the occa-sional selection of the best unacceptable paths while maintaining roughly thesame complexity. Note that GSECps can also be viewed as the generalizationof switch and examine combining with post-examine selection (SECps) schemestudied in Section 2.3 [26], since GSECps reduces to SECps when the number ofpaths to be combined is one.

Specifically, the mode of operation of GSEC can be summarized as follows. Thereceiver will sequentially estimate the instantaneous SNR of the available diver-sity paths and compare them with the fixed threshold γT to determine the accep-tance of a path. Let La stand for the number of acceptable paths. If at least Lc

paths are found to be acceptable, i.e. La ≥ La , the receiver will combine the firstLc of them in the MRC fashion. Otherwise, the receiver will combine all the La

acceptable paths, where La < Lc , and the strongest Lc − La unacceptable paths.As we can see, when there are enough acceptable paths, i.e. La ≥ Lc , GSECpswill be equivalent to GSEC. On the other hand, when La < Lc , the receiver isessentially combining the Lc best paths of the L available paths, as is the casewith GSC. Figure 4.6 presents a sample operation of the GSECps scheme.


We focus on the statistics of the combined SNR with GSECps over i.i.d. Rayleighfading environment, for which the closed-form expressions for PDF, CDF andMGF of γc are obtained. Note that depending on whether or not there are Lc

acceptable paths, we can apply the total probability theorem and write the CDFof γc in the following form

Fγc(x) = Pr[γc < x] (4.30)

= πLc· Pr[γc < x|La ≥ Lc ] + Pr[γc < x,La < Lc ],

where πLcis the probability that there are at least Lc acceptable paths, which

was given in (4.22), Pr[γc < x|La ≥ Lc ] is the conditional probability of the event


γ1

γ2

γ3

γ4

γ5

γ1 < γT γ2 > γT γ3 < γT

γ4 < γT γ5 < γT

γ4 largestamong γ1,γ3, γ5

γ1

γ2

γ3

γ4

γ5

γ1

γ2

γ3

γ4

γ5

γ1

γ2

γ3

γ4

γ5

γ1

γ2

γ3

γ4

γ5

γ1

γ2

γ3

γ4

γ5

Figure 4.6 Sample operation of a GSECps-based diversity combiner.

γc < x given that there are at least Lc acceptable paths, and Pr[γc < x,La < Lc ]is the joint probability of the events γc < x and La < Lc . Note that when La ≥Lc , GSECps combines Lc acceptable paths. As such, γc is the sum of Lc pathSNRs that are greater than γT . Therefore, Pr[γc < x|La ≥ Lc ] can be shown tobe given by [6, eq. (38)]

Pr[γc < x|La ≥ Lc ] = 1 − e−x −L c γ T

γ

Lc −1∑k=0

1k!

(x − LcγT

γ

)k

, x ≥ LcγT . (4.31)

Note that since La ≥ Lc , the combined SNR γc is always greater than LcγT .To calculate Pr[γc < x,La < Lc ], we first note that since there are less than Lc

acceptable paths, the receiver will in effect combine Lc strongest paths. There-fore, we have

γc =Lc∑i=1

γi:L , (4.32)

where γi:L is the ith largest SNR of all L path SNRs. Let ΓLc −1 denote thesum of the first Lc − 1 largest SNR, i.e. ΓLc −1 =

∑Lc −1j=1 γj :L . We can rewrite

Pr[γc < x,La < Lc ] while noting that La < Lc holds if and only if the Lcth


largest SNR of all L path SNRs, γLc :L , is less than γT , as

Pr[γc < x,La < Lc ] = Pr[ΓLc −1 + γLc :L < x, γLc :L < γT ], (4.33)

which can be computed with the joint PDF of γLc :L and ΓLc −1 , denotedby pγL c :L ,ΓL c −1 (y, z). Based on the order statistics results in a previous chap-ter, pγL c :L ,ΓL c −1 (y, z) for i.i.d. Rayleigh fading environment can be obtained as[27, eq. (11)]

pγL c :L ,ΓL c −1 (y, z) =L−Lc∑j=0

(−1)jL![z − (Lc − 1)y]Lc −2

(L − Lc − j)!(Lc − 1)!(Lc − 2)!j!γLce−

z + ( j + 1 ) y

γ ,

y ≥ 0, z ≥ (Lc − 1)y.(4.34)

Noting that the support of the joint PDF, Pr[γc < x,La < Lc ] can be written as

Pr[ΓLc −1 + γLc :L < x, γLc :L < γT ] =∫ minγT , x

L c

0

∫ x−y

(Lc −1)ypγL c :L ,ΓL c −1 (y, z) dz dy.

(4.35)After substituting (4.34) into (4.35), carrying out the integration and appropriatesimplifications, we have

Pr[γc < x,La < Lc ] =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Fγc(LcγT ) +

L−Lc∑j=1

Lc −2∑k=0

A(j,k)

×[e−

j γ T + L c γ Tγ

(1 − Γ

(Lc − 1 − k,

x − LcγT

γ

))−(

Γ(Lc − 1 − k,

LcγT

γ

)− Γ

(Lc − 1 − k,

x

γ

))]+(

L

Lc

)[(Γ(Lc,

LcγT

γ

)− Γ

(Lc,

x

γ

))− e−

L c γ Tγ

(1 − Γ

(Lc,

x − LcγT

γ

))], x ≥ LcγT ;

Fγc(x), x < LcγT ,

(4.36)where

Fγc(x) =

(L

Lc

)1 − Γ(Lc, x/γ) +

L−Lc∑l=1

(−1)Lc + l−1(

L − Lc

l

)(Lc

l

)Lc −1

×[1 − e−(1+ l/Lc )(x/γ )

1 + l/Lc−

Lc −2∑m=0

(−l

Lc)m

(1 − Γ(m + 1, x/γ)

)],

(4.37)

A(j,k) =(−1)jL!Lc

k

(L − Lc − j)!(Lc − 1)!j!(−j)k+1 , (4.38)


and Γ(n, x) is the incomplete Gamma function defined by

Γ(n, x) =1

(n − 1)!

∫ ∞

x

tn−1e−tdt. (4.39)

Combining (4.22), (4.31), and (4.36), we obtain the closed-form expression of theCDF of the output SNR γc and, equivalently, the outage probability of GSECps.

Starting from the CDF of the combined SNR with GSECps, we can routinelyderive the PDF and MGF. Specifically, after differentiation and some manipula-tions, the PDf of γc for i.i.d. Rayleigh fading channels is given by

pγc(x)

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(L

Lc

)[xLc −1e−x/γ

γLc (Lc − 1)!+

1γ

L−Lc∑l=1

(−1)Lc + l−1(

L − Lc

l

)(Lc

l

)Lc −1

× e−x/γ

(e−lx/Lc γ −

Lc −2∑m=0

1m!

(−lx

Lcγ

)m)]

, x < LcγT

πLc

[1

(Lc − 1)!γLc(x − LcγT )Lc −1e−

x −L c γ Tγ

]

+L−Lc∑j=1

Lc −2∑k=0

A(j,k)γk+1−Lc

(Lc − 2 − k)!e−x/γ

(e

−j γ Tγ (x − LcγT )Lc −2−k − xLc −2−k

)

+(

L

Lc

)1

(Lc − 1)!γLc

e−xγ(xLc −1 − (x − LcγT )Lc −1), x ≥ LcγT (4.40)

where πLcand A(j,k) were defined in (4.22) and (4.38), respectively. The MGF

of γc , denoted by Mγc(t), can be shown to be given, for the i.i.d. Rayleigh fading

case, by

Mγc(t) =

(L

Lc

)1 − Γ(Lc, (1 − tγ)LcγT /γ)

(1 − tγ)Lc+

L−Lc∑l=1

(−1)Lc + l−1(

L − Lc

l

)

×(

Lc

l

)Lc −1[I(l, t) −

Lc −2∑m=0

(−l

Lc

)m (1 − Γ(m + 1, (1 − tγ)LcγT /γ))(1 − tγ)m+1

]

+L−Lc∑j=1

Lc −2∑k=0

A(j,k)

[e−

L c γ T + j γ Tγ

eLc γT t

(1 − tγ)Lc −1−k− Γ(Lc − 1 − k, (1 − tγ)LcγT /γ)

(1 − tγ)Lc −1−k

]

+(

L

Lc

)[Γ(Lc, (1 − tγ)LcγT /γ)

(1 − tγ)Lc− eLc γT (t− 1

γ)

(1 − tγ)Lc

]+ πLc

eLc γT t

(1

1 − tγ

)Lc

,

(4.41)

where

I(l, t) =1

γt − Lc + lLc

[exp

(LcγT (tLcγ − Lc − l)

Lcγ

)− 1

], (4.42)


πLcand A(j,k) were defined in (4.22) and (4.38), respectively. With these closed-

form results, we can readily evaluate the error performance of different modula-tion schemes with GSECps.

4.5.2 Complexity analysis

In this subsection, we calculate the average number of path estimations and com-parisons needed by GSECps during path selection process over the i.i.d Rayleighfading channel. This study allows a thorough complexity comparison amongGSC, GSEC and GSECps.

Note that compared with conventional GSEC, GSECps adds a ranking processwhen La < Lc . Since the ranking process only involves comparisons, the aver-age number of path estimations needed by GSECps, denoted by N , will be thesame as GSEC, which was given in (4.29) in a previous section [6, eq. (27)]. ForRayleigh fading channels, (4.29) specializes to

NE =L∑

k=Lc

k

(k − 1k − Lc

)[1 − e−

γ Tγ]k−Lc e−

γ T L cγ

+ L

L∑k=L−Lc +1

(L

k

)[1 − e−

γ Tγ]k

e−γ T (L −k )

γ .

(4.43)

To calculate the average number of comparisons of GSECps, we first note thatevery path examination involves one path estimation and one comparison. IfLa ≥ Lc , according to the operation of GSECps, the receiver stops examiningpaths whenever it has found Lc acceptable paths. Therefore, in this case, thenumber of comparisons will be equal to the number of path examinations. Hence,the average number of comparisons for the La ≥ Lc case, denoted by NA , canbe shown to be given by

NA =L∑

k=Lc

k

(k − 1k − Lc

)[Fγ (γT )

]k−Lc[1 − Fγ (γT )

]Lc . (4.44)

On the other hand, when La < Lc , GSECps will first examine all L availablepaths. Therefore, the number of comparisons associated with path examinationsis L. Then GSECps ranks the L − La unacceptable paths to select Lc − La pathswith the highest instantaneous SNRs. It is not hard to show that the number ofcomparisons associated with the ranking process is

∑Lc −La

k=1 (L − La − k). There-fore, for any La , 0 ≤ La < Lc , the number of comparisons is L +

∑Lc −La

k=1 (L −La − k). Hence, the average number of comparisons for the La < Lc case, denotedby NB , is given by

NB =Lc −1∑j=0

Pr[La = j][L +

Lc −j∑k=1

(L − j − k)], (4.45)


where Pr[La = j] is the probability that there are exactly j acceptable paths.It can be shown, with the assumption of i.i.d. fading paths, that Pr[La = j] ismathematically given by

Pr[La = j] =(

L

j

)[Fγ (γT )

]L−j [1 − Fγ (γT )]j

. (4.46)

By combining these two mutually exclusive cases, we obtain the expression forthe overall average number of comparisons needed by GSECps over i.i.d. fadingchannels, denoted by NC , as

NC = NA + NB

=L∑

k=Lc

k

(k − 1k − Lc

)[Fγ (γT )

]k−Lc[1 − Fγ (γT )

]Lc

+Lc −1∑La =0

[L +

Lc −La∑k=1

(L − La − k)](

L

La

)[Fγ (γT )

]L−La[1 − Fγ (γT )

]La .

(4.47)

For Rayleigh fading channels, NC specializes to

NC =L∑

k=Lc

k

(k − 1k − Lc

)[1 − e−

γ Tγ]k−Lc e−

γ T L cγ

+Lc −1∑La =0

[L +

Lc −La∑k=1

(L − La − k)](

L

La

)[1 − e−

γ Tγ]L−La e−

γ T L aγ .

(4.48)

For GSEC, there is no ranking process and, as such, the average number ofcomparisons needed is equal to the number of path examinations, which has beengiven in (4.43). For GSC, the number of comparisons is always

∑Lc

k=1(L − k) toselect the strongest Lc paths through comparisons.


In Fig. 4.7, we study the operation complexity of GSECps in comparison withGSEC and GSC. We set L = 4 and Lc = 2. In Fig. 4.7(a), we plot the averagenumber of path estimations needed by the three schemes as a function of normal-ized switching threshold, γT /γ. As expected, GSECps needs the same numberof path estimations as GSEC and both schemes require fewer path estimationsthan GSC when the threshold is not too large. Only when γT is larger than γ

by more than 5 dB will GSECps and GSEC need to estimate all four paths.In Fig. 4.7(b), the average number of comparisons needed by these schemes

during the path selection process is plotted as a function of the normalizedswitching threshold, γT /γ. Note that GSC always performs roughly five com-parisons to find the two best paths among the four available while the receiverwith GSEC performs at most four comparisons between the path SNR and the


−15 −10 −5 0 5 10 152

3

4

5

6

7

8

9

Normalized Switching Threshold, γT/γ−, dB

Ave

rage

Num

ber

of C

ompa

rison

s, M (b)

−15 −10 −5 0 5 10 152

2.5

3

3.5

4

4.5

Normalized Switching Threshold, γT/γ−, dB

Ave

rage

Num

ber

of P

ath

Est

imat

ions

, N

(a)

GSCGSECpsGSEC

GSCGSECpsGSEC

Figure 4.7 Operation complexity of GSECps in comparison with GSC and GSEC(L = 4 and Lc = 2): (a) average number of path estimations, (b) average number ofcomparisons [25]. c© 2008 IEEE.

switching threshold. The number of comparisons needed by GSECps, however,increases from 2 to 9 in the worst case, which includes 4 comparisons between theinstantaneous path SNR and the switching threshold and 5 more comparisons tofind the two best unacceptable paths. While GSECps may need four more com-parisons than GSC in the worst case, these extra comparisons involve a fixedthreshold, and are therefore easy to implement. Moreover, as will be shown inthe next subsection, GSECps can provide nearly the same performance as GSCwhen γT is slightly less than γ, where the number of comparisons needed byGSECps is much less than GSC.

Figure 4.8 plots the average error rate of BPSK with GSEC, GSECps andGSC as a function of the average SNR per path with γT = 3 dB. Again, we setL = 4 and Lc = 2. It is easy to see that when the channel condition is poor, i.e.γ is small, GSECps has the same error performance as GSC. That is because itis more likely that there are not enough acceptable paths and GSECps operatesthe same as GSC by selecting the best unacceptable paths. As γ becomes larger,i.e. the channel condition is improving, the performance advantage of GSECpsover GSEC decreases. Note that in this case, the probability of La ≥ Lc increasesand GSECps behaves more like GSEC.


−10 −5 0 5 10 15 2010

−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Average SNR per Path, γ−, dB

Ave

rage

Err

or R

ate,

Ps(E

)

GSECGSECpsGSC

Figure 4.8 Average error rate of BPSK with GSC, GSEC, and GSECps over L = 4i.i.d. Rayleigh fading channels as function of the average SNR per path (γT = 3 dBand Lc = 2) [25]. c© 2008 IEEE.

In Fig. 4.9, we plot the average error rate of BPSK with the three schemesunder consideration as a function of the normalized switching threshold, γT /γ.It indicates that when γT is small, GSECps has the same error performance asGSEC because there are usually enough acceptable paths and GSECps operatesmore like GSEC. As γT increases, the performance of GSEC degrades while theperformance of GSECps becomes the same as that of GSC.

Considering Fig. 4.9 together with Fig. 4.7, we can study the trade-off involvedamong GSECps, GSEC, and GSC. Comparing GSECps with GSEC, we observethat both schemes require the same number of path estimations while GSECpsoffers better error performance at the cost of slightly more comparison opera-tions. In particular, if γT /γ is set to be −7 dB, where GSEC reaches its besterror performance, GSECps still offers a 60% decrease in average error rate whilerequiring 0.05 more comparisons on average. Comparing GSECps with GSC, wecan see that if we set γT /γ = −3 dB, GSECps has nearly the same error perfor-mance as GSC, but needs fewer path estimations and comparisons. In conclu-sion, GSECps achieves better performance–complexity trade-off than both GSECand GSC.


−20 −15 −10 −5 0 5 1010

−5

10−4

10−3

10−2

Normalized Switching Threshold,γT/γ−,dB

Ave

rage

Err

or R

ate,

Ps(E

)

GSECGSECpsGSC

Figure 4.9 Average error rate of BPSK with GSC, GSEC, and GSECps over L = 4i.i.d. Rayleigh fading channels as function of the normalized switching threshold γT /γ(Lc = 2) [25]. c© 2008 IEEE.

4.6 Summary

In this chapter, we studied four representative advanced diversity combiningschemes for a diversity-rich environment. The common basic idea is to reducethe combiner hardware complexity and/or processing power by only combininga proper selected path subset in the optimal MRC fashion. We derived the exactstatistics of their combiner output SNR, which are in turn applied to the per-formance analysis of these schemes over a generalized fading channel. Wheneverfeasible, we also quantified the associated complexity of each scheme in terms ofthe average number of active MRC branches and the average number of path esti-mations. Selected numerical examples were presented and discussed to illustratethe trade-off among different schemes.


Win and Winters [11] present a virtual branch-based method to analyze theperformance of the GSC scheme over fading channels. The analysis of GSC over


general correlated fading has also been addressed (see for example [15,28]). TheGSC scheme has also been designed based on the log-likelihood ratio of eachdiversity branch [29]. The idea of a hybrid combining scheme was also consideredfor the EGC scheme in [30]. A modified version of the AT-GSC scheme, whicheliminates the error state of no path selection, was proposed and studied by Chenand Tellambura [31]. Xiao and Dong propose a modified version of the NT-GSCscheme and present a new analytical method for the performance analysis ofboth GSC and NT-GSC [23]. Femenias [32] propose a generalized sort, switch,and examine combining (GSSEC) scheme and compared it with GSC and GSECschemes.

References

[1] W. C. Jakes, Microwave Mobile Communication, 2nd ed. Piscataway, NJ: IEEE Press,1994.


[3] N. Kong, T. Eng and L. B. Milstein, “A selection combining scheme for RAKEreceivers,” in Proc. IEEE Int. Conf. Univ. Personal Comm. ICUPC’95, Tokyo, Japan,November 1995, pp. 426–429.

[4] M. Z. Win and Z. A. Kostic, “Virtual path analysis of selective Rake receiver in densemultipath channels,” IEEE Commun. Letters, vol. 3, no. 11, pp. 308–310, November1999.

[5] Y. Roy, J.-Y. Chouinard and S. A. Mahmoud, “Selection diversity combining with mul-tiple antennas for MM-wave indoor wireless channels,” IEEE J. Select. Areas Commun.,vol. SAC-14, no. 4, pp. 674–682, May 1998.

[6] H.-C. Yang and M.-S. Alouini, “Generalized switch and examine combining (GSEC):a low-complexity combining scheme for diversity rich environments,” IEEE Trans.

Commun., vol. COM-52, no. 10, pp. 1711–1721, October, 2004.[7] K. J. Kim, S. Y. Kwon, E. K. Hong and K. C. Whang, “Comments on ‘Comparison of

diversity combining techniques for Rayleigh-fading channels’,” IEEE Trans. Commun.,vol. COM-46, no. 9, pp. 1109–1110, September 1998.

[8] M.-S. Alouini and M. K. Simon, “Performance of coherent receivers with hybridSC/MRC over Nakagami-m fading channels,” IEEE Trans. Veh. Technol., vol. VT-48, no. 4, pp. 1155–1164, July 1999.

[9] M. Z. Win and J. H. Winters, “Analysis of hybrid selection/maximal-ratio combining inRayleigh fading,” IEEE Trans. Commun., vol. COM-47, no. 12, pp. 1773–1776, Decem-ber 1999. See also Proceedings of IEEE International Conference on Communications

(ICC’99), Vancouver, British Columbia, Canada, pp. 6–10, June 1999.[10] M.-S. Alouini and M. K. Simon, “An MGF-based performance analysis of generalized

selective combining over Rayleigh fading channels,” IEEE Trans. Commun., vol. COM-48, no. 3, pp. 401–415, March 2000.

[11] M. Z. Win and J. H. Winters, “Virtual branch analysis of symbol error probability forhybrid selection/maximal-ratio combining in Rayleigh fading,” IEEE Trans. Commun.,vol. COM-49, no. 11, pp. 1926–1934, November 2001.

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5 Adaptive transmissionand reception

5.1 Introduction

Most conventional diversity techniques target the worst-case scenario [1]. Thegeneral principle is to perform a fixed set of combining operations on differentlyfaded replicas such that the combiner output signal exhibits a better quality.This basic design principle manifests itself in the conventional MRC, EGC, andSC schemes as well as those more recent advanced combining schemes discussedin the previous chapter, such as GSC, T-GSC, and GSEC. While this approachhas proven to be very effective in improving the performance of wireless commu-nication systems, especially when the channel experiences deep fades, it may leadto an inefficient utilization of the receiver processing resource when the channelbecomes more favorable. Note that the same set of combining operations willstill be performed even though the system performance may be acceptable withfewer combining operations in this case. This observation motivates the recentinterest in adaptive diversity combining for receiver power-saving purposes.

The basic idea of adaptive combining schemes is to adaptively utilize thediversity combiner resource in such a way that the combiner output signal sat-isfies a certain quality requirement. The generic structure of the adaptive diver-sity receiver is shown in Fig. 5.1. Specifically, the receiver will just performenough combining operations such that the quality of the combiner output sig-nal becomes acceptable. For example, minimum selection GSC (MS-GSC) [2–4]is one of the first adaptive combining schemes. With MS-GSC, the receiver com-bines the least number of best diversity branches such that the SNR of the com-biner output signal exceeds a certain predetermined threshold. Other sampleadaptive combining schemes include output threshold MRC (OT-MRC), outputthreshold GSC (OT-GSC) [5], and minimum estimation and combining GSC(MEC-GSC) [6].

It is worth mentioning that these adaptive combining schemes can also beviewed as the generalization of the conventional SSC scheme [1]. Recall thatwith the SSC scheme, the receiver continues to use its current diversity branchas long as its SNR is above a certain threshold (i.e. no combining operation).When the SNR of the current branch is below the threshold, the receiver switchesto the other branch (the simplest combining operation) and uses it for datareception. As such, SSC is one of the simplest adaptive combining schemes.

98 Chapter 5. Adaptive transmission and reception

γc

Diversity

Combiner

γ1

γ2

γLγc ≥ γT

Detector

Figure 5.1 Structure of an adaptive diversity receiver.

The aforementioned adaptive combining schemes, such as MS-GSC or OT-MRC,will perform more complex combining operations if necessary. Note also thatthese adaptive schemes differ from those threshold-based combining schemesdiscussed in the previous chapter, e.g. the T-GSC and GSEC schemes [7–9], inthat the threshold checking is performed on the combiner output rather thanon each individual branch. SSC/SEC schemes appear in both categories as theoutput SNR with these schemes is the SNR of the current used branch.

In the following sections, we first present and study three representative adap-tive combining schemes, namely OT-MRC, MS-GSC, and OT-GSC. We carryout a thorough and accurate performance versus complexity trade-off analysis oneach of them with the help of the order statistics results. After that, we illustratethe general applicability of the adaptive combining idea by considering adaptivetransmit diversity systems [10] and RAKE finger management schemes [11]. Wewill show that the adaptive combining principle can bring significant complexitysaving at the cost of little or no performance loss. Finally, we investigate the jointdesign of adaptive diversity combining with adaptive modulation. Specifically,noting that both adaptive modulation and adaptive combining utilize some pre-determined threshold in their operation, we naturally combine them and arriveat three different joint adaptive modulation and diversity combining (AMDC)schemes [12]. The performance and efficiency of each schemes are accuratelyquantified and illustrated using selected numerical examples.

5.2 Output-threshold MRC

With OT-MRC, the receiver tries to increase the combined SNR γc above theoutput threshold γT by gradually changing from a low-order MRC scheme toa higher-order MRC scheme. For the sake of convenience and clarity, we adopt

Output-threshold MRC 99

Startγc = 0, l = 0

l = l + 1

Estimate γl

Update combined SNRγc = γc + γl

γc > γT

Stop

Yes

No

l = LNo Yes

Figure 5.2 Flow chart for OT-MRC mode of operation.

a discrete-time implementation in the following presentation. More specifically,short guard periods are periodically, usually at the rate of channel coherent time,inserted into the transmitted signal. During these guard periods, the receiverreaches the appropriate diversity combining decisions for the subsequent databurst.

In each guard period, starting from the single-branch case, the OT-MRC com-biner successively estimates additional diversity paths, activates MRC branchesand applies MRC to the estimated paths in order to raise the combined SNRabove the threshold γT . The flow chart in Fig. 5.2 illustrates the mode of oper-ation of OT-MRC. In particular, after the guard interval starts, the combinerchecks the SNR of the first available diversity path, denoted by γ1 . If it is abovethe threshold γT , then the combiner simply uses this path for data reception, i.e.γc = γ1 . This is exactly the same as the no-diversity case. If γ1 < γT , the com-biner estimates a second diversity path, whose SNR is denoted by γ2 , activatesanother MRC branch, and applies MRC to these two paths. If the resulting com-bined SNR γc = γ1 + γ2 is above γT , then the combiner just acts as a dual-branchMRC. Otherwise (i.e. γ1 + γ2 < γT ), a third path is estimated and another MRCbranch is activated. This process stops whenever the combined SNR γc exceedsthe threshold γT . Note that an L-branch MRC receiver will be used only in theworst case when the combined SNR γc is still below γT after the activation ofL − 1 MRC branches.


Based on the OT-MRC mode of operation, we can see that the receiver doesnot always utilize all the L available diversity paths. If the MRC-combined SNRafter the activation of the first l MRC branches is above the threshold, thereceiver does not need to estimate the remaining L − l diversity paths. Corre-spondingly, only l MRC branches need to be active during the data reception.These features constitute the main power saving of OT-MRC over the traditionalMRC combiner, where all L diversity paths always have to be estimated and allL MRC branches are always active during data reception. This comes, of course,at the cost of a certain performance loss, which will be accurately quantified inthe following section.


To analyze the performance of OT-MRC, we need a statistical characterizationof the combiner output SNR. In this subsection, we derive the cumulative distri-bution function (CDF), the probability density function (PDF), and the momentgenerating function (MGF) of the combiner output SNR with OT-MRC.

Based on the mode of operation of OT-MRC, we note that the events that l

diversity paths (or equivalently, an l-branch MRC) are used for data reception,l = 1, 2, · · · , L, are mutually exclusive. Applying the total probability theorem,we can write the CDF of the combined SNR, FMRC

γc(·), in a summation form as

Fγc(x) = Pr[γc < x]

=L∑

l=1

Pr

⎡⎣γc =

l∑j=1

γj & γc < x

⎤⎦ , (5.1)

where γc =∑l

j=1 γj corresponds to the event that an l-branch MRC schemeis used for data reception. We observe that with OT-MRC (i) a single branchreceiver is used when the SNR of the first diversity path is above the thresh-old (i.e. γ1 ≥ γT ), (ii) an l-branch MRC (with 2 ≤ l ≤ L − 1) is used when thecombined SNR of an l − 1-branch MRC is below the threshold whereas the com-bined SNR of an l-branch MRC is above the threshold (i.e.

∑l−1j=1 γj < γT and∑l

j=1 γj ≥ γT ), and (iii) an L-branch MRC is used when the combined SNR of anL − 1-branch MRC is below the threshold (i.e.

∑L−1j=1 γj < γT ). Mathematically

speaking, we then have

Pr

⎡⎣γc =

l∑j=1

γj

⎤⎦ =

⎧⎪⎪⎨⎪⎪⎩

Pr[γ1 ≥ γT ], l = 1;

Pr[∑l−1

j=1 γj < γT &∑l

j=1 γj ≥ γT ], 2 ≤ l ≤ L − 1;

Pr[∑L−1

j=1 γj < γT ], l = L.

(5.2)


Using (5.2) in (5.1), we can rewrite the CDF of the combined SNR with OT-MRCafter some manipulations as

Fγc(x) = Pr[γT ≤ γ1 & γ1 < x]

+L−1∑l=2

Pr

⎡⎣ l−1∑

j=1

γj < γT ≤l∑

j=1

γj &l∑

j=1

γj < x

⎤⎦

+Pr

⎡⎣L−1∑

j=1

γj < γT &L∑

j=1

γj < x

⎤⎦

=

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

Pr[γT ≤ γ1 < x]

+∑L−1

l=2 Pr[∑l−1

j=1 γj < γT ≤∑l

j=1 γj < x]

+ Pr[∑L−1

j=1 γj < γT &∑L

j=1 γj < x], γT < x;

Pr[∑L

j=1 γj < x], γT ≥ x.

(5.3)

Under the assumption of i.i.d. fading paths, we can rewrite (5.3) in terms of thesingle-branch CDF, Fγ (·), and the PDF of an l-branch MRC output SNR (i.e.∑l

j=1 γj ), pl−M RCγc

(·), as

Fγc(x) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Fγ (x) − Fγ (γT )

+L−1∑l=2

∫ γT

0p(l−1)−M RC

γc(y)(Fγ (x − y) − Fγ (γT − y))dy

+∫ γT

0p(L−1)−M RC

γc(y)Fγ (x − y)dy, γT < x;∫ x

0pL−M RC

γc(y)dy, γT ≥ x.

(5.4)

Differentiating Fγc(x) in (5.4) with respect to x, we obtain a generic formula for

the PDF of the combined SNR as

pγc(x) =

⎧⎪⎪⎨⎪⎪⎩

pγ (x) +L∑

l=2

∫ γT

0p(l−1)−M RC

γc(y)Fγ (x − y)dy, γT < x;

pL−M RCγc

(x), γT ≥ x,

(5.5)

where pγ (·) is the single branch SNR PDF. Applying the CDF and PDF of anl-branch MRC output SNR summarized in Table 5.1, we can obtain the CDFand PDF of the combined SNR γc for various fading models of interest.

As an example, for the Rayleigh fading case, the CDF of the combined SNRwith OT-MRC is given, after integration and some simplifications, by the fol-lowing closed-form expression

Fγc(x) =

⎧⎨⎩1 − Ae−

xγ , γT < x;

1 −∑L

l=11

(l−1)!

(xγ

)l−1e−

xγ , γT ≥ x,

(5.6)


Table 5.1 Closed-form expressions of pl−M RCγc

(x) and F l−M RCγc

(x) for the three popularfading models.

Model pl−M RCγc

(x) F l−M RCγc

(x)

Rayleigh 1(l−1)!

xl−1

γ l e−xγ 1 − e−

xγ∑l−1

i=01i!

(xγ

)i

Rician K +1γ e−lK− (K + 1 )

γx 1 − Ql

(√2lK,

√2(K +1)

γ x

)(

K +1lK γ x

) l−12

Il−1

(2√

lK (K +1)γ x

)Nakagami

(mγ

)lmxl m −1

Γ(lm ) e−m

γx 1 − Γ(lm , m

γx)

Γ(lm )

where

A =L∑

l=1

1(l − 1)!

(γT

γ

)l−1

. (5.7)

Similarly, the PDF of the combined SNR with OT-MRC is obtained as

pγc(x) =

Aγ e−

xγ , γT < x;

1(L−1)!

xL −1

γ L e−xγ , γT ≥ x.

(5.8)

With the statistics of the combined SNR of OT-MRC derived, we can accu-rately evaluate the performance of OT-MRC. Specifically, the average error rateof OT-MRC can be calculated by averaging the conditional error rate of an inter-ested modulation scheme, PE (γc), over the PDF of the combined SNR γc , fγc

(x),given in (5.5). For the important special case of BPSK over Rayleigh fading, wecan show, after proper substitution and manipulations, that the average BERwith OT-MRC is given by the following closed-form expression

Pb =∫ ∞

0Q(

√2x)pγc

(x)dx (5.9)

=12−√

γ

1 + γ

L−1∑l=0

1l!

(γT

γ

)l

Q

(√21 + γ

γγT

)

+1

2√

π

(1

1 + γ

)l [Γ(

l +12

)− Γ

(l +

12,1 + γ

γγT

)],

where Q(·) is the Gaussian Q-function.In Fig. 5.3(b), we plot the average bit error rate of BPSK with OT-MRC in a

Rayleigh fading environment (given in (5.9)) as a function of the SNR thresholdγT . For reference purposes, the error rate of no diversity and conventional MRCcases are also plotted. As we can see, when the threshold γT is small, OT-MRCgives roughly the same performance as the no-diversity case. Conversely, whenthe threshold γT increases, the error rate of OT-MRC decreases and becomesthe same as that of the conventional MRC eventually. This is because as γT


−15 −10 −5 0 5 10 15

104

103

102

101

SNR threshold, γT, in dB

Ave

rage

Bit

Err

or P

roba

bilit

y P b(E

)

(b)

No DiversityOT−MRC MRC

−15 −10 −5 0 5 10 150

1

2

3

4

5

6

SNR threshold, γT, in dB

Ave

rage

Num

ber

of A

ctiv

e B

ranc

hes

NA (a)

MRCOT−MRCNo Diversity

Figure 5.3 Performance of OT-MRC in a Rayleigh fading environment: (a) average biterror rate of BPSK as function of average SNR per path; (b) outage probability asfunction of outage threshold with γ = 10 dB (L = 3 and γT = 6 dB) [5]. c© 2005IEEE.

increases, more paths need to be used to raise the combined SNR above γT andthis of course leads to a better performance.

5.2.2 Power saving analysis

We now quantify the power savings of OT-MRC scheme by calculating the aver-age number of path estimations during guard period, or equivalently, the averagenumber of active MRC branches during data reception.

Note that with OT-MRC, l = 1, 2, · · · , L, MRC branches may be active duringdata reception. Since they are mutually exclusive events, we can write the averagenumber of active MRC branches, denoted by NA , into a finite summation formas

NA =L∑

l=1

l Pr

⎡⎣γc =

l∑j=1

γj

⎤⎦ , (5.10)

where Pr[γc =∑l

j=1 γj ] is the probability that an l-branch MRC is used forthe subsequent data reception, which was given in (5.2). With the help of theassumption of i.i.d. fading paths, (5.2) can be rewritten in terms of the common


SNR CDF per branch Fγ (·) and the combined SNR CDF of an i-branch MRCF l−M RC

γc(·) as

Pr

⎡⎣γc =

l∑j=1

γj

⎤⎦ =

⎧⎪⎪⎨⎪⎪⎩

1 − Fγ (γT ), l = 1;

F(l−1)−M RCγc (γT ) − F l−M RC

γc(γT ), 2 ≤ l ≤ L − 1;

F(L−1)−M RCγc (γT ), l = L,

(5.11)

Substituting (5.11) into (5.10), the average number of active MRC branches withOT-MRC can be finally obtained after some simplifications as

NA = 1 +L−1∑l=1

F l−M RCγc

(γT ). (5.12)

Note that (5.12) also gives the average number of path estimations needed byOT-MRC during the guard period.

For the Rayleigh fading environment, using an appropriate expression forF (l)(·) from Table 5.1, we obtain the average number of active MRC branchesneeded with OT-MRC for the Rayleigh fading case as

NA = 1 +L−1∑l=1

[1 − e−

γ Tγ

l−1∑i=0

1i!

(γT

γ

)i]

. (5.13)

Figure 5.3(a) shows the average number of active MRC branches with OT-MRC in Rayleigh fading environment with L = 5. We can see that as the SNRthreshold γT increases, the average number of active MRC branches increasesfrom 1 to 5, just as expected. When the threshold γT is small, a single-branchSNR will be larger than γT . As the threshold γT increases, it becomes moredifficult to raise the combined SNR above γT . Note that for conventional MRC,all 5 branches are always active. So with a moderate choice of γT , OT-MRC canprovide considerable power saving over conventional MRC by maintaining feweractive branches on average. Hence, Fig. 5.3 gives a clear view of the trade-offof power consumption versus performance for OT-MRC. The choice of the SNRthreshold γT is of critical importance in this trade-off. By looking at Fig. 5.3(a)and 5.3(b) simultaneously, we can see that if γT is set, for example, to 10 dBin this scenario, OT-MRC gives the same error performance as the conventionalMRC but it just needs less than 4 active MRC branches on average. Therefore,we can conclude that OT-MRC can achieve the optimal MRC performance whileoffering a notable amount of power savings.

5.3 Minimum selection GSC

Minimum selection GSC (MS-GSC) was first proposed in [2] as a power-savingimplementation of the GSC scheme. With MS-GSC, the receiver combines theleast number of best diversity paths such that the combiner output SNR isabove a certain threshold. Both the simulation study in [2] and the preliminary

Minimum selection GSC 105

Startγc = 0, l = 0

l = l + 1

Estimate and rank γi, i = 1 2 · · · L

Update combined SNRγc = γc + γl:L

γc > γT

Stop

Yes

No

l = Lc

No Yes

Figure 5.4 Flow chart for MS-GSC mode of operation.

theoretical analysis in [3] show that MS-GSC can save a considerable amountof processing power by keeping fewer MRC branches active on average. In whatfollows, we present the mode of operation of MS-GSC and derive the statisticsof the combined SNR, which involves the new results of order statistics fromthe previous chapter. The statistical results are then applied to the exact per-formance analysis of MS-GSC over fading channels.

5.3.1 Mode of operation

As is usually the case in practice, we assume a discrete-time implementationapproach where combining decisions are made during a short guard period beforethe transmission of data burst. With MS-GSC, the receiver combines the leastnumber of best branches such that the SNR of combined signal, denoted by γc ,is greater than a preselected output threshold, γT . The flow chart in Fig. 5.4explains the mode of operation of MS-GSC, where γl:L denotes the SNR of thelth strongest path. In particular, after the guard period starts, the receiver firstestimates and ranks the SNR of all diversity paths. Then, starting from the bestpath, i.e. the path whose SNR is equal to γ1:L , the receiver tries to increase thecombined SNR γc above the threshold γT by combining an increasing number ofdiversity paths. More specifically, if γ1:L ≥ γT , then only the strongest diversitypath will be used, i.e. γc = γ1:L . In this case, the receiver acts as an L-branch


selection combiner. If γ1:L < γT , then the receiver will activate another MRC-branch, find the second strongest path and apply MRC to the two strongestpaths. If the output SNR, now equal to γ1:L + γ2:L , is greater than γT , only two-branch MRC will be used for data reception, i.e. γc = γ1:L + γ2:L . Similarly, ifthe output SNR of i − 1-branch MRC

∑i−1j=1 γj :L is less than γT and the output

SNR of i-branch MRC∑i

j=1 γj :L is greater than γT , then i strongest paths willbe combined for the succeeding data burst reception. This process is continueduntil either the combined SNR is above γT or all Lc MRC branches are activated.In the latter case, the receiver will act as a traditional L/Lc GSC combiner, i.e.the Lc strongest paths are combined in an MRC fashion.

The mode of operation of MS-GSC can be mathematically summarized as

γc = γ1:L iff γ1:L ≥ γT ;

γc = γ1:L + γ2:L iff γ1:L < γT & γ1:L + γ2:L ≥ γT ;...

γc =i∑

j=1

γj :L iffi−1∑j=1

γj :L < γT &i∑

j=1

γj :L ≥ γT ;

...

γc =Lc∑j=1

γj :L iffLc −1∑j=1

γj :L < γT . (5.14)


With the help of the order statistics results from the previous chapter, we nowderive the statistical characterization of the combined SNR with MS-GSC. Fromthe mode of operation of MS-GSC, we can see that the events γc =

∑ij=1 γ(j )

are mutually exclusive. Applying the total probability theorem, we can write theCDF of γc as

Fγc(x) = Pr[γc < x]

=Lc∑i=1

Pr[γc =i∑

j=1

γj :L & γc < x]. (5.15)

Applying the mode of operation of MS-GSC as summarized in (5.14), we have

Fγc(x) = Pr[γ1:L ≥ γT & γc = γ1:L < x]

+Lc −1∑i=2

Pr[i−1∑j=1

γj :L < γT & γc =i∑

j=1

γj :L ≥ γT & γc < x]

+Pr[Lc −1∑j=1

γj :L < γT & γc =Lc∑j=1

γj :L < x]. (5.16)


Note that only when γc =∑Lc

j=1 γ(j ) can the combined SNR be smaller than γT .We can rewrite the CDF of γc as

Fγc(x) =

⎧⎪⎪⎨⎪⎪⎩

FΓ1 (x) − FΓ1 (γT )

+∑Lc

i=2 F(i)γc (x) + FΓL c

(γT ), x ≥ γT ;

FΓL c(x), 0 ≤ x < γT ,

(5.17)

where FΓ i(·) is the CDF of the sum of the first i largest path SNRs and F

(i)γc (x)

denotes the probability Pr[∑i−1

j=1 γ(j ) < γT & γT ≤∑i

j=1 γ(j ) < x]. The statisticsof the sum of the first i largest path SNRs has been extensively studied inthe previous chapter during the discussion of the conventional GSC scheme.Specifically, the closed-form expression of FΓ i

(·) for i.i.d. Rayleigh fading case isgiven by [13, eq. (9.332)]

FΓ i(x) =

L!(L − i)!i!

1 − e−

xγ

i−1∑k=0

1k!

(x

γ

)k

+L−i∑l=1

(−1)i+ l−1 (L − i)!(L − i − l)!l!

(i

l

)i−1

×[(

1 +l

i

)−1 [1 − e−(1+ l

i ) xγ

]

−i−2∑

m=0

(− l

i

)m(

1 − e−xγ

m∑k=0

1k!

(x

γ

)k)]

. (5.18)

Let Γi−1 denote the sum of the first i − 1 largest SNRs, i.e. Γi−1 =∑i−1

j=1 γj :L .

Then, the probability F(i)γc (x) can be rewritten as

F (i)γc

(x) = Pr[Γi−1 < γT & γT ≤ γi:L + Γi−1 < x], (5.19)

which can be calculated using the joint PDF of γi:L and Γi−1 . pγi :L ,Γ i−1 (·, ·).Specifically, with the help of Fig. 5.5, F

(i)γc (x) can be calculated as

F (i)γc

(x) =

⎧⎪⎪⎨⎪⎪⎩∫ i−1

i xi−1

i γT

∫ yi−1

γT −y pγi :L ,Γ i−1 (z, y)dzdy

+∫ γT

i−1i x

∫ x−y

γT −y pγi :L ,Γ i−1 (z, y)dzdy, γT ≤ x < ii−1 γT ;

FΓ i−1 (γT ) − FΓ i(γT ), i

i−1 γT ≤ x.

(5.20)


i−1Γ

=

= (i−1)

0

γ

(b)(a)

0T

γ

i−1Γ Tγ

i:Lγ

i−1Γ

=

= (i−1)

Tγ

i−1Γ Tγ

i:L

i−1Γ

i:Lγ

i−1T

γ

Tγ

Tγ

i:Lγ

i−1Γ =+

i−1Γ

i:Lγ

+i−1T

γ

Tγ

i−1Γ i:Lγ x=+

Tγ

i:Lγ

i−1Γ =

Figure 5.5 The integration region of the joint PDF of Γi−1 and γi :L for calculating (a)F

(i)γ c (x), x ≥ iγT /(i − 1) and Pr[N = i]; (b) F

(i)γ c (x), γT ≤ x < iγT /(i − 1) [4]. c© 2006

IEEE.

After substituting (5.20) into (5.17), we obtain the generic expression of the CDFof the combined SNR γc with MS-GSC as

Fγc(x) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

FΓL c(x), 0 ≤ x < γT ;

FΓ1 (x) − FΓ1 (γT ) + FΓL c(γT )

+∑Lc

i=2

(∫ i−1i x

i−1i γT

∫ yi−1


+∫ γT

i−1i x

∫ x−y

γT −y pγi :L ,Γ i−1 (z, y)dzdy)

, γT ≤ x < Lc

Lc −1 γT ;...

FΓ1 (x) − FΓ1 (γT ) + FΓ l(γT )

+∑l

i=2

(∫ i−1i x

i−1i γT

∫ yi−1


+∫ γT

i−1i x

∫ x−y

γT −y pγi :L ,Γ i−1 (z, y)dzdy)

, l+1l γT ≤ x < l

l−1 γT ;...

FΓ1 (x), 2γT < x.

(5.21)

The generic expression for the PDF of the combined SNR with MS-GSC canbe obtained by differentiating (5.21) with respect to x and noting that

ddx

F (i)γc

(x) =∫ γT

i−1i x

pγi :L ,Γ i−1 (x − y, y)dy, γT ≤ x <i

i − 1γT , (5.22)

as

pγc(x) =

⎧⎪⎪⎨⎪⎪⎩

pΓ1 (x) +∑Lc

i=2

∫ γTi−1

i xpγi :L ,Γ i−1 (x − y, y)dy

× (U(x − γT ) − U(x − ii−1 γT )), x ≥ γT ;

pΓL c(x), 0 ≤ x < γT ,

(5.23)


where U(·) is the unit step function. It follows that the generic expression of theMGF of the combined SNR with MS-GSC is given by

Mγc(s) =

∫ γT

0pΓL c

(x)esxdx +∫ ∞

γT

pΓ1 (x)esxdx

+Lc∑i=1

∫ ii−1 γT

γT

[∫ γT

i−1i x

pγi :L ,Γ i−1 (x − y, y)dy

]esxdx. (5.24)

The joint PDF of pγi :L ,Γ i−1 (·, ·) has been extensively studied in a previous chap-ter. Specifically, for i.i.d. Rayleigh fading environments, its closed-form expres-sion was given in (3.31) and is reproduced here for convenience

pγl :L ,yl(x, y) =

L−l∑j=0

(−1)jL!(L − l − j)!(l − 1)!(l − 2)!j!γl

[y − (l − 1)x](l−2)e−y + ( j + 1 )x

γ ,

x ≥ 0, y ≥ (l − 1)x. (5.25)

After proper substitution and integration, we can obtain the statistics of thecombined SNR with MS-GSC in closed form. For example, we can show that thesummand in (5.23) is given, after successive integration by parts, by

∫ γT

i−1i x

pγi :L ,Γ i−1 (x − y, y)dy =L!

(L − i)!(i − 1)!i!γe−

xγ

(iγT − (i − 1)x

γ

)i−1

(5.26)

+L−i∑j=1

L!(−1)j−i+1

(L − i − j)!i!j!γ

(i

j

)i−1

e−( i + j )x

i γ

×[1 − e−

j ( ( i−1 )x −i γ T )i γ

i−2∑k=0

1k!

(j((i − 1)x − iγT )

iγ

)k]

,

γT ≤ x <i

i − 1γT .

Note that the PDF of the sum of the first i order statistics pΓ i(·) is given by [13,

eq. (9.325)]

pΓ i(x) =

L!(L − i)!i!

e−xγ

[xi−1

γi(i − 1)!

+1γ

L−i∑l=1

(−1)i+ l−1 (L − i)!(L − i − l)!l!

(i

l

)i−1

×(

e−l xi γ −

i−2∑m=0

1m!

(− lx

iγ

)m)]

. (5.27)

Therefore, the PDF of the combined SNR γc with MS-GSC over i.i.d. Rayleighfading paths can be shown, by specializing (5.23) with (5.26) and (5.27), to be


given in the following closed-form expression

pγc(x) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Lγ

(1 − e−

xγ

)L−1e−

xγ

+Lc∑i=2

L!

(L − i)!(i − 1)!i!γe−

xγ

(iγT − (i − 1)x

γ

)i−1

+L−i∑j=1

L!(−1)j−i+1

(L − i − j)!i!j!γ

(i

j

)i−1

exp(− (i + j)x

iγ

)

×[1 − e−

j ( ( i−1 )x −i γ T )i γ

i−2∑k=0

1k!

(j((i − 1)x − iγT )

iγ

)k]

× (U(x − γT ) − U(x − ii−1 γT )), x ≥ γT ;

L !(L−Lc )!Lc ! e

− xγ

[xL c −1

γ L c (Lc −1)!

+ 1γ

L−Lc∑l=1

(−1)Lc + l−1 (L − Lc)!(L − Lc − l)!l!

(Lc

l

)Lc −1

×(

e−l x

L c γ −Lc −2∑m=0

1m!

(− lx

Lcγ

)m)]

, 0 ≤ x < γT .

(5.28)

Furthermore, after substituting the closed-form expression for the Rayleigh PDFof γc given in (5.28) into (5.24) and carrying out the integration with the helpof the definition of incomplete Gamma function [14, sec. 8.35], we obtain theclosed-form expression for the MGF of γc as

Mγc(s) = L

L−1∑j=0

(L − 1

j

)(−1)j e

−(

1 + jγ

−s)

γT

1 + j − γs

+l∑

i=2

(L

i

)[ i−1∑m=0

(1 − i)m

(i − 1 − m)!

(iγT

γ

)i−1−m

Gi,0(s)

+L−i∑j=1

(L − i

j

)(−1)j−i+1

(i

j

)i−1(

e−(

1 + j / i

γ−s

)γT − e

−(

1 + j / i

γ−s

)i

i−1 γT

1 + j/i − γs

−i−2∑k=0

k∑m=0

(i−1

i j)m

(−j γT

γ

)k−m

Gi,j (s)

(k − m)!

)]+(

L

l

)[Fl(s) +

L−l∑i=1

(L − l

i

)

×(−1)l+i−1(

l

i

)l−1⎛⎝1 − e

−(

1 + i / l

γ−s

)γT

1 + i/l − γs−

l−2∑m=0

(− i

l

)m

Fm+1(s)

⎞⎠]

,

(5.29)

where

Gi,j (s) = ejγ Tγ

Γ(m + 1,(

1+jγ − s

)γT ) − Γ(m + 1,

(1+jγ − s

) (i

i−1

)γT )

m!(1 + j − γs)m+1 ,

(5.30)


0 5 10 1510

−6

10−5

10−4

10−3

10−2

10−1

100

Outage Threshold, γth

, dB

Out

age

Pro

babi

lity,

Pou

t

Lc = 1

Lc = 2

Lc = 3

Lc = 4

8/4 − GSCSimulation

Figure 5.6 Outage probability, or equivalently CDF of the combined SNR, withMS-GSC over i.i.d. Rayleigh fading paths (L = 8, γ = 3 dB, and γT = 6 dB) [4].c©2006 IEEE

Fl(s) =Γ(l) − Γ(l, (1/γ − s) γT )

(l − 1)!(1 − γs)l, (5.31)

and Γ(·) and Γ(·, ·) are the complete and the incomplete gamma functions, respec-tively. It can be shown that when γT = ∞, (5.29) reduces to the MGF of theoriginal GSC over i.i.d. Rayleigh fading paths, as given in [13, eq. (9.430)]. Theseclosed-form results can be readily applied to the error rate analysis of MS-GSCover Rayleigh fading environment.

Figure 5.6 plots the outage probability of MS-GSC over eight i.i.d. Rayleighfading paths for different number of MRC branches. The outage probability ofconventional 8/4-GSC is also plotted for comparison. As we can see, the outageperformance improves as the number of MRC branches increases, especially whenγth is smaller than γT . In fact, comparing the outage performance of an MS-GSC of Lc = 4 case with that of conventional GSC, we can see that they havethe same outage performance if γth ≤ γT . Intuitively, this is correct because thecombined SNR with MS-GSC can only be smaller than γT if the combined SNRof the corresponding conventional MRC is smaller than γT . When γth > γT , theoutage performance of MS-GSC degrades rapidly in comparison with the originalGSC scheme because the MS-GSC combiner only tries to increase the combinedSNR above γT . We can conclude from this figure that the SNR threshold γT for


0 2 4 6 8 10 12 14 16 18 2010

−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

Average SNR per Path, γ, dB

Ave

rage

Bit

Err

or P

roba

bilit

y, P

b E

Lc = 1

Lc = 2

Lc = 3

Lc = 4

6/4 − GSC

Figure 5.7 Average bit error rate of BPSK with MS-GSC over i.i.d. Rayleigh fadingpaths (L = 6 and γT = 10 dB) [4]. c© 2006 IEEE.

MS-GSC should be chosen to be larger than or equal to the outage thresholdγth in practice. As a validation of the analytical result, simulation results forthe outage probability of MS-GSC for Lc = 3 case are also presented. Note thatthese simulation results match perfectly our analytical results.

Figure 5.7 shows the average error rate of BPSK with MS-GSC over six i.i.d.Rayleigh fading paths for different number of MRC branches. The error rateof the conventional 6/4-GSC is also plotted for reference. As we can see, theerror performance improves as Lc increases, especially in the low average SNRregion. Comparing the error rate of the MS-GSC for Lc = 4 case and that ofconventional GSC, we observe that they have the same error performance inthe low average SNR region, but the performance of MS-GSC is not as goodin the medium to large SNR region. This BER behavior of MS-GSC can beexplained by the following. When the average SNR is small compared to theoutput threshold, the receiver will always combine the Lc best paths becausecombining fewer paths will not result in a combined SNR above the requiredoutput threshold. As a result, the combiner is actually working as a conventionalGSC. On the other hand, when the average SNR is large compared to the outputthreshold, the receiver will most likely combine the best path because it usuallysuffices to increase the combined SNR above the output threshold. As a result,the combiner is actually working as a selection combiner (SC).


−5 0 5 10 1510

−4

10−3

10−2

Output Threshold, γT, dB

Ave

rage

Bit

Err

or P

roba

bilit

y, P

b E

6 − SCApproximationExact Result6/4 − GSC

Figure 5.8 Average bit error rate of BPSK with MS-GSC over i.i.d. Rayleigh fadingpaths as function of output threshold γT (L = 6, Lc = 4, and γ = 3 dB) [4]. c© 2006IEEE.

This interpretation is further verified in Fig. 5.8 where we plot the average errorrate of BPSK with 6/4-MS-GSC as a function of the output threshold γT . As wecan see, the error rate of MS-GSC decreases from the error rate of six-branch SCto that of conventional 6/4-GSC as the output threshold increases from –5 dBto 15 dB. In Fig. 5.8, we also compare the approximate error rate result forMS-GSC in [3] with the exact result. It is observed that the approximation in [3]gives considerable larger error rate values.

5.3.3 Complexity savings

Note that unlike the traditional GSC scheme, where the receiver always needsto find the Lc best paths, MS-GSC needs only to find enough best paths suchthat the combined SNR is above the threshold γT . This feature of MS-GSCconstitutes the major complexity savings over conventional GSC. However, theMS-GSC receiver needs to introduce a threshold check at the combiner output.As the threshold checks are much easier to perform than path SNR ordering, MS-GSC requires less computing power than the original GSC for path selection. Inaddition, the receiver with MS-GSC needs to keep fewer MRC branches active onaverage during the data reception stage. As such, the valuable processing powerof the receiver is saved with MS-GSC. As a quantification of the power savings


with MS-GSC, we derive the average number of active MRC branches with MS-GSC in this section. A distribution of the number of active MRC branches withMS-GSC has been obtained in [3]. Based on the new results on order statisticspresented in the previous chapter, we offer a more compact solution for thedistribution of the number of active MRC branches and its average.

The number of active MRC branches with MS-GSC, denoted by n, takes valuesfrom 1 to Lc . From the mode of operation of MS-GSC, we have that n = 1 ifand only if γ1:L > γT , n = i, 2 ≤ i ≤ Lc − 1 if and only if

∑i−1j=1 γj :L < γT and∑i

j=1 γj :L ≥ γT , and n = Lc if∑Lc −1

j=1 γj :L < γT . Therefore, the probability massfunction (PMF) of n can be written as

Pr[n = i] =

⎧⎪⎪⎨⎪⎪⎩

Pr[γ1:L > γT ], i = 1;

Pr[∑i−1

j=1 γj :L < γT &∑i

j=1 γj :L ≥ γT ], 2 ≤ i ≤ Lc − 1;

Pr[∑Lc −1

j=1 γj :L < γT ], i = Lc.

(5.32)

Note that the probability Pr[∑i−1

j=1 γj :L < γT &∑i

j=1 γj :L ≥ γT ] is equivalent toPr[Γi−1 < γT & Γi−1 + γi:L ≥ γT ]. Therefore, this probability can be calculatedby integrating the joint PDF of Γi−1 and γi:L over the shaded region shown inFig. 5.5(a). Since the region of integration is exactly the difference of the regionsΓi−1 < γT and Γi−1 + γi:L < γT , we can instead calculate this probability byusing the CDFs of Γi−1 and Γi−1 + γi:L as the following

Pr[Γi−1 < γT & Γi−1 + γi:L ≥ γT ] = FΓ i−1 (γT ) − FΓ i−1 +γi :L (γT )

= FΓ i−1 (γT ) − FΓ i(γT ), (5.33)

where FΓ i(·) = F∑ i

j = 1 γ( j )(·), i = 1, 2, · · · , Lc − 1, is the CDF of the sum of the

first i order statistics of L random variables. Consequently, the PMF of n canbe shown to be given by

Pr[N = i] =

⎧⎪⎪⎨⎪⎪⎩

1 − FΓ1 (γT ), i = 1;

FΓ i−1 (γT ) − FΓ i(γT ), 2 ≤ i ≤ Lc − 1;

FΓL c −1 (γT ), i = Lc.

(5.34)

The average number of combined paths with MS-GSC can then be easily obtainedas

NA =Lc∑i=1

iPr[N = i]

= 1 +Lc −1∑i=1

FΓ i(γT ). (5.35)

Note that the closed-form expression of FΓ i(·) for i.i.d. Rayleigh fading was given

in (5.18). As such, both the PMF of N and its average N can be calculated withclosed-form results.

In Figs. 5.9 and 5.10, we study the performance versus power consump-tion trade-off involved in MS-GSC. Figure 5.9 shows the average error rate of

Output-threshold GSC 115

−5 0 5 10 1510

−4

10−3

10−2

10−1


Ave

rage

Bit

Err

or P

roba

bilit

y, P

b E

4 − SCL = 4L = 5L = 6L = 77/4 − GSC

Figure 5.9 Average bit error rate of BPSK with MS-GSC over i.i.d. Rayleigh fadingpaths as function of output threshold γT (Lc = 4 and γ = 3 dB) [4]. c© 2006 IEEE.

MS-GSC as function of the output threshold with four MRC branches and dif-ferent number of diversity paths. As expected, the performance of MS-GSCimproves as the number of available diversity paths L increases. In addition,the error rate of MS-GSC decrease from that of the L-branch SC to that of theL/4-GSC scheme as γT increases. Figure 5.10 shows the corresponding averagenumber of active MRC branches with MS-GSC as the function of the outputthreshold. As we can see, the average number of active MRC branches decreasesas the number of available diversity paths increase, but increases as the outputthreshold increases since it becomes more difficult to raise the combined SNRabove the output threshold. Considering both Figs. 5.9 and 5.10, we can see thatif the output threshold γT is set to 10 dB, MS-GSC has the same error rateperformance as the conventional GSC while requiring fewer active branches onaverage. In particular, for the L = 6 and Lc = 4 case, MS-GSC needs on average3.3 active MRC branches, which translates to a power saving of 17.5% comparedto conventional GSC.

5.4 Output-threshold GSC

The OT-MRC scheme saves the receiver processing power by estimating andcombining a minimum number of paths. It still requires the same hardwarecomplexity as conventional MRC. On the other hand, the MS-GSC scheme can


−5 0 5 10 151

1.5

2

2.5

3

3.5

4


Ave

rage

Num

ber

of A

ctiv

e M

RC

Bra

nche

s, N

L = 4L = 5L = 6L = 7

Figure 5.10 Average number of active MRC branches with MS-GSC as a function ofoutput threshold γT (Lc = 4 and γ = 3 dB) [4]. c© 2006 IEEE.

reduce the receiver hardware complexity with the GSC principle and the num-ber of combining paths through minimum best selection. However, the MS-GSCreceiver always needs to estimate all L available diversity paths. In this section,we present another power-saving variant of GSC [5], which essentially combinesthe desired properties of both OT-MRC and MS-GSC schemes. The resultingoutput-threshold GSC (OT-GSC) scheme is shown to provide considerable pro-cessing power saving compared with conventional GSC while maintaining thesame diversity benefit.

The OT-GSC scheme estimates and combines additional diversity paths onlyif necessary. When the number of estimated paths is larger than the numberof available MRC branches, the receiver applies the best selection to reduce thehardware complexity. The flow chart in Fig. 5.11 illustrates the mode of operationof OT-GSC. In particular, the operation of OT-GSC combiner can be dividedinto two stages: an MRC stage followed by a GSC stage.

MRC stage. During the MRC stage, the combiner acts as an Lc -branch OT-MRC combiner. In particular, the combiner tries to improve the combined SNRΓ above the threshold γT by gradually changing its configuration from a single-branch MRC to an Lc -branch MRC. If all Lc branches of the MRC combinerhave been activated but the combined SNR is still below γT , then the receiveroperation enters the GSC stage.


Startγc = 0, l = 0

l = l + 1

Estimate γl

Update combined SNRγc = γc + γl

γc > γTYesNo

l = Lc

No Yes

Rank SNR of combined pathsγi:Lc

, i = 1, 2, · · · , Lc

l = l + 1

Estimate γl

γl > γLc:Lc

Update GSC output SNRγc = γc − γLc:Lc

+ γl

l = LNo

γc > γTNo

No

Yes

Yes

Yes

Stop

MRC

GSC

Figure 5.11 Flow chart for the OT-GSC mode of operation.


GSC stage. During the GSC stage, the combiner tries to improve the combinedSNR Γ above the SNR threshold γT by gradually changing its configuration froman Lc + 1/Lc GSC to an L/Lc GSC, where l/Lc GSC denotes a combiner thatcombines in an MRC fashion the Lc strongest paths out of the l available paths.In particular, knowing that the combined SNR after MRC combining of the firstLc diversity paths is still below γT , the combiner checks the SNR of Lc + 1/Lc

GSC output. As such, the combiner first estimates another diversity path andthen checks if its SNR is greater than the minimum SNR of the Lc used paths.If it is greater, the newly estimated path is used as an MRC input in place ofthe path which has minimum SNR. If the resulting combined SNR is above γT ,the combiner stops the branch update. On the other hand, if either the SNRof the newly estimated path is less than the minimum SNR of the currentlyused paths or if the resulting combined SNR is still below the threshold afterpath replacement, then the combiner looks into the combined SNR of Lc + 2/Lc

GSC by estimating another diversity path. This process is continued until eitherthe combined SNR Γ is above γT or all L available diversity paths have beenchecked. Note again that only in the worst case will the receiver use an L/Lc

GSC combiner for reception.From the operation of OT-GSC, we can see that unlike conventional GSC,

where all L diversity paths need to be estimated and ranked in every guardperiod and all Lc MRC branches need to be active all the time, the receiver withthe OT-GSC scheme uses the available resources adaptively. In particular, bothpath estimation and MRC branch activation are performed only if necessary.These features of OT-GSC constitute the major power saving of this new GSCscheme over conventional GSC.


In this section, we accurately quantify the power savings of OT-GSC by derivingclosed-form expressions for both average number of path estimations and averagenumber of active MRC branches.

Average number of path estimationsBased on the mode of operation of the OT-GSC scheme, the number of pathestimations performed during a guard period varies from 1 to L. Therefore,conditional on the number of path estimations in a guard period, we can write theaverage number of path estimations, denoted by NE , in the following summationform

NE =L∑

l=1

πl l, (5.36)

where πl is the probability that exactly l diversity paths are estimated during aguard period and it is equal to the probability that either an l-branch MRC forl ≤ Lc or an l/Lc -GSC for l > Lc is used for the subsequent data reception. As


a single branch receiver is used if the SNR of the first diversity path is above γT

while an l-branch MRC receiver is used if the combined SNR of an l − 1-branchMRC is below γT whereas the combined SNR of an l-branch MRC is above γT ,we have under the i.i.d. fading paths assumption:

πl =

1 − Fγ (γT ), l = 1;

F(l−1)−M RCγc (γT ) − F l−M RC

γc(γT ), 1 < l ≤ Lc,

(5.37)

where F l−M RCγc

(·) is the CDF of the combined SNR with l-branch MRC combiner,the closed-form expression of which for three popular fading channel model ofinterest are given in Table 5.1. On the other hand, for Lc < l ≤ L − 1, an l/Lc -branch GSC is used if the combined SNR of an (l − 1)/Lc -branch GSC is belowγT whereas the combined SNR of an l/Lc -branch GSC is above γT . In addition,if the combined SNR of an (L − 1)/Lc -branch GSC is below γT , then an L/Lc -branch GSC is used. Based on these observations and under the i.i.d. fading pathassumption, it can be shown that

πl =

F

(l−1)/Lc −GSCγc (γT ) − F

l/Lc −GSCγc (γT ), Lc < l < L;

F(L−1)/Lc −GSCγc (γT ), l = L,

(5.38)

where Fl/Lc −GSCγc (·) is the CDF of the combined SNR with l/Lc branch GSC,

i.e. the CDF of the sum of the Lc largest path SNR out of l estimated ones. Sub-stituting (5.37) and (5.38) into (5.36), it can be shown after some manipulationsand simplifications, that the average number of path estimations is given by

NE = 1 +Lc∑l=1

F l−M RCγc

(γT ) +L−1∑

l=Lc +1

F l/Lc −GSCγc

(γT ). (5.39)

For the i.i.d. Rayleigh fading case, substituting F l−M RCγc

(·) from Table 5.1 (for

the Rayleigh case) and Fl/Lc −GSCγc (·) as given in (5.18) into (5.39) leads to the

following closed-form expression for NE in i.i.d. Rayleigh fading

NE = 1 +Lc∑l=1

[1 − e−

γ Tγ

l−1∑i=0

1i!

(γT

γ

)i]

+L−1∑

l=Lc +1

(l

Lc

)1 − e−

γ Tγ

Lc −1∑i=0

1i!

(γT

γ

)i

+l−Lc∑i=1

(−1)Lc +i−1(

l − Lc

i

)(Lc

i

)Lc −1

×[(

1 +i

Lc

)−1 [1 − e(1+ i

L c ) γ Tγ

]

−Lc −2∑m=0

(− i

Lc

)m(

1 − e−γ Tγ

m∑k=0

1k!

(γT

γ

)k)]

. (5.40)


−15 −10 −5 0 5 10 151

1.5

2

2.5

3

3.5

4

4.5

5

Normalized Switching Thresholds, γT/ γ−, dB

Ave

rage

Num

ber

of P

ath

Est

imat

es, N

E

Lc=1

Lc=2

Lc=3

Lc=4

Lc=5

Figure 5.12 Average number of path estimations with OT-GSC as a function of thenormalized SNR threshold, γT /γ (L = 5) [5]. c© 2005 IEEE.

In Fig. 5.12, we show the average number of path estimations with OT-GSC ina guard period (given in (5.40)) as a function of the normalized SNR threshold,γT /γ. Note that the average number of path estimations NE increases from 1to L as the threshold increases, just as expected, since it becomes more difficultto raise the combined SNR above the threshold. Note also that NE decreases asthe number of available MRC branches Lc increases, but the difference becomesnegligible after Lc becomes greater than 2.

Average number of active MRC branchesWith OT-GSC, if the combined SNR exceeds the SNR threshold during the MRCstage, fewer than Lc MRC branches might be active during the subsequent databurst reception. As a result, a considerable amount of receiver processing powerwill be saved. Conditional on the number of active MRC branches during thedata burst reception, we can write the average number of active MRC branches,denoted by NA , in the following summation form

NA =Lc∑l=1

πl l, (5.41)


−15 −10 −5 0 5 10 151

1.5

2

2.5

3

3.5

4

4.5

5

Normalized Switching Thresholds, γT/ γ−, dB

Ave

rage

Num

ber

of A

ctiv

e M

RC

Bra

nche

s, N

A

Lc=1

Lc=2

Lc=3

Lc=4

Lc=5

Figure 5.13 Average number of active MRC branches with OT-GSC as a function ofthe normalized SNR threshold, γT /γ (L = 5) [3]. c© 2004 IEEE.

where πl is the probability that exactly l MRC branches are active during adata burst reception. For l < Lc , this is equal to the probability that an l-branchMRC is used during data reception. Based on the results of the previous sub-section and noting that once the operation of the combiner reaches the GSCstage, all Lc MRC branches need to be active, it can be shown that NA is givenby

NA = 1 +Lc −1∑l=1

F l−M RCγc

(γT ). (5.42)

For the i.i.d. Rayleigh fading case, after substituting the Rayleigh case ofF l−M RC

γc(·) from Table 5.1 in (5.42), we have the closed-form expression for NA

given by

NA = 1 +Lc −1∑l=1

[1 − e−

γ Tγ

l−1∑i=0

1i!

(γT

γ

)i]

. (5.43)

In Fig. 5.13, we show the average number of active MRC branches duringdata reception (given in (5.43)) as a function of the normalized SNR thresh-old, γT /γ. Note that the average number of active MRC branches NA increases


from 1 to Lc as the normalized SNR threshold increases, just as expected byintuition.


We now derive the statistics of the combined SNR with OT-GSC, which willfacilitate its performance analysis over fading channels. We again focus on thei.i.d. Rayleigh fading environment. Note that the events that Le , Le = 1, 2, ..., L,diversity branches are examined are mutually exclusive. Applying the total prob-ability theorem, we can write the CDF of the combined SNR γc in a summationform as

Fγc(x) = Pr[γc < x] =

L∑k=1

Pr[γc < x,Le = k], (5.44)

where Pr[γc < x,Le = k] is the joint probability of the events that γc < x andthe receiver has examined exactly k paths. Based on the mode of operation ofOT-GSC, we note that when Le ≤ Lc , the OT-GSC operation terminates at theMRC stage. Moreover, it can be observed that Le paths are estimated, Le ≤ Lc ,if and only if the MRC-combined SNR of the first Le − 1 branches is less thanthe output-threshold γT but the combined SNR of the first Le branches is aboveγT . Therefore, Pr[γc < x,Le = k], k ≤ Lc , can be shown to be given by

Pr[γc < x,Le = k] =

Pr[γT ≤ γc = γ1 < x], k = 1

Pr[γT ≤ γc =∑k

j=1 γj < x,∑k−1

j=1 γj < γT ], 2 ≤ k ≤ Lc.

(5.45)

When Le > Lc , the OT-GSC operation terminates at the GSC stage. In particu-lar, for Lc + 1 ≤ Le ≤ L − 1, an Le/Lc -GSC is used when the combined SNR ofthe Le/Lc -GSC is above the output-threshold γT whereas the output SNR of the(Le − 1)/Lc -GSC is less than γT . Finally, Le = L if and only if the combined SNRof the (L − 1)/Lc -GSC is below the output threshold. Mathematically speaking,we have, for k > Lc

Pr[γc < x,Le = k] =Pr[γT ≤ γc =

∑Lc

j=1 γj :k < x,∑Lc

j=1 γj :k−1 < γT ], k ≤ L − 1

Pr[γc =∑Lc

j=1 γj :L < x,∑Lc

j=1 γj :L−1 < γT ], k = L,

(5.46)

where γj :k denotes the jth largest SNR among k estimated path SNRs. Substi-tuting (5.45) and (5.46) into (5.44), we can rewrite the CDF of the combined


SNR with OT-GSC as

Fγc(x) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Pr[∑Lc

j=1 γj :L < γT ]

+ Pr[γT ≤ γ1 < x] +Lc∑

k=2

Pr[γT ≤k∑

j=1

γj < x,

k−1∑j=1

γj < γT ]

︸︷︷︸F1 (x)

+L∑

k=Lc +1

Pr[γT ≤Lc∑j=1

γj :k < x,

Lc∑j=1

γj :k−1 < γT ]

︸︷︷︸F2 (x)

, x ≥ γT

Pr[∑Lc

j=1 γj :L < x], x < γT .

(5.47)Note that the probability Pr[

∑Lc

j=1 γj :L < x] is equal to the CDF of the combined

SNR of L/Lc -GSC evaluated at x, i.e. Fl/Lc −GSCγc (x). For i.i.d. Rayleigh fading

channels, the closed-form expression of Fl/Lc −GSCγc (x) is given in (5.18). The term

F1(x) defined in (5.47) can be rewritten in terms of the single-branch SNR CDFFγ (·) and the PDF of an l-branch MRC output SNR f l−M RC

γc(·) as

F1(x) = Fγ (x) − Fγ (γT ) +Lc∑

k=2

∫ γT

0fk−M RC

γc(y)

(Fγ (x − y) − Fγ (γT − y)

)dy

(5.48)For Rayleigh fading, after proper substitution from Table 5.1, carrying out theintegrations and some manipulation, F1(x) can be shown to be given by

F1(x) =Lc∑

k=1

γTk−1

(k − 1)!γk−1

[e−

γ Tγ − e−

xγ]. (5.49)

To derive F2(x), we first note that∑Lc

j=1 γj :k−1 can be written as∑Lc

j=1 γj :k−1 =ΓLc −1/k−1 + γLc :k−1 , where γLc :k−1 is the Lcth largest SNR among k − 1 esti-mated path SNRs and ΓLc −1/k−1 =

∑Lc −1j=1 γj :k−1 , i.e. ΓLc −1/k−1 is the sum of

the largest Lc − 1 branch SNRs. Based on the mode of operation of OT-GSC,the weakest path is replaced at the GSC stage if and only if the instantaneousSNR of the newly examined branch γk is larger than γLc :k−1 . Also note that inthis case ΓLc −1/k−1 remains unchanged, we can write the summand of F2(x) as

Pr[γT ≤Lc∑j=1

γj :k < x,

Lc∑j=1

γj :k−1 < γT ]

= Pr[γT ≤ ΓLc −1/k−1 + γk < x,ΓLc −1/k−1 + γLc :k−1 < γT ].

(5.50)

Because all branch SNRs are i.i.d. random variables, γk is independent of bothΓLc −1/k−1 and γLc :k−1 . As such, we can compute the probability in (5.50) withthe joint PDF of ΓLc −1/k−1 and γLc :k−1 , denoted by pγL c :k −1 ,ΓL c −1 / k −1 (y, z), and


the CDF for γk , Fγ (·). In particular, F2(x) can be written as

F2(x) =L∑

k=Lc +1

∫ γ TL c

0

∫ γT −y

(Lc −1)ypγL c :k −1 ,ΓL c −1 :k −1 (y, z) (5.51)

×(Fγ (x − z) − Fγ (γT − z))dzdy.

For i.i.d. Rayleigh fading channels, it can be shown in that the joint PDFpγL c :L ,ΓL c −1 / L

(y, z) was given in (5.25). After substituting (5.25) into (5.51), car-rying out the integration and appropriate simplifications, we obtain the closed-form expression of F2(x) as

F2(x) =[e−γ Tγ − e−

xγ ]

L∑k=Lc +1

k−1−Lc∑j=0

A(k − 1, j)

×[B(Lc, j) + (− γLc

j + 1)Lc e−

( j + 1 )γ Tγ L c

],

(5.52)

where

A(k, j) =(−1)j k!

Lc(k − Lc − j)!(Lc − 1)!j!γLc, (5.53)

and

B(Lc, j) =Lc −1∑i=0

(−1)iγTLc −1−i

(Lc − 1 − i)!( j+1γLc

)i+1. (5.54)

Finally, after substituting (5.49), (5.52) and (5.18) into (5.47), we arrive at theclosed-form expression of the CDF of OT-GSC output SNR, or equivalently, theoutage probability of OT-GSC, over i.i.d. Rayleigh fading channels.

Starting from the CDF of γc with OT-GSC, we can routinely obtain its PDFand MGF. In particular, after differentiation and some manipulations, the PDFof γc with OT-GSC for i.i.d. Rayleigh fading channels is given by

pγc(x) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

1γ

e−xγ

Lc∑k=1

γTk−1

(k − 1)!γk−1 +L∑

k=Lc +1

k−1−Lc∑j=0

A(k − 1, j)

×[B(Lc, j) + (− γLc

j + 1)Lc e−

( j + 1 )γ Tγ L c

],

x ≥ γT

(L

Lc

)[xLc −1e−x/γ

γLc (Lc − 1)!+

1γ

L−Lc∑l=1

(−1)Lc + l−1(

L − Lc

l

)

×(

Lc

l

)Lc −1

e−x/γ

(e−lx/Lc γ −

Lc −2∑m=0

1m!

(−lx

Lcγ

)m)]

,

x < γT

(5.55)


0 5 10 1510

9

108

107

106

105

104

103

102

Average SNR per Path, γ,dB

Ave

rage

Err

or P

roba

bilit

y

OT−GSC AnalysisMS−GSCGSCOT−GSC Simulation

Figure 5.14 Average error rate of BPSK with OT-GSC, MS-GSC and GSC over L = 5i.i.d. Rayleigh fading channels as function of the average SNR per branch(Lc = 3 andγT = 8 dB) [15]. c© 2007 IEEE.

where A(k, j) and B(Lc, j) are defined in (5.53) and (5.54), respectively. It followsthat the MGF of γc can be obtained as

Mγc(t) =

(L

Lc

)1 − Γ(Lc, (1 − tγ)γT /γ)

(1 − tγ)Lc+

L−Lc∑l=1

(−1)Lc + l−1(

L − Lc

l

)

×(

Lc

l

)Lc −1[I(l, t) −

Lc −2∑m=0

(−l

Lc

)m × (1 − Γ(m + 1, (1 − tγ)γT /γ))(1 − tγ)m+1

]

+e−

( 1−γ t )γ Tγ

1 − γt

Lc∑k=1

γTk−1

(k − 1)!γk−1 +L∑

k=Lc +1

k−1−Lc∑j=0

A(k − 1, j)

×[B(Lc, j) + (− γLc

j + 1)Lc e−

( j + 1 )γ Tγ L c

],

(5.56)

where

I(l, t) =1

γt − Lc + lLc

[exp

(γT (tLcγ − Lc − l)

Lcγ

)− 1

]. (5.57)

These closed-form results can be readily applied to the performance analysis ofOT-GSC over fading channels.

Figure 5.14 compares the average error rate of BPSK with OT-GSC, MS-GSCand GSC as a function of average SNR per branch with the output-threshold


−8 −3 2 7 1210

−4

10−3

10−2

10−1

Normalized Output Threshold, γT/γ,dB

Ave

rage

Err

or P

roba

bilit

y

OT−GSCMS−GSCGSC6−branch SC1−branch MRC

Figure 5.15 Average error rate of BPSK with OT-GSC, MS-GSC and GSC over L = 6i.i.d. Rayleigh fading channels as function of the normalized output-threshold γT /γ(Lc = 3 and γ = 3 dB) [15]. c© 2007 IEEE.

γT for both OT-GSC and MS-GSC set to be 8 dB. As a verification of ouranalytic result, simulation results for the average error rate of OT-GSC are alsopresented. Note that these simulation results match the analytic results perfectly.We can also see, when the channel condition is poor, i.e. γ is small comparedto γT , these schemes have the same error performance. This behavior can beexplained as follows. When the average branch SNR is small compared to theoutput threshold, the receiver with OT-GSC and MS-GSC will usually estimateall available branches and combine the best Lc ones because applying MRC/GSCto a subset of available branches will not be able to raise the combined SNR overγT . Therefore, both MS-GSC and OT-GSC are actually working in the sameway as a conventional GSC. As γ becomes larger, i.e. the channel condition isimproving, OT-GSC has a higher error probability than both GSC and MS-GSC. This can be explained by noting that in this case, it is more likely that theinstantaneous SNR of the first branch is greater than γT and, as such, OT-GSCreduces to a single-branch receiver while MS-GSC tends to work as a selectioncombiner.

In Fig. 5.15, we plot the average error rate of BPSK with OT-GSC, MS-GSCand GSC schemes as function of the normalized output threshold γT /γ. As wecan see, the error rate of OT-GSC decreases from that of a single-branch MRC to

Adaptive transmit diversity 127

−10 −8 −6 −4 −2 0 2 4 6 8 10 120

1

2

3

4

5

6

7

Ave

rage

Pat

h E

stim

atio

n

−10 −8 −6 −4 −2 0 2 4 6 8 10 120

1

2

3

4

Normalized Output Threshold, γT/γ, dB

Ave

rage

Num

ber

of C

ombi

ned

Pat

hs

GSCOT−GSCMS−GSC

GSCMS−GSCOT−GSC

Figure 5.16 Average number of path estimations and combined paths with OT-GSC,MS-GSC and GSC over L = 6 i.i.d. Rayleigh fading channels as function ofnormalized output-threshold γT /γ (Lc = 3 and γ = 3 dB) [15]. c© 2007 IEEE.

that of a conventional 6/3 GSC when the normalized output threshold increasesfrom –12 dB to 12 dB, whereas that of MS-GSC degrades from a 6-branch SC.To better illustrate the new trade-off of complexity versus performance offeredby OT-GSC, we present in Fig. 5.16 the average number of path estimations andcombined paths with OT-GSC, MS-GSC and GSC schemes based on the analyt-ical results given in previous sections. It can be seen that while both GSC andMS-GSC always require L path estimations, OT-GSC requires much fewer pathestimations, especially when the normalized SNR is small. Considering Figs. 5.15and 5.16 together, we can observe that the OT-GSC scheme can offer the sameperformance as GSC with a fewer number of combined paths and path estima-tions. In comparison with MS-GSC, OT-GSC still saves certain amount of pathestimations while combining more paths to achieve the same error performance.

5.5 Adaptive transmit diversity

The idea of adaptive diversity combining can also apply to transmit diver-sity scenarios. The main advantage of transmit diversity is that diversity gainoffered by multiple antenna at the base station can be explored for the downlink


transmission even with single-antenna receivers. In general, transmit diversitysystems can be classified into open-loop and closed-loop depending on whetheror not the channel state information feedback is required. The most popularopen-loop schemes are those based on orthogonal space-time block code (STBC)[16–20], which can achieve full diversity gain with simple linear processing at thereceiver, and as such receive considerable research and industrial interest. On theother hand, OSTBC will suffer a rate loss when the number of transmit anten-nae is greater. Moreover, the more the transmit antennae are used, the largerthe SNR loss occurring due to the power spreading over multiple antennae.

A viable solution to these issues is antenna selection at the transmitter side[21–26]. It has been shown that while requiring a certain amount of closed-loopfeedback, these antenna selection schemes incur much less of a rate/power losscompared to conventional STBC-based transmit diversity systems. Nevertheless,these antenna selection schemes still require the estimation of wireless channelscorresponding to every transmit antenna. The receiver needs to feed back theindex of all selected channels. In this section, we illustrate how the idea of adap-tive combining can be applied to the transmit diversity systems by consideringa new class of closed-loop transmit diversity system [10, 27, 28]. Specifically, asubset of the available transmit antennae is used for data transmission withSTBC. The transmitter will replace the antenna leads to the lowest-path SNRwith an unused transmit antenna if the output SNR of the space-time decoder isbelow a certain threshold. The resulting OSTBC system with transmit antennareplacement offers performance comparable to the corresponding selection-basedtransmit diversity systems with reduced feedback load.

5.5.1 Mode of operation

The system model of transmit diversity with antenna replacement is illustratedin Fig. 5.17. The transmitter is equipped with a total of N + L antennae, whereN antennae are used for orthogonal space-time transmission and L antennaeare used for replacement. The receiver compares the output SNR of the space-time decoder with a fixed SNR threshold, denoted by γT . If the output SNR∑N

i=1 γi is larger than or equal to γT , no antenna replacement is necessary andthe receiver informs the transmitter to keep using the current antennae. Onthe other hand, if

∑Ni=1 γi is smaller than the threshold γT , the receiver asks the

transmitter to replace the L antennae leading to the weakest received signal withthe L remaining antennae. Therefore, the receiver needs to feedback either onebit of information to indicate no need for antenna replacement or 1 + log2(N CL ),where N CL = N !

(N −L)!L ! bits of information inform the transmitter that the indexof transmitting antennae needs to be replaced. It is not difficult to show that theaverage feedback load of the proposed scheme is

NA = 1 · Pr[ N∑

i=1

γi ≥ γT

]+ (1 + log2(N CL )) · Pr

[ N∑i=1

γi < γT

]. (5.58)


Data

1

N+1

.

.

.

Switch Logic

2

Informationsource

Feedback information

Receiver

ChannelEstimator

Comparator

Space-timeEncoder

Control

Threshold

SNR

N

.

.

.

Figure 5.17 System model of transmit diversity systems [10].

For the i.i.d. Rayleigh fading scenario, it can be shown that the average feedbackload specializes to

NA = 1 + log2(N CL )

(1 − e−

γ Tγ

N −1∑i=0

1i!

(γT

γ

)i)

. (5.59)

It can be seen that the proposed system has the maximum feedback load whenthe number of switching antennae L ≈ N/2. In Fig. 5.18, we plot the averagefeedback load of the proposed system as a function of γT for different valuesof L and fixed average SNR, γ = 15 dB, while keeping the total number oftransmit antennas (N + L) fixed to 6. As we can see, for L < N cases, thefeedback load increases from 1 bit to log2(N CL ) + 1 bits when the thresholdγT increases, as expected. When N = L = 3, the systems need only one bitof feedback information to indicate whether all transmit antennae should bereplaced or not. We note that different threshold values will lead to differenttrade-offs of error performance and feedback load. To illustrate this, we alsoplot the average error rate of the proposed system on the same figure. As wecan see, the error performance improves and eventually saturates to a constantvalue as the threshold γT increases. Based on this figure, we can that see ifwe select the threshold properly, the proposed scheme can achieve its best pos-sible error performance with very low feedback load. Note that the proposedscheme also enjoys low hardware complexity in terms of pilot channels and RFchains.


5 10 15 20 25

10−6

10−5

10−4

Switching Threshold, γT [dB]

Bit

Err

or R

ate

5 10 15 20 250.5

1

1.5

2

2.5

3

3.5

4

Ave

rage

Fee

dbac

k Lo

ad, N

A

Relation between BER and Average Feedback load(N+L=6, E[γ]=15 [dB])

3 ant. STBC with 3 SW ant.4 ant. STBC with 2 SW ant.5 ant. STBC with 1 SW ant.

Figure 5.18 Comparison of average feedback load of the proposed systems withN + L = 6 transmit antennae over i.i.d. Rayleigh fading channels.

5.5.2 Statistics of received SNR

We now derive the statistics of the received SNR of the antenna replacementscheme. We first consider the case of L = N . In this case, all transmit antennaewill be replaced when the output SNR

∑Ni=1 γi is smaller than the threshold γT .

The proposed scheme operates in a similar way to the traditional SSC scheme,while the output SNR of each branch is equal to

∑Nj=1 γj . Therefore, we can

write the CDF of the combined SNR as

Fγc(x) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

Pr[γT ≤N∑

j=1

γj < x] + Pr[ N∑

j=1

γj < γT &N∑

j=1

γj < x

], x ≥ γT ,

Pr[ N∑

j=1

γj < γT &N∑

j=1

γj < x

], 0 ≤ x < γT .

(5.60)

Following the similar analytical approach for SSC, we can then obtain the closed-form expression of the PDF of γc under the i.i.d. Rayleigh fading assumption


as

pγc(x) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

(2 − e−

γ Tγ

L−1∑k=0

1k!

(γT

γ

)k)xL−1

(L − 1)!γLe−

xγ , x ≥ γT ,

(1 − e−

γ Tγ

L−1∑k=0

1k!

(γT

γ

)k)xL−1

(L − 1)!γLe−

xγ , 0 ≤ x < γT .

(5.61)

We now consider the case 1 ≤ L < N . Let γi:N denote the ith largest SNRwith γ1:N > γ2:N > ... > γN :N . Based on the mode of operation of the proposedscheme, the combined SNR γc is given by

γc =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

N∑i=1

γi,N∑

i=1

γi ≥ γT ,

N −L∑i=1

γi:N +L∑

j=1

γj ,

N∑i=1

γi < γT .

(5.62)

Let Γj denote the sum of the largest j SNRs among the N ones, i.e. Γj =∑ji=1 γi:N . We can write the CDF of γc as

Fγc(x) =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

Pr[γT ≤ ΓN < x]

+ Pr[ΓN < γT & ΓN −L +

∑Lj=1 γj < x

], x ≥ γT ,

Pr[ΓN < γT & ΓN −L +

L∑j=1

γj < x

], 0 ≤ x < γT .

(5.63)

Applying the Bayesian rule and the i.i.d. fading channel assumption, we canrewrite the joint probability in (5.63) as

Pr[ΓN < γT & ΓN −L +

L∑j=1

γj < x

]

= Pr[ΓN −L +

L∑j=1

γj < x

∣∣∣∣ΓN < γT

]Pr[ΓN < γT ]

= Pr[γs < x]Pr[ΓN <γT ], (5.64)

where we define a new random variable γs ≡∑L

j=1 γj + (ΓN −L |ΓN < γT ). ThePDF of the combined SNR can be obtained by taking the derivative of (5.63)and is given by

pγc(x) =

pΓN

(x) + Pr[ ΓN < γT ]pγs(x), x ≥ γT ,

Pr[ ΓN < γT ]pγs(x), 0 ≤ x < γT ,

(5.65)

where pγs(x) denotes the PDF of the γs . Since γs is the sum of the two ran-

dom variables∑L

j=1 γj and (ΓN −L |ΓN < γT ), we can calculate pγs(x) in the


convolution form as

pγs(x) =

∫ x

0p∑L

j = 1 γj(x − z)pΓN −L |ΓN <γT

(z)dz. (5.66)

Noting that the PDF p∑Lj = 1 γj

(x − z) is readily available in closed form, we focuson the following derivation of the conditional PDF pΓN −L |ΓN <γT

(x).We start with the CDF of ΓN −L |ΓN < γT , which can be written as

FΓN −L |ΓN <γT(x) =

1Pr[ΓN < γT ]

Pr[ΓN −L < x,ΓN < γT ], (5.67)

where the joint probability can be calculated as

Pr[ΓN −L < x,ΓN < γT ] = Pr[ΓN −L < x,ΓN −L + γN −L+1:N + zl < γT ] (5.68)

=∫ x

0

∫ γT −y

0

∫ γT −y−γ

0pΓN −L ,γN −L −1 :N ,zl

(y, γ, z)dzdγdy.

In (5.69), zl denotes the sum of the L − 1 smallest SNRs among total N ones,i.e. zl =

∑Ni=N −L+2 γi:N and pΓN −L ,γN −L −1 :N ,zl

(y, γ, z) denotes the joint PDF ofΓN −L , γN −L−1:N and zl , which has been investigated in a previous chapter. Foran i.i.d. Rayleigh fading environment, the three-dimensional PDF is available inclosed form. Consequently, the conditional PDF pΓN −L |ΓN <γT

(x) can be writtenas

pΓN −L |ΓN <γT(x) =

d

dx

Pr[ΓN −L < x,ΓN < γT ]Pr[ΓN < γT ]

=1

Pr[ΓN < γT ]

∫ min[γT −x, xN −L ]

0

∫ min[γT −x−γ ,(L−1)γ ]

0

× pΓN −L ,γN −L −1 :N ,zl(y, γ, z)dzdγ, 0 ≤ x < γT . (5.69)

Finally, after substituting (5.69) into (5.66) and then into (5.65), we have theintegral closed form for the PDF of the combined SNR, γc , given by

pγc(x) =⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

pΓN(x) +

∫ γT

0p∑N + L

l = N + 1 γl(x − y)

∫ min[γT −y , yN −L ]

0

∫ min[γT −y−γ ,(L−1)γ ]

0

pΓN −L ,γN −L −1 :N ,zl(y, γ, z)dzdγdy, x ≥ γT ,

∫ x

0p∑N + L

l = N + 1 γl(x − y)

∫ min[γT −y , yN −L ]

0

∫ min[γT −y−γ ,(L−1)γ ]

0

×pΓN −L ,γN −L −1 :N ,zl(y, γ, z)dzdγdy, 0 ≤ x < γT .

(5.70)

To verify the validity of this new analytical result, we compare the PDF in(5.70) with the Monte Carlo simulation result in Fig. 5.19. It is clear that thereis an excellent match between the analytical result and the Monte Carlo simu-lation. Using the PDF of the combined SNR obtained, we can study the outage


5 10 15 20 25 30 35 40 45 500

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Combined SNR γt

Pro

babi

lity

Den

sity

Fun

ctio

n

TheoreticalSimulation

Figure 5.19 Comparison of the analytical results for the PDF of the combined SNRand the Monte Carlo simulation.

probability and the average bit error rate performance of the proposed schemeover i.i.d. Rayleigh fading.

Figure 5.20 plots the outage probability of the proposed diversity system withthe fixed threshold, γT = 2 dB. We vary the number of antennae for replacementwhile keeping four transmit antennae for orthogonal STBC transmission and oneantenna for reception. Note that the L = 0 case corresponds to traditional open-loop transmit diversity. We note that when L < N , the outage performance ofthe proposed diversity improves with increasing number of switching antennaeL, especially when the outage threshold is smaller than the switching threshold.However, if L = N , the outage performance actually degrades considerably forthe smaller value of the switching threshold.

Figure 5.21 shows the average BER of the proposed diversity system withfixed switching threshold γT = 16 dB. Again, we vary the number of antennaefor replacement from 0 to N = 4. It is interesting to see from this figure thatas the number of antennae used for switching increases, the error performanceof the proposed scheme improves. When L = N , however, the error performancealso degrades significantly, while still slightly better than no antenna for thereplacement case.

In Fig. 5.22, we investigate the average BER performance of different transmitconfigurations while fixing the total number of antennae at the transmitter N + L

to 6. The total transmitting power is also fixed irrespective of the number of


−5 0 5 1010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Outage Threshold, γth

[dB]

Out

age

Pro

babi

lity

L = 0L = 1L = 2L = 3L = N

Figure 5.20 Comparison of the outage probability of N -fold MRC for N = 4 and theproposed systems with L = 1, 2, 3 and N = 4.

0 2 4 6 8 10 12 1410

−7

10−6

10−5

10−4

10−3

10−2

SNR(dB)

Bit

Err

or R

ate

L = 0L = 1L = 2L = 3L = N

Figure 5.21 Average BER of BPSK of the proposed systems with threshold γT = 16(dB) and N -fold MRC over i.i.d. Rayleigh fading channels.

RAKE finger management over the soft handoff region 135

0 2 4 6 8 10 12 1410

−6

10−5

10−4

10−3

10−2

10−1

100

SNR[dB]

Bit

Err

or R

ate

6 ant. STBC5 ant. STBC with 1 SW ant.4 ant. STBC with 2 SW ant.3 ant. STBC with 3 SW ant.

Figure 5.22 Average BER of BPSK of the proposed systems with N + L = 6 transmitantennae and OSTBC over i.i.d. Rayleigh fading channels.

antennae used for orthogonal STBC transmission. As we can see, as L increasesand as long as L < N , the proposed transmit diversity system offers improvedperformance in the case that all six antennas are used for ST transmission. Inparticular, the configuration with four transmitting antennae and two switchingantennae offers about 1 dB gain more than the 6-antenna orthogonal STBCsystem and more than a half dB gain over the configuration with five transmittingantennae and one switching antenna.

5.6 RAKE finger management over the soft handoff region

Another application of the concept of adaptive combining is the figure manage-ment for a RAKE receiver operating in the handover region. RAKE receiversare commonly used in wideband wireless systems, such as WCDMA and UWBsystems, to mitigate the effect of fading. RAKE receivers also facilitate the softhandover (SHO) in the handover region. Specifically, the mobile receiver cancombine resolvable paths from both the serving base station (BS) and the tar-get BS, which can greatly reduce the probability of dropped connections whenthe mobile receiver is at the cell boundary [29, sect. 9.5.1]. On the other hand,serving the mobile user with more than one BS will incur a significant amount


of system overhead, usually termed SHO overhead. Note that user data signalsneed to be sent to and from all involved BSs to enjoy the diversity benefit.

In this section, we apply the concept of adaptive combining to develop severallow-complexity finger management schemes and investigate their performanceand complexity [11, 30]. The main idea is to minimize the SHO overhead byletting the receiver scan the additional resolvable paths from the target BS only ifthe received signal based solely on the serving BS is of unsatisfactory quality. Wewill show that the proposed schemes can reduce the unnecessary path estimationsand the SHO overhead compared to the conventional GSC scheme [31,32] whichalways uses the best resolved paths from both BSs.

5.6.1 Finger management schemes

We consider the mobile unit which is equipped with an Lc finger RAKE receiverand is capable of despreading signals from different BSs using different fingers,and thus facilitating the SHO process. Without loss of generality, we assumethat there are L resolvable paths from the serving BS and La paths from thetarget BS. We focus on the receiver operation when the mobile unit is movingfrom the coverage area of its serving BS to that of a target BS. As the mobileunit enters the SHO region, the RAKE receiver relies at first on the L resolvablepaths gathered from the serving BS and as such starts with Lc/L-GSC. If we letΓi:j be the sum of the i largest SNRs among j ones, i.e. Γi:j =

∑ik=1 γk :j where

γk :j is the kth-order statistics, then the total received SNR after GSC is givenby ΓLc :L . At the beginning of every time slot, the receiver compares the receivedSNR, ΓLc :L , with a certain target SNR, denoted by γT . If ΓLc :L is greater thanor equal to γT , the mobile continues to rely solely on its current serving BS andSHO is not initiated.

On the other hand, whenever ΓLc :L falls below γT , the mobile will initiateSHO by estimating and combining resolvable paths from the target BS using theavailable figures. We consider two finger update strategies in this work, i.e. fullGSC [11] or block change [30]. With full GSC, the RAKE receiver reassigns itsLc fingers to the Lc strongest paths among the total L + La available resolvablepaths from both the serving and the target BSs (i.e. the RAKE receiver uses(L + La)/Lc -GSC). Now the total received SNR is given by ΓLc /L+La

. Based onthe above mode of operation, we can see that the final combined SNR, denotedby γc , with full GSC scheme, is given mathematically by

γc =

ΓLc /L+La

, 0 ≤ ΓLc /L < γT ;

ΓLc /L , ΓLc /L ≥ γT .(5.71)

With the block change scheme, the RAKE receiver compares the sums of twogroups of path SNRs: the sum of the Ls smallest paths among the Lc currentlyused paths from the serving BS (i.e.

∑Lc

i=Lc −Ls +1 γi:L ) and the sum of the Ls

strongest paths from the target BS (i.e.∑Ls

i=1 γi:La). Then, the receiver replaces


the Ls smallest paths which are currently used with the best group. Note thatno replacement occurs if the sum of the Ls strongest additional paths from thetarget BS is less than the sum of the Ls weakest paths among the Lc strongestpaths from the serving BS. For simplicity, if we let

Y =Lc −Ls∑

i=1

γi:L , Z =Lc∑

i=Lc −Ls +1

γi:L ,

and W =Ls∑i=1

γi:La,

then, based on the above mode of operation, the final combined SNR, γc , withblock change is then given by

γc =

Y + maxZ,W, 0 ≤ Y + Z < γT ;

Y + Z, Y + Z ≥ γT .(5.72)

Note that the block change scheme only needs to compare the sum of twogroups of Ls path SNRs and, as such, avoids reordering all the paths, which isessential for the full GSC scheme. Therefore, a further reduction in SNR compar-isons and SHO overhead can be obtained. In the following analysis, we focus onthe block change scheme and compare its performance with the full GSC schemewhenever appropriate.


Based on the mode of operation of the block change scheme summarized in(5.72), the CDF of the combined SNR γc , Fγc

(x), can be written as

Fγc(x) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

Pr [Y + maxZ,W < x] , 0 ≤ x < γT ;

Pr [γT ≤ Y + Z < x]

+Pr[Y + maxZ,W < x,

Y + Z < γT ], x ≥ γT

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Pr [Z ≥ W,Y + Z < x]

+ Pr [Z < W,Y + W < x] , 0 ≤ x < γT ;

Pr [γT ≤ Y + Z < x]

+Pr[Z ≥ W,Y + Z < γT ]

+Pr[Z < W,Y + W < x,

Y + Z < γT ], x ≥ γT .

(5.73)

In (5.73), Pr [γT ≤ Y + Z < x] can be calculated easily using the CDF of thecombined SNR of L/Lc -GSC. Noting that W is independent of Y and Z, we can


calculate the other joint probabilities in (5.73) as

Pr[Z ≥ W,Y + Z < x] (5.74)

=∫ x

0

∫ x−y

0

∫ x−y

w

pY ,Z (y, z)pW (w)dzdwdy,

Pr[Z < W,Y + W < x] (5.75)

=∫ x

0

∫ x−y

0

∫ w

0pY ,Z (y, z)pW (w)dzdwdy,

Pr[Z ≥ W,Y + Z < γT ] (5.76)

=∫ γT

0

∫ γT −y

0

∫ γT −y

w

pY ,Z (y, z)pW (w)dzdwdy,

Pr[Z < W,Y + W < x, Y + Z < γT ] (5.77)

=∫ γT

0

∫ x−y

0

∫ minw,γT −y

0pY ,Z (y, z)pW (w)dzdwdy.

It is easy to see that pW (w) is simply the PDF of the combined SNR withLa/Ls -GSC. The joint PDF of Y and Z, pY ,Z (y, z) is the joint PDF of twopartial sums of ordered random variables. Based on the result of the previouschapters, pY ,Z (y, z) can be calculated as

pY ,Z (y, z) =∫ z

L s

0

∫ yL c −L s

zL s

pA,γL c −L s :L ,B ,γL c :L (y − α, α, z − β, β)dαdβ,

y >Lc − Ls

Lsz, (5.78)

where pA,γl :L ,B ,γk :L (a, α, b, β) is the joint PDF of four random variables definedas the following

A=∑L c −L s −1

i = 1 γi :L︷︸︸︷γ1:L , · · · , γLc −Ls −1:L , γLc −Ls :L︸︷︷︸

Y =∑L c −L s

i = 1 γi :L

,

B=∑L c −1

i = L c −L s + 1 γi :L︷︸︸︷γLc −Ls +1:L , · · · , γLc −1:L , γLc :L︸︷︷︸

Z=∑ L c

i = L c −L s + 1 γi :L

, γLc +1:L , · · · , γL :L .

(5.79)For i.i.d. Rayleigh fading channels, we can obtain the closed-form expression

for the joint PDF, pA,γl :L ,B ,γk :L (a, α, b, β), as

pA,γl :L ,B ,γk :L (a, α, b, β) (5.80)

=L!e−(a+α+b+β )/γ (1 − e−β/γ )L−k [a − (l − 1)α]l−2

(L − k)!(k − l − 1)!(k − l − 2)!(l − 1)!(l − 2)!γk

×k−l−1∑j=0

(k − l − 1

j

)(−1)j [b − β(k − l − j − 1) − αj]k−l−2

×U(α)U(α − β)U(a − (l − 1)α)

×U(b − β(k − l − j − 1) − αj),

0 < (k − l − 1)β < b < (k − l − 1)α.


Note that the joint PDF of Y and Z (5.78) involves only finite integrations ofelementary functions and, as such, can be easily calculated with mathematicalsoftware, such as Mathematica. Also note that even though (5.78) is valid onlywhen l ≥ 2 and k ≥ l + 2, all other cases can be obtained easily by followingsimilar steps used for (5.80).

Finally, considering (5.78) together with (5.74)–(5.77), the CDF of γt , Fγc(x),

in (5.73) can be obtained. Differentiating (5.73) with respect to x, we can obtain,after some manipulations, the following generic expression for the PDF of thecombined SNR, γc , as

pγc(x) =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

∫ x

0

(pY ,Z (y, x − y)FW (x − y)

+pW (x − y)∫ x−y

0 pY ,Z (y, z)dz)dy, 0 ≤ x < γT ;

pY +Z (x)

+∫ γT

0

(pW (x − y)

∫ γT −y

0 pY ,Z (y, z)dz)dy, x ≥ γT ,

(5.81)

where pY ,Z (·, ·) is defined in (5.78), pY +Z (·) is the PDF of the com-bined SNR with L/Lc -GSC, pW (·) and FW (·) are the PDF and CDF ofthe combined SNR with La/Ls -GSC, respectively, for both of which theclosed-form expression for i.i.d. Rayleigh fading environment is available (see(5.27) and (5.18)).

With the statistics of the combined SNR derived, we can evaluate the per-formance of the block change scheme and compare it with that of the full GSCscheme through numerical example. In Fig. 5.23, we consider the effect of theswitching threshold on the performance by plotting the average BER of BPSKversus the average SNR per path, γ, of the block change scheme proposed andthe full GSC scheme in [11] for various values of γT over i.i.d. Rayleigh fadingchannels when L = 5, La = 5, Lc = 3, and Ls = 2. From this figure, it is clearthat the higher the threshold, the better the performance, as we expect intu-itively. Note that when γ becomes larger, the combined SNR is typically largeenough in a way that the receiver does not need to rely on the additional pathsfrom the target BS. Hence, we can observe that in good channel conditions (i.e.γ is relatively large compared to γT ), both schemes become insensitive to varia-tions in γT . Also note that when the switching threshold is small, both schemeshave almost the same performance since the additional paths are not necessary.On the other hand, in the case of large threshold values, the full GSC schemeshows better performance, as with the full GSC scheme, instead of compar-ing and replacing blocks, the Lc largest paths are selected among the L + La

ones.In Fig. 5.24, we vary the block size, Ls , with two values of γT . We can see

that for the low threshold, the variations of the block size do not affect theperformance since in this case no replacement is needed. However, when thethreshold is set high, we can observe the performance difference according tothe value of Ls . For our chosen set of parameters, the best performance which


−10 −8 −6 −4 −2 0 2 4 6 8 1010

−6

10−5

10−4

10−3

10−2

10−1

100

Average SNR per Path [dB]

Ave

rage

BE

R

Block ChangeFull GSC

γT = 10 dB

γT = 5 dB

γT = 0 dB

γT = −5 dB

Figure 5.23 Average BER of BPSK versus the average SNR per path, γ, of the BlockChange and the full GSC schemes for various values of γT over i.i.d. Rayleigh fadingchannels when L = 5, La = 5, Lc = 3, and Ls = 2 [30]. c© 2008 IEEE.

is very close to that of the full GSC scheme can be acquired when Ls = 2. Thisis because if Ls = 1, we have little benefit from the additional paths, while ifLs = 3, we have more chances to lose the better paths during the replacementprocess.


As shown in the previous section, because the paths with the strongest SNRvalues are selected whenever SHO is initiated, the full GSC scheme always pro-vides a better performance than the block change scheme. However, the blockchange scheme enjoys a lower complexity, as it avoids the need for a full reorder-ing process of all available paths from the serving and the target BSs. In thissection, we investigate this complexity trade-off issue by quantifying the averagenumber of path estimations, the average number of SNR comparisons, and theSHO overhead.


−10 −8 −6 −4 −2 0 2 4 6 8 1010

−6

10−5

10−4

10−3

10−2

10−1

100

Average SNR per Path [dB]

Ave

rage

BE

R

Block Change, L

s=1

Block Change, Ls=2

Block Change, Ls=3

Full GSC

γT = −10 dBγ

T = 10 dB

Figure 5.24 Average BER of BPSK versus the average SNR per path, γ, of the BlockChange and the full GSC schemes for various values of Ls and γT over i.i.d. Rayleighfading channels when L = 5, La = 5 and Lc = 3 [30]. c© 2008 IEEE.

With the proposed scheme, the RAKE receiver estimates the L paths in thecase of ΓLc /L ≥ γT or L + La in the case of ΓLc /L < γT . Hence, the averagenumber of path estimations of the block change scheme is the same as that ofthe full GSC scheme [11, eq. (25)].

As another complexity measure, we calculate the average number of requiredSNR comparisons. Noting that the average number of SNR comparisons for j/i-GSC, denoted by Cj/i−GSC , can be obtained as1

Cj/i−GSC =min[i,j−i]∑

k=1

(j − k), (5.82)

1 With the traditional sorting approach, we need k − 1 comparisons to find the kthlargest/smallest one after the previous k − 1 largest/smallest ones have been found. We fol-low this traditional approach in order to perform an accurate complexity comparison whilenoting that with a quick sorting algorithm for n paths, we just need O(log(n)) complexity.


we can express the average number of SNR comparisons for the full GSC schemeand the block change scheme as

CF ull = Pr[ΓLc /L ≥ γT

]CL/Lc −GSC (5.83)

+Pr[ΓLc /L < γT

]C(L+La )/Lc −GSC

and

CBlock = CL/Lc −GSC + Pr[ΓLc /L < γT

](5.84)

×(CLc /Ls −GSC + CLa /Ls −GSC + 1

),

respectively.Since the SHO is attempted whenever ΓLc /L is below γT , the probability of the

SHO attempt is same as the outage probability of Lc/L-GSC evaluated at γT ,i.e. FΓL c / L

(γT ). The SHO overhead, denoted by β, is commonly used to quantifythe SHO activity in a network and is defined as [33, eq. (9.2)]

β =N∑

n=1

nFn − 1, (5.85)

where N is the number of active BSs and Fn is the average probability thatthe mobile unit uses n-way SHO. Based on the mode of operation of the blockchange scheme, we can express its SHO overhead, β, as

β =

FY +Z (γT ) Pr [Z < W |Y + Z < γT ] , Ls < Lc ;

0, Ls = Lc.(5.86)

Note that FY +Z (γT ) is the CDF of Lc/L-GSC output SNR evaluated at γT .Since W is independent to Z and Y , we can calculate the conditional probability,Pr[Z < W |Y + Z < γT ], in (5.86) as

Pr [Z < W |Y + Z < γT ] (5.87)

=∫ ∞

0FZ |Y +Z<γT

(x)pW (x)dx,

where the conditional CDF in (5.87) can be obtained as

FZ |Y +Z<γT(x) =

Pr [Z < x, Y + Z < γT ]Pr [Y + Z < γT ]

(5.88)

=1

FY +Z (γT )×⎧⎪⎪⎨

⎪⎪⎩∫ x

0

∫ γT −z

(Lc −Ls )z/Ls

fY ,Z (y, z)dydz, 0 ≤ x < Ls

LcγT ;∫ Ls γT /Lc

0

∫ γT −z

(Lc −Ls )z/Ls

fY ,Z (y, z)dydz, x ≥ Ls

LcγT ,

After successive substitutions from (5.88) to (5.86), we can finally obtain theanalytical expression of the SHO overhead.


−2 0 2 4 6 8 10 12 140

5

10

15

20

25

30

Output Threshold, γT [dB]

Ave

rage

Num

ber

of S

NR

Com

paris

ons

L

c/(L+L

a)−GSC

Full GSCBlock Change, L

s=1

Block Change, Ls=2

Block Change, Ls=3

(a) Average Number of SNR Comparisons

−2 0 2 4 6 8 10 12 1410

3

102


Ave

rage

BE

R

L

c/(L+L

a)−GSC


s=1

Block Change, Ls=2

Block Change, Ls=3

(b) Average BER

−2 0 2 4 6 8 10 12 14

0

0.2

0.4

0.6

0.8

1


SH

O O

verh

ead

(β)

L

c/(L+L

a)−GSC


s=1

Block Change, Ls=2

Block Change, Ls=3

Simulation Results

(c) SHO Overhead

Figure 5.25 Complexity trade-off versus the output threshold, γT , of the block changeand the full GSC schemes, and conventional GSC for various values of Ls over i.i.d.Rayleigh fading channels with L = 5, La = 5, Lc = 3, and γ = 0 dB [30]. c© 2008IEEE.

In Fig. 5.25, we plot (a) the average number of SNR comparisons, (b) theaverage BER, and (c) the SHO overhead versus the output threshold, γT , ofthe block change and the full GSC schemes for various values of Ls over i.i.d.Rayleigh fading channels when L = 5, La = 5, Lc = 3, and γ = 0 dB. For compar-ison purposes, we also plot those for conventional (L + La)/Lc -GSC. Note that


the full GSC scheme is acting as (L + La)/Lc -GSC when the output thresholdbecomes large. Hence, we can observe from all the subfigures that the full GSCscheme converges to GSC as γT increases.

Recall that the block change scheme has the same path estimation load asthe full GSC scheme. However, from Fig. 5.25(a), we can see that the blockchange scheme leads to a great reduction of the SNR comparison load comparedto the full GSC scheme. For example, let us consider the case that γT > 8 dBand Ls = 2. In this case, the reduction of the SNR comparison load is maxi-mized compared to the full GSC scheme. However, from Fig. 5.25(b), we canobserve in the same SNR region a very slight performance loss of the blockchange scheme compared to the full GSC as well as the conventional GSCschemes.

For the SHO overhead, simulation results are also presented in Fig. 5.25(c)to verify our analysis. It is clear from this figure that the receiver has a higherchance to use 2-way SHO as Ls decreases. This is because as Ls decreases,the probability that the sum of the Ls smallest paths among the Lc currentlyused paths from the serving BS is less than the sum of the Ls strongest pathsfrom the target BS is increasing and as such, we have a higher chance to replacegroups. From this figure together with Fig. 5.25(b), we can quantify the trade-offbetween the SHO overhead and the performance. Again, let us consider the casethat γT > 8 dB and Ls = 2. In this case, note the reduction of SHO overhead atthe expense of a slight performance loss in comparison to the full GSC scheme.If we increase the number of fingers, i.e. Lc = 4, we can observe more reductionsin complexity with a slight performance loss.

5.7 Joint adaptive modulation and diversity combining

Adaptive modulation has been widely utilized to achieve efficient and reliablecommunications over time-varying wireless fading channels. The basic idea ofadaptive modulation, as summarized in a previous chapter, is to match the modu-lation parameters with the instantaneous fading channel conditions such that thetransmission rate is maximized while satisfying a certain instantaneous error raterequirement [34–36]. For the constant-power variable-rate uncoded M -ary QAMscheme studied in [35], the value range of the instantaneous SNR is divided intoN + 1 regions with properly selected threshold γTn

, n = 0, 1, · · · , N . The modu-lation scheme 2n -QAM is used during the data reception if received SNR γ is inthe nth region, where γTn

≤ γ < γTn + 1 . More thorough treatment on adaptivetransmission can be found in [37].

We can naturally observe that both adaptive modulation and adaptive diver-sity combining schemes utilize some predetermined threshold in their operation.Based on this observation, we look into some joint design of adaptive modula-tion and diversity combining in this section. With the resulting joint adaptive

Joint adaptive modulation and diversity combining 145

AMC controller

DiversityTransmitter

Block fading channel

AMC controller

AMC mode selection

DetectorCombiner

...

Figure 5.26 System model of joint adaptive modulation and diversity combining.

modulation and diversity combining (AMDC) schemes, the receiver jointly deter-mines the most appropriate modulation mode and diversity combiner structurebased on the current channel conditions and the desired BER requirements. Inthe meantime, the receiver simultaneously uses the thresholds set for the adap-tive modulation mode selection to guide the operation of the adaptive combiningschemes. The system model of the proposed joint design is illustrated in Fig. 5.26.The proposed AMDC systems can efficiently explore the bandwidth and powerresource by transmitting at higher data rate and/or combining the least num-ber of diversity paths under favorable channel conditions. On the other hand,the AMDC system also responds to channel degradation with an increase in thenumber of combined paths and/or a reduction in the data rate.

While our proposed ideas are applicable to many output threshold based adap-tive combining schemes [2–6, 38], for the sake of clarity, we focus on MS-GSCwith Lc = L, i.e. the receiver implements the same number of MRC branchesas the number of available diversity paths [4]. Depending on the primary objec-tive of the joint design, we can arrive at an AMDC scheme with a high pro-cessing power efficiency (termed power-efficient AMDC), an AMDC schemewith a high bandwidth efficiency (termed bandwidth-efficient AMDC), and anAMDC scheme with a high bandwidth efficiency and improved power efficiencyat the cost of a higher error rate than the bandwidth-efficient AMDC scheme(termed bandwidth-efficient and power-greedy AMDC) [12], all of which sat-isfy the desired BER requirement. For all the three AMDC schemes underconsideration, we quantify through accurate analysis their processing powerconsumption (quantified in terms of average number of combined diversitypaths), spectral efficiency (quantified in terms of average number of transmittedbits/s/Hz), and performance (quantified in terms of average BER). In addition,some selected numerical examples are presented to illustrate the mathematicalformalism.


5.7.1 Power-efficient AMDC scheme

The primary objective of the power-efficient AMDC scheme is to minimize theprocessing power consumption of the diversity combiner, i.e. to minimize theaverage number of combined/active diversity paths during the data burst recep-tion. Once this primary objective is met, this scheme tries to afford the largestpossible spectral efficiency while meeting the required target BER. Based onthese objectives, the diversity combiner will perform just enough combining oper-ations such that at least the lowest adaptive modulation mode, e.g. BPSK, willexhibit an instantaneous BER smaller than the predetermined target value. Inparticular, the receiver tries to increase the output SNR γc above the thresholdfor BPSK, denoted by γT1 , by performing MS-GSC diversity. After estimatingand ranking the L available diversity paths, the combiner first checks if the SNRof the strongest path is greater than γT1 . If so, the combiner uses the outputsignal from just the strongest path. If not, the combiner checks the combinedSNR of the first two strongest paths. If the combined SNR is still less than γT1 ,then the combiner checks the combined SNR of the first three strongest paths.This process is continued until either (i) the combined SNRs become greaterthan γT1 , or (ii) all L available paths have been combined. In the first case, thereceiver starts to determine the modulation mode to be selected by checking inwhich interval the resulting output SNR falls. In particular, the receiver sequen-tially compares the output SNR with respect to the thresholds, γT2 , γT3 , · · · ,γTN

. Whenever the receiver finds that the output SNR is smaller than γTn + 1

but greater than γTn, it selects the modulation mode n for the subsequent data

burst and feeds back that particular modulation mode to the transmitter. If thecombined SNR of all L available branches is still below γT1 , the receiver may askthe transmitter to either (i) transmit using the lowest modulation mode in vio-lation of the target instantaneous BER requirement (option 1), or (ii) buffer thedata and wait until the next guard period for more favorable channel conditions(option 2).

While this mode of operation is different than that of the power-efficient mul-tiple thresholds minimum selection combining (MT-MSC) scheme, which wasdescribed and simulated in [39], it can be shown easily that these two schemesare actually equivalent from an output SNR and performance standpoint. Assuch, an alternative presentation/implementation of our proposed power-efficientAMDC scheme can be found in the flow chart illustrating the mode of operationof the power efficient MT-MSC given in [39, fig. 1].

Statistics of output SNRBased on the mode of operation described above, we can see that the receivedSNR, γc , of the power-efficient AMDC system is the same as the combined SNRof MS-GSC scheme with γT1 as the output threshold. In other words, the CDFof the received SNR, Fγc

(·), of the power-efficient AMDC based on MS-GSC is


given by

Fγc(γ) =

⎧⎪⎪⎨⎪⎪⎩

FM SC (γT 1 )γc (γ), for option 1;F

M SC (γT 1 )γc (γ), γ > γT1 ;

FM SC (γT 1 )γc (γT1 ), 0 < γ ≤ γT1

for option 2,(5.89)

where FM SC (γT 1 )γc (·) denotes the CDF of the combined SNR with L-branch

MS-GSC and using γT1 as an output threshold. The generic expression ofF

M SC (γT 1 )γc (·) was given in (5.21) with γT changed to γT1 and Lc = L, which

is available in closed-form for the i.i.d. Rayleigh fading environment in [4, eq.(30)].

Correspondingly, the PDF of the received SNR, pγc(·), is given by

pγc(γ) =

p

M SC (γT 1 )γc (γ), for option 1;

pM SC (γT 1 )γc (γ)U(γ − γT1 ) + p

M SC (γT 1 )γc (γT1 )δ(γ), for option 2,

(5.90)where δ(·) is the Delta function and p

M SC (γT 1 )γc (·) denotes the PDF of the com-

bined SNR with L-branch MS-GSC and using γT1 as an output threshold, whichis available in closed form for the i.i.d. Rayleigh fading environment in (5.23)with γT changed to γT1 and Lc = L.

Performance and processing power analysisThe power consumption for diversity combining can be quantified in terms of theaverage number of combined paths [4]. It can be shown that the average numberof combined paths with the power-efficient AMDC is given for option 1 by

NC = 1 +L−1∑i=1

FΓ i(γT1 ),

and for option 2 by

NC = 1 +L−1∑i=1

FΓ i(γT1 ) − LFL−M RC

γc(γT1 ),

where FΓ i(·) is the CDF of the combined SNR with L/i-GSC scheme (which

is given in closed-form for i.i.d. Rayleigh fading in (5.18)) and FL−M RCγc

(·) isthe CDF of the combined SNR with L-branch MRC scheme (which is given inclosed-form for i.i.d. Rayleigh fading in Table 5.1).

The average spectral efficiency of an adaptive modulation system can be cal-culated as [35, eq. (33)]

η =N∑

n=1

n Pn ,


where n is the number of bits carried by a symbol of modulation mode, Pn isthe probability that the nth constellation is used. For the power-efficient AMDCsystem based on MS-GSC, it can be shown that Pn is given by

Pn =

⎧⎪⎪⎨⎪⎪⎩

FM SC (γT 1 )γc (γTn + 1 ) − F

M SC (γT 1 )γc (γTn

), n ≥ 2;

FM SC (γT 1 )γc (γT2 ), n = 1

for option 1;

FM SC (γT 1 )γc (γTn + 1 ) − F

M SC (γT 1 )γc (γTn

), for option 2.

(5.91)

Therefore, the average spectral efficiency of the MS-GSC-based AMDC systemis given by

η =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

N −N∑

n=2

FM SC (γT 1 )γc (γTn

), for option 1;

N −N∑

n=1

FM SC (γT 1 )γc (γTn

), for option 2.

(5.92)

The average BER for adaptive modulation system can be calculated as [35, eq.(35)]

< BER >=1η

N∑n=1

n BERn, (5.93)

where BERn is the average error rate for constellation n, and which is given by

BERn =∫ γT n + 1

γT n

BERn (γ)pM SC (γT 1 )γc (γ)dγ, (5.94)

where BERn (γ) is the conditional BER of the constellation n over the AWGNchannel given that the SNR is equal to γ, an approximate expression of whichwas given in (2.59). Therefore, the average BER of the power-efficient AMDCbased on MS-GSC can be calculated as

< BER >=

⎧⎪⎪⎨⎪⎪⎩

1η

(∫ γT 20 BER1(γ) p

M SC (γT 1 )γc (γ) dγ

+∑N

n=2 n∫ γT n + 1

γT nBERn (γ) p

M SC (γT 1 )γc (γ) dγ

), for option 1;

1η

∑Nn=1 n

∫ γT n + 1γT n

BERn (γ) pM SC (γT 1 )γc (γ) dγ, for option 2.

(5.95)

5.7.2 Bandwidth-efficient AMDC scheme

The primary objective of the bandwidth-efficient AMDC scheme is to maximizethe spectral efficiency. As such, the receiver with this scheme performs the nec-essary combining operations so that the highest achievable modulation modecan be used while satisfying the instantaneous BER requirement. More specifi-cally, the receiver tries first to increase the output SNR γc above the thresholdof highest modulation mode 2N -QAM, i.e. γTN

, by employing an MS-GSC typeof diversity. After estimating and ranking all the available diversity paths, the


combiner sequentially checks the SNR of the strongest path, the combined SNRof the two strongest paths, the combined SNR of the three strongest paths, etc.Whenever the combined SNR is larger than γTN

, the receiver stops checkingand informs the transmitter to use 2N -QAM as the modulation mode for thesubsequent data reception. If the combined SNR of all the available paths isstill below γTN

, the receiver selects the modulation mode corresponding to theSNR interval in which the combined SNR falls into. In particular, the receiversequentially compares the output SNR with the thresholds, γTN −1 , γTN −2 , · · · ,γT1 . Whenever the receiver finds that the output SNR is smaller than γTn + 1 butgreater than γTn

, it selects the modulation mode n for the subsequent data burstand feeds back this selected mode to the transmitter. If, in the worst case, thecombined SNR of all the available paths ends up being below γT1 , the receiverhas the same two termination options as for the power-efficient scheme (i.e. totransmit using the lowest modulation mode (option 1) or to wait until the nextguard period (option 2)).

Statistics of output SNRBased on the mode of operation described above, we can see that the receivedSNR, γc , of the bandwidth-efficient AMDC system is the same as the combinedSNR of MS-GSC diversity with γTN

as the output threshold. Therefore, the CDFof the received SNR of this bandwidth-efficient AMDC scheme based on MS-GSCis given by

Fγc(γ) =

⎧⎪⎪⎨⎪⎪⎩

FM SC (γT N

)γc (γ), for option 1;F

M SC (γT N)

γc (γ), γ > γT1 ;

FM SC (γT N

)γc (γT1 ), 0 < γ ≤ γT1

for option 2,(5.96)

where FM SC (γT N

)γc (·) denotes the CDF of the combined SNR with L-branch MS-

GSC and γTNas an output threshold. Correspondingly, the PDF of the received

SNR is given by

pγc(γ) =

p

M SC (γT N)

γc (γ), for option 1;

pM SC (γT N

)γc (γ)U(γ − γT1 ) + p

M SC (γT N)

γc (γT1 )δ(γ), for option 2,(5.97)

where pM SC (γT N

)γc (·) denotes the PDF of the combined SNR with L-branch MS-

GSC and γTNas output threshold.

Performance and processing power analysisWith the statistics of the received SNR, as given in previous subsection, we cannow study the processing power and the performance of the bandwidth-efficientAMDC scheme as we did for the power-efficient AMDC system. For conciseness,we just list the analytical results in the following.


Average number of combined branches

NC =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

1 +L−1∑i=1

FΓ i(γTN

), for option 1;

1 +L−1∑i=1

FΓ i(γTN

) − LFγcL−M RC (γT1 ), for option 2.

(5.98)

Average spectral efficiency

η =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

N −N∑

n=2

FM SC (γT N

)γc (γTn

), for option 1;

N −N∑

n=1

FM SC (γT N

)γc (γTn

), for option 2.

(5.99)

Average BER

< BER >=

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

1η

(∫ γT 2

0BER1(γ) p

M SC (γT N)

γc (γ) dγ

+∑N

n=2 n

∫ γT n + 1

γT n

BERn (γ) pM SC (γT N

)γc (γ) dγ

), for option 1;

1η

∑Nn=1 n

∫ γT n + 1

γT n

BERn (γ) pM SC (γT N

)γc (γ) dγ, for option 2.

(5.100)

5.7.3 Bandwidth-efficient and power-greedy AMDC scheme

The bandwidth-efficient and power-greedy AMDC scheme can be viewed as amodified bandwidth-efficient scheme with better power efficiency at the cost ofa slightly higher error rate. Basically, the receiver combines the least numberof diversity branches such that the highest achievable modulation mode can beused while satisfying the instantaneous BER requirement. During this process,the receiver ensures that combining additional branches does not make a highermodulation mode feasible.

More specifically, the receiver tries first to increase the output SNR γc abovethe threshold of 2N -QAM, i.e. γTN

, by using the MS-GSC diversity combiningscheme. After estimating and ranking all the available diversity paths, the com-biner sequentially checks the SNR of the strongest path, the combined SNR ofthe two strongest paths, the combined SNR of the three strongest paths, etc.Whenever the combined SNR is larger than γTN

, the receiver selects 2N -QAMas the modulation mode and uses current combiner structure for data recep-tion during the subsequent data burst. If the combined SNR of all availablebranches is still below γTN

, the receiver determines the highest achievable mod-ulation mode by checking in which interval the combined SNR of these branchesfalls. The receiver sequentially compares the output SNR with respect to the


thresholds, γTN −1 , γTN −2 , · · · , γT1 . Whenever the receiver finds that the out-put SNR is smaller than γTn + 1 but greater than γTn

, it selects the modulationmode n for the subsequent data burst as it is the highest achievable one. Beforethe data transmission, the receiver selects the minimum combiner structure (i.e.with the minimum number of active branches) such that the output SNR isstill greater than γTn

. Basically, the receiver sequentially turns off the weakestbranches until a further branch turnoff will lead to an output SNR below γTn

. If,in the worst case, the combined SNR of all the available branches is below γT1 , thereceiver has the same two terminating options as for the two previously presentedschemes.

While this mode of operation is different than the mode of operation of thebandwidth-efficient MT-MSC scheme, which was described and simulated in [40],it can be shown easily that these two schemes are actually equivalent froman output SNR and performance standpoint. As such, an alternative presen-tation/implementation of our proposed bandwidth-efficient and power-greedyAMDC scheme can be found in the flow chart illustrating the mode of operationof the bandwidth-efficient MT-MSC given in [40, fig. 1].

Statistics of output SNRBased on its mode of operation described above, we can see that the output SNR,γc , of the bandwidth-efficient and power-greedy AMDC scheme is the same as thecombined SNR of MS-GSC with γTN

as the output threshold only for γc > γTN

case, i.e.

Fγc(x) = F

M SC (γT N)

γc (x), x > γTN. (5.101)

For the case of γT1 ≤ Γ < γTN, the statistics are more complicated to calcu-

late. Note that γTn≤ γc < γTn + 1 if and only if γTn

≤∑L

i=1 γi < γTn + 1 . It canbe shown that the CDF of the received SNR γc over the range of [γTn

, γTn + 1 ] ismathematically given by

Fγc(x) = FL−M RC

γc(γTn

) + Pr

[γTn

≤ γ1:L < x &L∑

k=1

γk < γTn + 1

]

+L∑

l=2

Pr

⎡⎣ l−1∑

j=1

γj :L < γTn≤

l∑j=1

γj :L < x &L∑

k=1

γk < γTn + 1

⎤⎦ , (5.102)

where FL−M RCγc

(·) denotes the CDF of the combined SNR with L-branchMRC and γj :L denotes the jth largest path SNR among the L availableones. Let yl denote the sum of the l − 1 largest ordered path SNRs, i.e. yl =∑l−1

j=1 γj :L , and let zl denote the sum of the L − l smallest ordered path SNRs,


i.e. zl =∑L

j= l+1 γj :L . The CDF can be rewritten as

Fγc(x) = F (L)(γTn

) + Pr[γTn

≤ γ1:L < x & γ1:L + z1 < γTn + 1

]+

L−1∑l=2

Pr[yl < γTn

≤ yl + γl:L < x & yl + γl:L + zl < γTn + 1

]+ Pr [yL < γTn

≤ yL + γL :L < x] , (5.103)

which can be calculated in terms of the joint PDFs of yl , γl:L and zl as

Fγc(x) = FL−M RC

γc(γTn

) +∫ x

γT n

∫ γT n + 1 −γ

0pγ1 :L ,z1 (γ, z)dzdγ (5.104)

+L−1∑l=2

∫ γT n

0

∫ min[y/(l−1),x−y ]

γT n −y

∫ γT n + 1 −y−γ

0pyl ,γl :L ,zl

(y, γ, z)dzdγdy

+∫ γT n

0

∫ min[y/(L−1),x−y ]

γT n −y

pyL ,γL :L (y, γ)dγdy, γTn≤ x < γTn + 1 .

For the case of γc < γT1 , it can be shown that the CDF of γc for the MS-GSC-based AMDC scheme is given by

Fγc(x) =

FL−M RC

γc(x), for option 1;

FL−M RCγc

(γT1 ), for option 2,0 < x < γT1 . (5.105)

Thus, we have obtained the generic expression of the CDF of the combined SNRfor the whole SNR value range. After differentiating Fγc

(x) with respect to x,we obtain a generic formula for the PDF of the output SNR, γc , as

pγc(x) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

pM SC (γT N

)γc (x), x > γTN

;∫ γT n + 1 −x

0pγ1 :L ,z1 (x, z)dz

+L−1∑l=2

(∫ γT n

l−1l x

∫ γT n + 1 −x

0pyl ,γl :L ,zl

(y, x − y, z)dzdy

× (U(x − γTn) − U(x − l

l−1 γTn)))

+∫ γT n

L −1L x

pyL ,γL :L (y, x − y)dy

×(U(x − γTn) − U(x − L

L−1 γTn)), γTn

≤ x < γTn + 1 ;pL−M RC

γc(x), for option 1;

δ(x)pL−M RCγc

(γT1 ), for option 2,0 < x < γT1 .

(5.106)

Note that the CDF and the PDF of the combined SNR are given in terms ofthe joint PDFs of yl , γl:L , and zl , which was studied extensively in the previouschapter. Specifically, it has been shown that these joint PDFs are available inclosed form for the i.i.d. Rayleigh fading case. Specifically, it has been shown


that Fyl ,γl :L ,zl(y, γ, z) is given by

pyl ,γl :L ,zl(y, γ, z) = pγl :L (γ) × pzl |γl :L =γ (z) × pyl |γl :L =γ ,zl =z (y)

=L!

(L − l)!(l − 1)!γL

[y − (l − 1)γ]l−2

(l − 2)!(L − l − 1)!e−

y + γ + zγ U(y − (l − 1)γ)

×L−l∑i=0

(L − l

i

)(−1)i(z − iγ)L−l−1U(z − iγ)

γ > 0, y > (l − 1)γ, z < (L − l)γ, (5.107)

where U(·) is the unit step function. The joint PDFs pγ1 :L ,z1 (γ, z) andpyL ,γL :L (y, γ) can be obtained as marginals of the joint PDF given in (5.107).

Performance and processing power analysisWe now study the performance and processing power of the bandwidth-efficientand power-greedy AMDC schemes based on the statistics of the received SNRthat we just derived. Note that the bandwidth-efficient and power-greedy AMDCscheme always uses the highest achievable modulation mode, and as such itwill have the same average spectral efficiency as the bandwidth-efficient AMDCscheme, which was analyzed in a previous section. In the following we focus onthe average number of combined branches and on the average BER.

Let Pl,n denote the probability that mode n is used with l combined branches.We can calculate the average number of combined branches with the bandwidth-efficient and power-greedy AMDC scheme by averaging over all possible valuesof l and n as

Nc =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

L∑l=1

N∑n=1

l Pl,n + LFL−M RCγc

(γT1 ), for option 1;

L∑l=1

N∑n=1

l Pl,n , for option 2.

(5.108)

Based on the mode of operation of the bandwidth-efficient and power-greedyschemes, it can be shown that Pl,n can be calculated for n = N as

Pl,N =

1 − F

L/1−GSCγc (γTN

), l = 1F

L/(l−1)−GSCγc (γTN

) − FL/l−GSCγc (γTN

), 1 < l ≤ L,(5.109)

and for n < N as

Pl,n =⎧⎪⎪⎪⎨⎪⎪⎪⎩

Pr[γTn

≤ γ1:L &∑L

k=1 γk :L < γTn + 1

], l = 1

Pr[∑l−1

j=1 γj :L < γTn≤∑l

j=1 γj :L &∑L

k=1 γk :L < γTn + 1

], 1 < l < L

Pr[∑L−1

j=1 γj :L < γTn≤∑L

j=1 γj :L < γTn + 1

], l = L,

(5.110)


which can be calculated using the joint PDFs of yl , γl:L and zl as

Pl,n =⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

∫ γT n + 1

γT n

∫ γT n + 1 −γ

0pγ1 :L ,z1 (γ, z)dzdγ, l = 1;∫ (l−1)γT n + 1 /l

(l−1)γT n /l

∫ y/(l−1)

γT n −y

∫ γT n + 1 −y−γ

0pyl ,γl :L ,zl

(y, γ, z)dzdγdy

+∫ γT n

(l−1)γT n + 1 /l

∫ γT n + 1 −y

γT n −y

∫ γT n + 1 −y−γ

0pyl ,γl :L ,zl

(y, γ, z)dzdγdy, 1 < l < L;∫ (L−1)γT n + 1 /L

(L−1)γT n /L

∫ y/(L−1)

γT n −y

pyL ,γL :L (y, γ)dγdy

+∫ γT n

(L−1)γT n + 1 /L

∫ γT n + 1 −y

γT n −y

pyL ,γL :L (y, γ)dγdy, l = L.

(5.111)Following the same approach as for the power-efficient AMDC while applying

the PDF of the received SNR given in (5.106), the average BER of a bandwidth-efficient and power-greedy AMDC scheme based on MS-GSC can be calculatedas

< BER >=⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

∫ γT 2

0BER1(γ) pγc

(γ) dγ +N∑

n=2

n

∫ γT n + 1

γT n

BERn (γ) pγc(γ) dγ

N −N∑

n=2

FM SC (γT N

)γc (γTn

)

, for option 1;

N∑n=1

n

∫ γT n + 1

γT n

BERn (γ) pγc(γ) dγ

N −N∑

n=1

FM SC (γT N

)γc (γTn

)

, for option 2.

(5.112)


We now illustrate the mathematical formalism that we developed in the previoussections through several numerical examples. In particular, we compare the pro-cessing power consumption, spectral efficiency, and BER performance of threeAMDC schemes. In the following numerical examples, we set the number of avail-able diversity branches L = 5, the number of adaptive modulation mode N = 4,and the SNR thresholds are set to satisfy the instantaneous BER requirement ofBER0 = 10−3 .

Note that the greater the average number of combined branches during databurst reception, the larger the average receiver processing power consump-tion. We investigate the power efficiency of the three AMDC schemes under


−5 0 5 10 15 20 25 300

1

2

3

4

5

6

Average SNR, γ, dB

Ave

rage

# o

f Com

bine

d B

ranc

hes

Option 1

−5 0 5 10 15 20 25 300

1

2

3

4

5

6

Average SNR, γ, dB

Ave

rage

# o

f Com

bine

d B

ranc

hes

Option 2

AM with MRCBandwidth Efficient AMDCBandwidth Efficient/Power Greedy AMDCPower Efficient AMDCAM without Diversity

AM with MRCBandwidth Efficient AMDCBandwidth Efficient/Power Greedy AMDCPower Efficient AMDCAM without Diversity

Figure 5.27 Average number of combined branches of three AMDC schemes with twooptions as a function of the average path SNR γ (L = 5, N = 4 andBER0 = 10−3 ) [12]. c© 2007 IEEE.

consideration in Fig. 5.27 by plotting the average number of combined branchesof these AMDC schemes with two options as a function of the average path SNRγ. For reference purposes, the average number of combined branches of adaptivemodulation without diversity and with L-branch MRC cases are also plotted. Weobserve that on average, the power-efficient AMDC scheme always combines theleast number of diversity branches for any value of γ, whereas the bandwidth-efficient AMDC scheme combines the most among the three schemes, but stillless than the adaptive modulation with L-branch MRC case. We also notice thatwhen the average path SNR is very small, the three AMDC schemes with option1 (i.e. transmitting using the lowest modulation mode in the worst case when thecombined SNR of all L available paths ends up being below γT1 ) consumes moreprocessing power than with option 2 (i.e. buffering the data and waiting until thenext guard period for a more favorable channel when the combined SNR of all L

available paths ends up being below γT1 ) by combining more diversity branches.The average spectral efficiency of three AMDC schemes with two options are

plotted as a function of the average path SNR γ in Fig. 5.28. The average spectralefficiency curves of adaptive modulation without diversity and with L-branchMRC cases are also included. We can see that the bandwidth-efficient AMDC and


−5 0 5 10 15 20 25 300

1

2

3

4

5

Average SNR, γ, dB

Ave

rage

Spe

c. E

ffici

ency

Option 1

−5 0 5 10 15 20 25 300

1

2

3

4

5

Average SNR, γ, dB

Ave

rage

Spe

c. E

ffici

ency

Option 2

AM with MRCBandwidth-Efficient AMDCBandwidth-Efficient/Power-Greedy AMDCPower-Efficient AMDCAM without Diversity

AM with MRCBandwidth-Efficient AMDCBandwidth-Efficient/Power-Greedy AMDCPower-Efficient AMDCAM without Diversity

Figure 5.28 Average spectral efficiency of three AMDC schemes with two options as afunction of the average path SNR γ (L = 5, N = 4 and BER0 = 10−3 ) [12]. c© 2007IEEE.

the bandwidth-efficient/power-greedy AMDC schemes have the same spectralefficiency as adaptive modulation with L-branch MRC, which is much greaterthan that of the power-efficient AMDC scheme over the medium value range ofγ. Thus by observing Figs. 5.27 and 5.28, there is a trade-off of spectral efficiencyand power consumption between power-efficient and bandwidth-efficient AMDCschemes. We also observe that when the average SNR is very small, three AMDCschemes with option 1 exhibit higher spectral efficiency than with option 2, atthe expense that the target BER is not met with option 1.2

Finally, we examine the BER performance of three AMDC schemes inFig. 5.29. This figure confirms that when the average SNR is very small, theaverage BER of the three AMDC schemes with option 1 become larger thanthe target BER0 = 10−3 in violation of the instantaneous BER requirement. Wecan also see that the bandwidth-efficient/power-greedy AMDC scheme has thepoorest error performance among the three AMDC schemes, but still satisfies

2 Note that the average spectral efficiency is viewed as a valid performance measure only ifthe BER requirement is satisfied, then the results for option 1 are valid only for an averageSNR greater than 1 dB.


−5 0 5 10 15 20 25 3010

−6

10−5

10−4

10−3

10−2

Average SNR, γ, dB

Ave

rage

BE

R

Option 1

−5 0 5 10 15 20 25 3010

−6

10−5

10−4

10−3

10−2

Average SNR, γ, dB

Ave

rage

BE

R

Option 2

Bandwidth-Efficient/Power-Greedy AMDCPower-Efficient-AMDCBandwidth-Efficient AMDC

Bandwidth-Efficient/Power-Greedy AMDCPower-Efficient AMDCBandwidth-Efficient AMDC

Figure 5.29 Average BER of three AMDC schemes with two options as a function ofthe average path SNR γ (L = 5, N = 4 and BER0 = 10−3 ) [12]. c© 2007 IEEE.

the BER requirement except over the low-SNR range with option 1. Compar-ing the error performance of the power-efficient AMDC and bandwidth-efficientAMDC, we find that over the low-SNR range, the bandwidth-efficient schemeslightly outperforms the power-efficient scheme, whereas for the medium to highSNR range, the power-efficient scheme performs better. This is because whenthe SNR is low, the bandwidth-efficient scheme needs to combine more diversitybranches and when the SNR is higher, the power-efficient scheme tends to settleon the lower adaptive modulation mode, which has better error protection.

Considering the three figures together, we can draw the following conclusions.

There is a trade-off of power consumption versus spectral efficiency betweenthe power-efficient AMDC scheme and the bandwidth-efficient AMDC scheme,which have comparable BER performance.

The bandwidth-efficient/power-greedy AMDC scheme offers better power-efficiency than the bandwidth-efficient AMDC scheme at the cost of slightlypoorer BER performance and with the same spectral efficiency.

Option 1 for the worst case scenario leads to a better spectral efficiency at theexpense of higher power consumption and in violation of the BER require-ment, whereas option 2 avoids the extra power consumption and BER require-ment violation but causes a certain amount of transmission delay.


5.8 Summary

In this chapter, we investigated the idea of adaptive combining through its appli-cations in various scenarios, including traditional reception diversity systems,transmit diversity systems, RAKE receiver over soft handover region, and jointdesign with adaptive modulation. In most cases, the joint distribution functionsof partial sums of ordered random variables were of critical importance in the per-formance and complexity analysis of the resulting systems. Through the accuratetrade-off analysis through selected numerical examples, we were able to confirmthat adaptive combining can bring significant complexity and processing powersavings to the wireless systems operating over fading channel, as a minimumor no loss in performance. As such, adaptive combining has huge applicationpotential in future wireless systems.


Minimum estimation and combining GSC (MEC-GSC) [6] extends the idea ofMS-GSC by introducing a switching and examining combining (SEC) stage.Lioumpas et al. [41] propose an adaptive GSC scheme, which essentially appliesthe adaptive combining principle to NT-GSC scheme. Bouida et al. combineadaptive combining based on MS-GSC with transmit power control in [42]. Thejoint design of adaptive modulation with OT-MRC scheme is presented in [43].Lee and Ko extend the joint AMDC scheme to a multi-channel environmentin [44]. Discrete transmit power control is considered together with joint AMDCscheme in [45].

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[1] W. C. Jakes, Microwave Mobile Communication, 2nd ed. Piscataway, NJ: IEEE Press,1994.

[2] S. W. Kim, D. S. Ha and J. H. Reed, “Minimum selection GSC and adaptive low-powerRAKE combining scheme,” in Proc. of IEEE Int. Symp. on Circuits and Systems.

(ISCAS’03), Bangkok, Thailand, vol. 4, May 2003, pp. 357–360.[3] P. Gupta, N. Bansal and R. K. Mallik, “Analysis of minimum selection H-S/MRC in

Rayleigh fading,” IEEE Trans. Commun., vol. COM-53, no. 5, pp. 780–784, May 2005.See also the conference version in Proc. of IEEE Int. Conf. on Commun. (ICC’05).

[4] H.-C. Yang, “New results on ordered statistics and analysis of minimum-selection gen-eralized selection combining (GSC),” IEEE Trans. Wireless Commun., vol. TWC-5,no. 7, July 2006. See also the conference version in Proc. of IEEE Int. Conf. on Com-

mun. (ICC’05).[5] H.-C. Yang and M.-S. Alouini, “MRC and GSC diversity combining with an output

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[7] M. K. Simon and M.-S. Alouini, “Performance analysis of generalized selection combin-ing with threshold test per branch (T-GSC),” IEEE Trans. Veh. Technol., vol. VT-51,no. 5, pp. 1018–1029, September 2002.

[8] A. Annamalai, G. Deora and C. Tellambura, “Unified analysis of generalized selectiondiversity with normalized threshold test per branch,” in Proc. of IEEE Wireless Com-

mun. and Networking Conf. (WCNC’03), New Orleans, Louisiana, vol. 2, March 2003,pp. 752–756.

[9] X. Zhang and N. C. Beaulieu, “SER and outage of thresholdbased hybridselection/maximal-ratio combining over generalized fading channels,” IEEE Trans.

Commun., vol. 52, no. 12, pp. 2143–2153, December 2004.[10] S. Choi, H.-C. Yang and Y.-C. Ko, “Performance analysis of transmit diversity systems

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[11] S. Choi, M.-S. Alouini, K. A. Qaraqe and H.-C. Yang, “Soft handover overhead reduc-tion by RAKE reception with finger reassignment,” IEEE Trans. on Commun., vol. 56,no. 2, pp. 213–221, February 2008.

[12] H.-C. Yang, N. Belhaj and M.-S. Alouini, “Performance analysis of joint adaptive mod-ulation and diversity combining over fading channels,” IEEE Trans. on Commun., vol.COM-55, no. 3, pp. 520–528, March 2007.


Channels, 2nd ed. New York, NY: John Wiley & Sons, 2004.[14] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed.

San Diego, CA: Academic Press, 1994.[15] H.-C. Yang and L. Yang, “Exact error rate analysis of output-threshold general-

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[16] V. Tarokh, N. Seshadri and A. R. Calderbank, “Spacetime codes for high data ratewireless communication: performance criterion and code construction,” IEEE Trans.

Inf. Theory, vol. 44, no. 2, pp. 744–765, March 1998.[17] G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading

environment when using multiple antennas,” Wirel. Pers. Commun., vol. 6, no. 3,pp. 311–335, March 1998.

[18] S. M. Alamouti, “A simple transmit diversity technique for wireless communications,”IEEE J. Sel. Areas Commun., vol. 16, no. 8, pp. 1451–1458, October 1998.

[19] D. Gesbert, M. Shafi, D. S. Shiu, P. Smith and A. Naguib, “From theory to practice: anoverview of MIMO spacetime coded wireless systems,” IEEE J. Sel. Areas Commun.,vol. 21, no. 3, pp. 281–302, April 2003.

[20] A. J. Paulraj, D. A. Gore, R. U. Nabar and H. Blcskei, “an overview of MIMO commu-nications. A key to gigabit wireless,” Proc. IEEE, vol. 92, no. 2, pp. 198–218, February2004.

[21] S. Thoen, L. V. der Perre, B. Gyselinckx and M. Engels, “Performance analysis ofcombined transmit-SC/receive-MRC,” IEEE Trans. Commun., vol. 49, no. 1, p. 5–8,January 2001.


[22] Z. Chen, “Asymptotic performance of transmit antenna selection with maximal-ratiocombining for generalized selection criterion,” IEEE Commun. Lett., vol. 8, no. 4,pp. 247–249, April 2004.

[23] A. F. Molisch, M. Z. Win and J. H. Winters, “Reduced-complexity transmit/receive-diversity systems,” IEEE Trans. Signal Processing, vol. 51, no. 11, pp. 2729–2738,November 2003.

[24] A. Ghrayeb and T. M. Duman, “Performance analysis of MIMO systems with antennaselection over quasi-static fading channels,” IEEE Trans. Veh. Technol., vol. 52, no. 2,pp. 281–288, March 2003.

[25] D. A. Gore and A. J. Paulraj, “MIMO antenna subset selection with space-time cod-ing,” IEEE Trans. Signal Processing, vol. 50, no. 10, pp. 2580–2588, October 2002.

[26] D. J. Love and R. W. Heath, “Diversity performance of precoded orthogonal space-timeblock codes using limited feedback,” IEEE Commun. Lett., vol. 8, no. 5, pp. 305–307,May 2004.

[27] S. Choi, Y.-C. Ko and E. J. Powers, “Optimization of switched MIMO systems overRayleigh fading channels,” IEEE Trans. Veh. Technol., vol. 56, no. 1, pp. 103–114,January 2007.

[28] K.-H. Park, Y.-C. Ko and H.-C. Yang, “Performance analysis of transmit diversitysystems with multiple antenna replacement,” IEICE Trans. on Commun., vol. E91-B,no. 10, pp. 3281–3287, October 2008.


[30] S. Choi, M.-S. Alouini, K. A. Qaraqe and H.-C. Yang, “Fingers replacement methodfor RAKE receivers in the soft handover region,” IEEE Trans. on Wireless Commun.,vol. 7, no. 4, pp. 1152–1156, April 2008.

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[32] M.-S. Alouini and M. K. Simon, “An MGF-based performance analysis of generalizedselective combining over Rayleigh fading channels,” IEEE Trans. Commun., vol. COM-48, no. 3, pp. 401–415, March 2000.

[33] H. Holma and A. Toskala, WCDMA for UMTS, revised ed. New York, NY: John Wiley& Sons, 2001.

[34] A. J. Goldsmith and S.-G. Chua, “Adaptive coded modulation for fading channels,”IEEE Trans. Commun., vol. COM-46, no. 5, pp. 595–602, May 1998.

[35] M.-S. Alouini and A. J. Goldsmith, “Adaptive modulation over Nakagami fading chan-nels,” Kluwer J. Wireless Commun., vol. 13, nos. 1–2, pp. 119–143, 2000.

[36] K. J. Hole, H. Holm, and G. E. Oien, “Adaptive multidimentional coded modulationover flat fading channels,” IEEE J. Select. Areas Commun., vol. SAC-18, no. 7, pp.1153–1158, July 2000.

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[38] R. K. Mallik, P. Gupta and Q. T. Zhang, “Minimum selection GSC in independentRayleigh fading,” IEEE Trans. Veh. Technol., vol. TVT-54, no. 3, pp. 1013–1021, May2005.

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[39] N. Belhaj, N. Hamdi, M.-S. Alouini and A. Bouallegue, “Low-power minimum esti-mation and combining with adaptive modulation,” in Proc. of the Eighth IEEE Inter-

national Symposium on Signal Processing and its Applications (ISSPA’2005), Sydney,Australia, August 2005.

[40] ——, “Adaptive modulation and combining for bandwidth efficient communicationover fading channels,” in Proc. of the IEEE Personal Indoor Mobile Radio Conference

(PIMRC’2005), Berlin, Germany, September 2005.[41] A. S. Lioumpas, G. K. Karagiannidis and T. A. Tsiftsis, “Adaptive generalized selection

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[42] Z. Bouida, N. Belhaj, M.-S. Alouini and K. A. Qaraqe, “Minimum selection GSC withdownlink power control,” IEEE Trans. on Wireless Commun., vol. TWC-7, no. 7, pp.2492–2501, July 2008.

[43] Y.-C. Ko, H.-C. Yang, S.-S. Eom and M.-S. Alouini, “Adaptive modulation and diver-sity combining based on output-threshold MRC,” IEEE Trans. Wireless Commun.,vol. TWC-6, no. 10, pp. 3728–3737, October 2007.

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[45] A. Gjendemsjø, H.-C. Yang, G. E. Øien and M.-S. Alouini, “Joint adaptive modulationand diversity combining with downlink power control,” IEEE Trans. Veh. Technol.,vol. VT-57, no. 4, pp. 2145–2152, July 2008.

6 Multiuser scheduling

6.1 Introduction

Multiple antenna techniques can provide significant diversity benefit to wire-less systems. In certain practical scenarios, however, it might be challenging toimplement multiple antennas at the wireless terminals. In such a case, we canstill extract the diversity benefit by exploring the different fading channels cor-responding to multiple users. Since users are separately located, different userchannel will most likely experience independent fading [1, 2]. At any given timeinstant, it is highly probable that at least one user channel will have a favorablechannel condition. The overall system performance will improve if the channelaccess is always granted to the users with the best instantaneous channel qual-ity, usually the one with the highest SNR, resulting in the so-called multiuserdiversity gain.

Both multiple antenna diversity and multiuser diversity try to improve theperformance of wireless systems over fading channels. Their approaches, how-ever, are conceptually different. Antenna diversity targets at eliminating deepSNR fades by combining multiple diversity paths together, whereas multiuserdiversity rides the SNR peaks of different users channels. As such, multiuserdiversity exploits multipath fading rather than reducing it. In certain cases, fad-ing might need to be intentionally introduced with some random beamformingapproach [3]. Multiuser diversity enjoys several inherent advantages, includingsimpler receiver structures, as a single antenna per receiver is sufficient, andnaturally independent fading channels, as users are usually geographically sep-arated. On the other hand, multiuser diversity may require additional systemresources to collect user channel state information, especially for non-reciprocalchannels. Furthermore, multiuser diversity may lead to unfairness across users inthe short term, even though it can guarantee long-term fairness through channelnormalization and/or utilizing historical throughput information.

In this chapter, we will analyze the performance of different multiuser diver-sity strategies and discuss their associate implementation complexity wheneverappropriate. We first review the basics of multiuser scheduling, including thecapacity benefit, fairness issue, and feedback load reduction. We then analyzethe single user scheduling schemes for the perspective of channel access statis-tics, such as channel access rate and access duration of an arbitrary user. After

Multiuser diversity 163

BS

user 1

user 2

user K

Figure 6.1 Sample multiuser system.

that, the study is extended to the multiuser scheduling scenario. We presentthree different schemes and analyze their performance and efficiency. The chap-ter concludes with a discussion about the power reallocation for parallel multiuserscheduling. We focus mainly on those schemes that have low complexity and, assuch, can be readily applied to practical multiuser wireless systems.

6.2 Multiuser diversity

We consider a generic cell with one base station and K active single-antennausers, as illustrated in Fig. 6.1. For the sake of clarity, we assume that thebase station has only one antenna and the system operates in the time divisionmultiple access (TDMA) fashion in this chapter (the extension to the multiple-antenna base station will be considered in the next chapter). During each TDMAtime slot, the system will schedule the user with the best channel condition totransmit. With the assumption of a frequency flat-fading channel model, thequality of user channels can be characterized solely by the instantaneous receivedSNR, denoted by γk . As such, the user with the largest instantaneous receivedSNR will be selected for data transmission during the a particular time slot [1,2].Mathematically speaking, user k∗ where

k∗ = arg maxk

γk, (6.1)

will be scheduled for transmission.1 It follows that the received SNR of the sched-uled user is the largest one of K user SNRs, which are inherently independentas users are randomly distributed in the coverage area. The CDF of the received

1 Note that we employ the absolute SNR value-based scheduling strategy here, which may leadto fairness issues. An alternative normalized SNR-based scheduling will be considered in thefollowing sections.

164 Chapter 6. Multiuser scheduling

SNR over the time slot is given by [4]

Fγk ∗ (x) =K∏

k=1

Fγk(x). (6.2)

If users experience identical fading, which may be possible with a proper powercontrol mechanism, the PDF of the received SNR at the scheduled user becomes

pγk ∗ (x) = Kpγ (x)[Fγ (x)]K−1 , (6.3)

which is the PDF of combiner output SNR with K branch selection combining.Therefore, multiuser diversity can achieve the same diversity gain as selectionbased antenna reception diversity. With the statistics of the received SNR at thescheduled user, we can evaluate the performance gain offered by multiuser diver-sity transmission [5]. In what follows, we use the system ergodic capacity as theperformance metric to evaluate different scheduling strategies. Mathematically,the system ergodic capacity can be calculated by averaging the instantaneouscapacity over the distribution of the received SNR as

Csys =∫ ∞

0log2(1 + γ)pγk ∗ (γ)dγ.

For i.i.d. Rayleigh fading, after appropriate substitution and carrying out inte-gration, the capacity with multiuser diversity transmission specializes to

Csys = K log2(e)K−1∑k=0

(−1)k (K − 1)!(k + 1)!(K − k − 2)!

e(k+1)/γ E1

(k + 1

γ

), (6.4)

where E1(·) is the exponential integral function of the first order [6], defined by

E1(x) =∫ ∞

1

e−xt

tdt, x ≥ 0, (6.5)

which is related to the exponential-integral function Ei(x) by E1(x) = −Ei(−x).The multiuser diversity gain can also be demonstrated with the asymptotic

behavior of the γk ∗ when K approaches infinity. For i.i.d. Rayleigh fading sce-nario, it can be shown that γk ∗ , which is the largest one of K random variables,has the limiting distribution of the Gumbel type. More specifically, the limitingCDF of γk ∗ as K approaches infinity is given by

limK→+∞

Fγ1 :K −bK(x) = exp

(−e−x

), (6.6)

where bK is the solution of the equation Fγ (lK ) = 1 − b/K, which is equal tolog K for the Rayleigh fading case. It follows that the SNR of the selected userincreases at the same rate with log K as K approaches infinity [3].

6.2.1 Addressing fairness

In a practical environment, the average channel gain corresponding to differentusers differs with the experienced path loss and shadowing process. If the power


control mechanism is not available or not perfect, then the users experience favor-able average fading conditions that might be scheduled much more frequently,which leads to unfairness to other users. In this scenario, we can improve fair-ness among users by taking into account the historical throughput informationof each user during the user selection stage [3]. The basic principle is that usersreceiving more channel access should be weighted less during the competition forchannel access during upcoming time slots. A popular strategy that implementsthis principle is the so-called proportional fair scheduling, which was shown tobe able to achieve long-term fairness [3]. With proportional fair scheduling, theselected users over the ith time slot, denoted by k∗(i), should have the maximumnormalized instantaneous capacity, i.e.

k∗(i) = arg maxk

Ck (i)Rk (i)

, (6.7)

where Ck (i) is the instantaneous rate of user k over slot i, given by

Ck (i) = log2(1 + γk (i)), (6.8)

and Rk (i) denotes the historical throughput of user k for up to the i − 1th timeslots, which can be updated with the following relationship

Rk (s) = Rk (s − 1), k = k∗, (6.9)

Rk ∗(s) = Rk ∗(s − 1) + Ck ∗(s − 1). (6.10)

Alternatively, we can schedule users based on their normalized SNR ratherthan the absolute SNR. Specifically, as users will have different average receivedSNR, denoted by γk , we can improve the fairness among users by schedulingusers based on their normalized SNRs, given by γk = γk

γ k[5]. User k∗where

k∗ = arg maxk

γk

γk

, (6.11)

will be scheduled for transmission during a particular time slot. It is easy toshow that the PDF of the normalized SNR γk under the Rayleigh fading channelmodel is commonly given by

pγk(x) = e−x . (6.12)

To derive the PDF of the scheduled user’s SNR γ∗k based on the normalized SNR

based user selection, we first consider its CDF, which can be shown to be given


by

Fγk ∗ (γ) = Pr[γk ∗ < γ] (6.13)

=K∑

i=1

Pr[γk ∗ < γ; γk ∗ = γi ]

=K∑

i=1

∫ γγ i

0pγk

(x)K∏

k=1,k =i

Fγk(x)dx.

After taking derivatives with respect to γ, the PDF of the received SNR at thescheduled user is obtained, after some manipulations, as

pγk ∗ (γ) =K∑

i=1

1γi

pγi(

γ

γi

)K∏

k=1,k =i

Fγk(

γ

γi

). (6.14)

For the Rayleigh fading model, after proper substitution and carrying out inte-gration, the system capacity with normalized SNR-based scheduling is given by

Csys = log2(e)K∑

i=1

K−1∑k=0

(−1)k

1 + k

(K − 1

k

)e(k+1)/γ i E1

(k + 1

γi

). (6.15)

6.2.2 Feedback load reduction

The availability of users’ instantaneous channel state information is essential tothe implementation of multiuser diversity transmission. For the downlink sce-nario, the base station needs to collect the channel SNRs corresponding to allusers in order to select the best user for data transmission. This will translateinto a huge amount of channel probing and/or channel quality feedback, whichconsume additional system resource. It is of great practical importance if we canreduce the feedback load while maintaining nearly the same multiuser diversitygain. In this context, the selective multiuser diversity scheduling scheme has beendemonstrated to be an effective solution among several other approaches [5].

The basic idea of selective multiuser diversity is to allow only those userswhose channel qualities are good enough, i.e. with received SNR above a cer-tain threshold γT , to feed back their channel state information. Note that withmultiuser diversity scheduling, the base station will select a single user for trans-mission. As such, only users with a good enough channel will have the chanceto be selected. Furthermore, the base station only requires the quality of sched-uled users for capacity-achieving rate adaptation. Intuitively, we can expect theselective multiuser diversity approach can achieve the same diversity gain as con-ventional multiuser diversity if at least only one user feeds back. On the otherhand, selective multiuser scheduling may lead to scheduling outage when no userfeeds back. It is not difficult to show that the probability of scheduling outage


under the assumption of i.i.d. faded user channels is

Pout =K∏

i=1

Fγi(γT ), (6.16)

where Fγi(·) denotes the CDF of received SNR at the ith user.

An easy solution to avoid the scheduling outage is to randomly probe andselect a user for data transmission when there is no user feedback. With minormodification, we can show that the SNR of the final scheduled user is given by

γk ∗ =

maxkγk, if no outage;

randkγk, if outage.(6.17)

It follows that the CDF of the received SNR γ∗k over i.i.d. fading channels can

be shown to be

Fγk ∗ (x) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

K∑k=1

K!k!(K − k)!

[Fγ (γT )]K−k

×[Fγ (x) − Fγ (γT )]k , x ≥ γT ;

[Fγ (γT )]K−1Fγ (x), x < γT .

(6.18)

The PDF of γ∗k can be routinely obtained after taking the derivative with respect

to x as

pγk ∗ (x) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

K∑k=1

K!(k − 1)!(K − k)!

[Fγ (γT )]K−k

×[Fγ (x) − Fγ (γT )]k−1pγ (x), x ≥ γT ;

[Fγ (γT )]K−1pγ (x), x < γT ,

(6.19)

which can be readily utilized to evaluate the system capacity. The average feed-back load (AFL) in terms of the average number of SNR feedback per schedulingtime slot can be determined as

N =K∑

k=1

k Pr[k users feedback] (6.20)

=K∑

k=1

K!(K − k)!(k − 1)!

[Fγ (γT )]K−k [1 − Fγ (γT )]k .

When the user channels are independent but not identically distributed, wecan utilize normalized SNR during feedback thresholding as well as the userselection process. In particular, user k will feed back if its normalized SNR γk =γk/γk is greater than a particular threshold γT . It follows that the probabilityof scheduling outage is given by

Pout = [Fγ (γT )]K , (6.21)

where Fγ (·) is the common CDF of the normalized SNR. The average feedbackload in terms of the average number of SNR feedback per scheduling time slot


can be determined as

N =K∑

k=1

K!(K − k)!(k − 1)!

[Fγ (γT )]K−k [1 − Fγ (γT )]k . (6.22)

The statistics of the received SNR at the scheduled user can also be obtained,and this is left as an exercise for the readers.

6.3 Performance analysis of multiuser selection diversity

In general, the viability of these multiuser scheduling schemes in practical sys-tems largely depends on the number of active users and the channel changingrate [1]. While having more users increases the multiuser diversity gain, the aver-age length of time that each individual user is picked to communicate decreases.On the other hand, if the channel varies too quickly, an accurate estimate ofthe channel strength would be difficult. However, the channel fading must befast enough so that the average time that any user accesses the channel is nottoo long to ensure a certain fairness among all users. Clearly, a key measure toevaluate the viability of the multiuser scheduling algorithms is to determine theaverage channel access time based on the fading rate and the number of users.In this section, we study the average access time (AAT) and the average accessrate (AAR) of individual users in a multiuser environment subject to the ergodicRayleigh fading [7]. From a practical perspective, the AAT can be used for set-ting the time-slot length, as if the time slot is longer compared to the AAT,the scheduler basically cannot track the channel variation fast enough and thescheduling gain will be seriously reduced. On the other hand, if the time slot istoo short compared to the AAT, there will be too much unnecessary feedback.We also introduce another quantity, which is average waiting time (AWT), toindicate how long on average a user has to wait for the next access. The AWTis important for the time-out timer consideration in the upper-layer protocols.

6.3.1 Absolute SNR-based scheduling

With absolute SNR-based scheduling, the scheduler selects user i in a time slotif and only if the instantaneous SNR of the user i is larger than that of all otherusers, i.e.

γi ≥ γj , j = 1, ..., L; j = i. (6.23)

Equivalently, we can write

γi ≥ γ∗, (6.24)

where γ∗ is the maximum SNR among all other users, i.e.

γ∗ = maxj=1,...,L,j =i

(γj ). (6.25)

Performance analysis of multiuser selection diversity 169

Therefore, the AAR of user i is precisely the average number of times the processr =

√γi/γ∗ = αi/α∗ crosses level 1 per unit time, where αi is the amplitude of

the complex channel gain for user i and α∗ is the largest amplitude of all otherusers. It follows that the AAR of user i can be evaluated as the average levelcrossing rate (LCR) [8, eq. (2.90)] of the process r at level 1, which is given by

Ni =∫ ∞

0rpr,r (1, r)dr, (6.26)

where r is the time derivative of the process r and pr,r (1, r) is the joint PDF ofr and r, which was given by [9, eq. (9); 10, eq. (15)]

pr,r (1, r) =∫ ∞

0

∫ ∞

−∞α2∗pαi

(α∗r)pαi(rα∗ + α∗r) (6.27)

×pα∗,α∗(α∗, α∗)dα∗dα∗.

For Rayleigh fading user channels, the PDF of the channel amplitude αi is givenby

pαi(α) =

2α

Ωiexp

(−α2

Ωi

), α ≥ 0, (6.28)

where Ωi is the short-term average channel power gain of the ith user, andthe time derivative of signal amplitude process αi follows normal distributionwith zero mean and is independent of the signal amplitude αi , with PDF givenby [11, eq. (1.3–34)]

pαi(α) =

1√2πσi

exp(− α2

2σ2i

), (6.29)

where σ2i = Ωiπ

2f 2i for isotropic scattering and fi is the maximum Doppler fre-

quency shift of the ith user, and finally, the joint PDF of α∗ and α∗ is givenby [12]

pα∗,α∗(x, x) =L∑

j=1,j =i

1√2πσj

exp

(− x2

2σ2j

)2x

Ωjexp

(− x2

Ωj

)(6.30)

×L∏

k=1,k =i,j

[1 − exp

(− x2

Ωk

)]

=L∑

j=1,j =i

∑τ ∈T L

i j

sign(τ)1√

2πσj

exp

(− x2

2σ2j

)2x

Ωjexp

(− x2

Ωj− τx2

),

where TLij is the set obtained by expanding the product∏L

k=1,k =i,j

[1 − exp

(− x2

Ωk

)]then taking the natural logarithm of each

term [13, 14], and sign(τ) is the corresponding sign of each term in the expan-sion. After proper substitution and carrying out integrations, we can obtain the


analytical expression of AAR for user i as

Ni =L∑

j=1,j =i

∑τ ∈T L

i j

sign(τ)π√2

√Ωif 2

i + Ωj f 2j

ΩiΩj

(1Ωi

+1Ωj

+ τ

)−3/2

. (6.31)

The AAT of user i, denote by Ti , is defined as the average time duration ofuser i’s channel access. It can be shown that the AAT of user i can be calculatedas

Ti =Pi

Ni(6.32)

where Pi denotes the probability that user i accesses the channel at any timeinstant or, equivalently, the average access probability (AAP) of user i. Theprobability Pi can be calculated as

Pi = Pr[αi ≥ α∗] =∫ ∞

0pαi

(x)∫ x

0pα∗(y)dydx. (6.33)

For the Rayleigh fading case under consideration, we can show after propersubstitution and manipulation that the access probability is given by

Pi =∫ ∞

0

2x

Ωiexp

(−x2

Ωi

) L∏k=1,k =i

[1 − exp

(− x2

Ωk

)]dx (6.34)

=∑

τ ′∈T Li

sign(τ)1

1 + τ ′Ωi.

Similarly, the AWT of user i, which characterizes the average time duration auser has to wait for the next access, can be calculated as

Wi =1 − Pi

Ni=

1Ni

− Ti. (6.35)

6.3.2 Normalized SNR-based scheduling

With normalized SNR-based scheduling, the base station schedules the user withthe largest normalized SNR, or equavalently, the user with largest normalizedchannel amplitude, defined as βk = αk/

√Ωk . The AAR and AAT of user i in

this case can be calculated using the up-crossing rate of the process

r = βi/β∗, (6.36)

where

β∗ = maxj=1,··· ,L,j =i

(βj ), (6.37)

at the level 1. The rate can be similarly calculated as the absolute SNR-basedcase but using the PDFs of βs and βs. As a result of normalization, the PDF of

Multiuser parallel scheduling 171

βk becomes

pβk(x) = 2x exp(−x2), (6.38)

the PDF of its time derivative βk is

pβk(x) =

1√2πσk

exp(− x2

2σ2k

), (6.39)

where σ2k = π2f 2

k , and the joint PDF of β∗ and β∗ becomes

pβ∗,β∗(x, x) =

L∑j=1,j =i

2x√2πσj

exp

(− x2

2σ2j

− x2

)[1 − exp(−x2)]L−2 . (6.40)

Following the exact same steps as in the absolute SNR-based scheduling case,we get the AAR of user i, Ni , as

Ni =L∑

j=1,j =i

π√2

√f 2

i + f 2j

L−2∑n=0

(L − 2

n

)(−1)n (n + 2)−3/2 . (6.41)

With normalized SNR-based scheduling, the access probability of user i

becomes 1/L. It follows that the AAT of user i is given by

Ti =1

NiL. (6.42)

After proper substitution and some rearrangement, we can rewrite Ti as

Ti =const∑L

j=1,j =i

√f 2

i + f 2j

, (6.43)

where const denotes some constant. We see that if the network has a fast-movinguser k among many other stationary users, we have

Ti =

const

(L−1)fk, i = k;

constfk

, i = k,(6.44)

which implies that the AAT of the fast-moving user is roughly 1/(L – 1) timesthat of other stationary users.

6.4 Multiuser parallel scheduling

In some wireless networks, we might need to schedule multiple users at thesame time. This applies to, for example, the TDMA systems where multipletime slots within channel coherence time Tc are to be allocated to users, or theFDMA systems where multiple frequency subchannels with one channel coher-ence bandwidth Bc are to be allocated. Parallel orthogonal channel access isalso feasible in wideband CDMA systems or ultra-wideband (UWB) systems. In


this scenario, we need efficient scheduling algorithms to carry out the user selec-tion. While there exist some optimization-based approaches in the literature,they often require the solution of a multi-dimensional nonlinear optimizationproblem, which make them impractical in a real-world environment [15]. In thissection, we present several low-complexity schemes and investigate their perfor-mance and complexity.

Note that all the user selection schemes can be implemented based on boththe absolute SNRs or the normalized SNRs of user channels. In this section,for the sake of clarity, we focus on the absolute SNR-based approach with theassumption that with proper power control algorithm, the received SNRs atdifferent users experience independent and identical fading. The generalizationto non-identical fading is straightforward. In the following discussion, we alsoassume the multiuser channels are orthogonal, in either time, frequency, or codedomain, and their mutual interference and external interference are ignored.

6.4.1 Generalized selection multiuser scheduling (GSMuS)

GSMuS [16] was originated from the conventional GSC diversity systems. Withthis scheme, the system schedules a fixed number of the best users among allK active users in each scheduling interval. Specifically, the scheduler ranksthe instantaneous SNRs of all K users in the descending order, denoted byγ1:K ≥ γ2:K ≥ · · · ≥ γK :K , where γk :K is the kth largest user SNR. The schedulerwill then choose the Ks users with the largest SNRs for simultaneous transmis-sion in the next available time slots. We can immediately see that similar to theconventional multiuser diversity scheme, GSMuS requires the channel informa-tion feedback from all users. The analysis of GSMuS scheme is simpler than thatfor GSC diversity systems as we only need the marginal statistics of the first Ks

largest SNR, instead of their sum. Specifically, when user channels experiencei.i.d. fading, the PDF of the kth largest user SNR can be easily obtained fromthe classical order statistical result as

pγk :K (x) =K!

(K − k)!(k − 1)![Pγ (x)]K−k [1 − Pγ (x)]k−1pγ (x), (6.45)

where Pγ (x) and pγ (x) are the common CDF and PDF of user SNRs, availablefor popular fading channel models.

For the general independent and non-identical distributed (i.n.d.) fading chan-nel case, the MGF of the kth largest user SNR, denoted by Mγk :K (s), can beobtained as [17, eq. (6)]

Mγk :K (s) =∑

n1 ,··· ,nk −1n1 <n2 < ···<nk −1

∑nk

∫ ∞

0e−sxpnk

(x) (6.46)

[k−1∏l=1

[1 − Fnl(x)]

][K∏

l ′=k+1

Fn ′l(x)

]dx,


where ni ∈ 1, 2, · · · ,K, i = 1, · · · ,Ks , are the index of the ith best user, fnk(x)

is the PDF of the kth best user SNR, and Fnl(x) is the CDF of the lth largest

user SNR. Note that summation∑

n1 ,··· ,nk −1n1 <n2 < ···<nk −1

is carrying over all possible

index sets of the largest k − 1 user SNRs out of the total K users,∑

nkover the

possible indexes of the kth selected user and there are total K !(K−k)!(k−1)! terms

in the double summations. It follows that the PDF of the Kth largest user SNRis given by

pγk :K (s) =∑

n1 ,··· ,nk −1n1 <n2 < ···<nk −1

∑nk

pnk(x)

[k−1∏l=1

[1 − Fnl(x)]

][K∏

l ′=k+1

Fn ′l(x)

], (6.47)

which reduces to (6.45) with the i.i.d. assumption.With the above PDFs of the scheduled users’ SNR, we can readily evaluate the

throughput and error rate performance of GSMuS scheme. We can also calculatethe channel access statistics of an arbitrary user, including the AAR and theAAT defined in the previous section [16]. In particular, the AAR of user k canbe calculated as the up-crossing rate at level 1 of the process rk = αk/α∗, whereαk is the channel amplitude of user k and α∗ = αKs :K−1 is the Ksth largestchannel amplitude among the other K − 1 users (excluding user k). Note thatwith GSMuS, user k will be selected whenever its channel amplitude becomesone of the Ks largest among all users. Applying the result of LCR, the AAR ofuser k can given by

Nk =∫ ∞

0rprk ,rk

(1, r)dr, (6.48)

where rk is the time derivative of the process rk and frk ,rk(1, r) is the joint PDF

of rk and rk , which can be calculated in terms of the PDF of αk , αk , and thejoint PDF of α∗ and α∗ = αKs :K−1 as in (6.27). Assuming that the αKs :K−1 andαKs :K−1 are independent, it can be shown that the joint PDF of fα∗,α∗(x, x) isgiven by

pα∗,α∗(x, x) =∑

n1 ,··· ,nK s −1n1 <n2 < ···<nK s −1

∑nK s

pαn K s(x)

[Ks −1∏l=1

[1 − Fαn l(x)]

](6.49)

×[

K∏l ′=Ks +1

Fαn ′l

(x)

]pαn K s

(x),

where ni ∈ 1, · · · , k − 1, k + 1, · · · ,K (ni = k), i = 1, · · · ,Ks , are the indexof the ith best user SNR among K − 1 users. Note that the summation∑

n1 ,··· ,nK s −1n1 <n2 < ···<nK s −1

is carried over all possible index sets of the largest Ks − 1

branch SNRs and∑

nK sover the possible indexes of the Ksth user, out of the

total K − 1 users, except for the kth user. For the i.n.d. Rayleigh fading userchannel scenario, after proper substitutions and carrying out the integration, the


AAR of user k can be shown to be given by

Nk =∑

n1 ,··· ,nK s −1n1 <n2 < ···<nK s −1

∑nK s

∑τ ∈T K −1

1 ···K s

sign(τ)π√2

(6.50)

×

√Ωkf 2

k + ΩnK sf 2

nK s

ΩkΩnK s

(1

Ωk+

Ks∑l=1

1Ωnl

+ τ

)−3/2

,

where TK−11···Ks

denote the set obtained by expanding the product∏K−1l ′=Ks +1

[1 − exp

(− x2

Ωn ′l

)]then taking the natural logarithm of each

term [13, 14], and sign(τ) is the corresponding sign of each term in theexpansion.

Based on the above notation, the AAP of user k can be calculated as itschannel amplitude αk becomes larger than the Ksth largest channel ampli-tude among the other K − 1 users α∗ = αKs :K−1 . Mathematically speaking, wehave

Pk =∫ ∞

0pαk

(x)∫ x

0pαK s :K −1 (y)dydx. (6.51)

After substituting (6.28) and (6.47) and carrying out integration and manipula-tion, we can obtain the closed-form expression of the AAP of user k for Rayleighfading case as

Pk =∑

n1 ,··· ,nK s −1n1 <n2 < ···<nK s −1

∑nK s

∑τ ′∈T K −1

1 ···K s

sign(τ ′)Ωk

ΩnK s(1 + (τ ′ +

∑Ks −1l=1

1Ωn l

)Ωk ).

(6.52)The AAT of user k then can be readily calculated as the ratio of the AAP overAAR, i.e. Tk = Pk/Nk .

6.4.2 On–off based scheduling (OOBS)

The OOBS scheme is inspired by the AT-GSC diversity combining scheme [18]and the on–off type of schemes discussed in [5, 19]. The difference with the AT-GSC scheme is that the OOBS scheme will process the selected users in parallelwithout combining them together [20]. The mode of operation of the OOBSscheme can be summarized as follows. At the beginning of each scheduling timeinterval, the BS sends a pilot signal to all users. These users will estimate thereceived SNR γi using the received signal and compare it to a preselected thresh-old SNR, denoted by γT . A user will feed back its SNR information to the BSonly if its SNR are above the threshold γT . Only these acceptable users are thenscheduled by the BS for the subsequent transmission time slot. If no user has anacceptable SNR, the BS simply waits a time period, in the order of the channelcoherence time, before starting a new round of scheduling. Note that with the


GSMuS scheme, the BS needs always to collect the SNR feedbacks from all K

users to determine the Ks best users for scheduling.We can see that the OOBS scheme will lead to a random time-varying feedback

load and the number of scheduled user is also randomly varying. As only the userswith acceptable channel condition, i.e. the users with SNR greater than the SNRthreshold γT , will feed back their channel SNRs, the AFL of the OOBS schemeover i.i.d. fading environment is the same as that of the SMuS scheme discussedin the previous section, which is given by

AFL =K∑

k=0

k

(K

k

)[1 − Fγ (γT )]k [Fγ (γT )]K−k

= K [1 − Fγ (γT )] . (6.53)

For the i.i.d. Rayleigh fading case, (6.53) specializes to, after substituting in thecommon CDF of Rayleigh faded received SNR,

N = K exp(−γT

γ

). (6.54)

Therefore, the OOBS scheme requires, on average, fewer feedbacks than theGSMuS scheme. Note that the average number of scheduled user with the OOBSscheme is equal to the number of acceptable users. In a practical system, thenumber of users that can be scheduled for simultaneous transmission may belimited by other constraints.

We now evaluate the performance of the OOBS scheme over an i.i.d. fadingenvironment in terms of the more practical performance measures of averagespectrum efficiency (ASE). Based on the mode of operation of the OOBS scheme,the received SNR of scheduled users has a truncated PDF from above at γT . Itcan be shown that the PDF of the received SNR of a scheduled user is given by

pγs(γ) =

pγ (γ )

1−Fγ (γT ) γ ≥ γT ;

0 otherwise,(6.55)

where pγ (·) and Fγ (·) are the common PDF and CDF of user SNRs. For i.i.d.the Rayleigh fading special case, (6.55) specializes to

pγs(γ) =

⎧⎨⎩

1γ exp

(− γ−γT

γ

), γ ≥ γT ;

0 otherwise,(6.56)

where γ are the common average SNR for all users.We assume the constant-power variable-rate adaptive M-QAM scheme dis-

cussed in earlier chapters. The threshold values γTnfor different constellation

sizes and instantaneous BER requirements were summarized in Table 2.1. The


ASE of a scheduled user with the OOBS scheme can be calculated as [21, eq. (33)]

ASEu =N∑

n=1

RnPn , (6.57)

where RnNn=1 are the spectral efficiencies of N available constellation sizes and

Pn is the probability of using the nth constellation, which can be obtained as

Pn =∫ γT n + 1

γT n

pγs(γ) dγ. (6.58)

The ASE of all scheduled users can then be determined by multiplying theaverage number of scheduled users to the ASE of a single scheduled user, as

ASEt = [K (1 − Pγs(γT ))]

N∑n=1

Rn

∫ γT n + 1

γT n

pγs(γ) dγ. (6.59)

Inserting (6.55) into (6.59), we have

ASEt = KN∑

n=1

Rn

∫ γT n + 1

γT n

pγ (γ) U (γ − γT ) dγ. (6.60)

For the i.i.d. Rayleigh fading assumption, inserting (6.56) into (6.60) andcarrying out integration, we can write

ASEt = KN∑

n=1

RnP ′n , (6.61)

where

P ′n =

⎧⎨⎩exp

(−max(γT ,γT n )

γ

)− exp

(− γT n + 1

γ

)if γTn + 1 ≥ γT ;

0 if γTn + 1 < γT .(6.62)

If the SNR threshold γT = γTqwhere q ∈ 1, 2, 3, . . . , N, then Rnq−1

n=1 = 0and (6.61) simplifies to

ASEt = K

[N −1∑n=q

Rn

exp

(−γTn

γ

)− exp

(−γTn + 1

γ

)

+RN exp(−γTN

γ

)]. (6.63)

6.4.3 Switched-based scheduling (SBS)

The SBS scheme is inspired by the switching-based diversity combiningschemes [22,23]. With the SBS scheme, the BS schedules a predetermined fixedtotal number of acceptable users, denoted by Ks , and if necessary, the best unac-ceptable users in a GSC fashion [20]. More specifically, at the beginning of each


scheduling period, the BS sends a pilot signal sequentially to each user in orderto request a channel state information from each user in a sequential manner.After receiving the pilot signal from the BS, each user estimates its SNR γi andfeeds it back to the BS. The BS will compare each received SNR with the pres-elected SNR threshold γT . If the user SNR γi is greater than γT , then the userchannel is considered acceptable and the user will be scheduled for transmission.This process of estimation, feedback, and comparison is repeated until the BSfinds Ks acceptable users, after which the data transmission will start withoutfurther probing. If the BS only finds Ka < Ks acceptable users after probing allK user channels, the BS will rank the K − Ka users with unacceptable channelsand schedules the best Ks − Ka unacceptable users along the already selectedKa acceptable users. As the SBS scheme may not always select the best users fortransmission, its performance will be worse than the GSC-based GSMuS schedul-ing scheme, especially when the threshold is relative small in comparison withaverage SNR. On the other hand, the SBS scheme can approach the performanceof the GSMuS scheme as γT increases. Note that that when γT goes to infinity,the SBS scheme becomes equivalent to the GSMuS scheme. The advantage ofthe SBS scheme is a lower feedback load. The GSMuS scheme needs always fullK feedbacks during each scheduling period. The feedback with the SBS schemevaries between Ks and K.

In the following, we derive the analytical expressions of the AFL and the ASEwith the SBS scheme over i.i.d. fading environment. This study allows for athorough trade-off study between SBS and GSMuS schemes.

To calculate the AFL, we consider two mutually exclusive cases depending onwhether there are Ks acceptable users or not. If there are less than Ks acceptableusers among the total K users, the SBS scheme needs to request SNR feedbacksfrom all K users. As a result, the number of feedback becomes K and the numberof unacceptable users is random and takes values from (K − Ks + 1) to K andfollows a binomial distribution. Since the diversity paths are assumed to be i.i.d.faded, the probability that there are less than Ks acceptable users can be easilywritten as

PB =K∑

l=K−Ks +1

(K

l

)[Fγ (γT )]l [1 − Fγ (γT )]K−l . (6.64)

When there are at least Ks acceptable users among the total K users, the SBSscheme will terminate the process of SNR feedback as soon as Ks acceptableusers are found. In this case, the number of feedbacks takes values from Ks toK and follows a negative binomial (or Pascal) distribution, with the probabilityof K feedbacks given by

Pk =(

k − 1k − Ks

)[1 − Fγ (γT )]Ks [Fγ (γT )]k−Ks , k = Ks, . . . ,K. (6.65)


Finally, by combining these two mutually exclusive cases, we can obtain theoverall average amount of feedback during the guard period as

AFL =K∑

k=Ks

k Pk + KPB

=K∑

k=Ks

k

(k − 1

k − Ks

)[1 − Fγ (γT )]Ks [Fγ (γT )]k−Ks

+ K

K∑l=K−Ks +1

(K

l

)[Fγ (γT )]l [1 − Fγ (γT )]K−l . (6.66)

For the i.i.d. Rayleigh fading case, after replacing Fγ (γ) with 1 − exp(− γ

γ

), we

obtain the following closed-form expression for AFL

AFL =K∑

k=Ks

k

(k − 1

k − Ks

)[exp

(−γT

γ

)]Ks[1 − exp

(−γT

γ

)]k−Ks

+ KK∑

l=K−Ks +1

(K

l

)[1 − exp

(−γT

γ

)]l [exp

(−γT

γ

)]K−l

. (6.67)

To calculate the ASE, we now derive the statistics of the received SNR of ascheduled user. Considering the cases that the number of acceptable users Ka isgreater or equal to Ks and that Ka is smaller than Ks separately, we can writethe PDF of the received SNR of a scheduled user with the SBS scheme as

pγs(γ) = Pr [Ka ≥ Ks ] pγs 1

(γ) + Pr [Ka < Ks ] pγs 2(γ) , (6.68)

where pγs 1(γ) is the PDF of a scheduled user when Ka ≥ Ks , the probability of

which is given by

Pr [Ka ≥ Ks ] =K∑

Ka =Ks

(K

Ka

)[1 − Fγ (γT )]Ka [Fγ (γT )]K−Ka , (6.69)

and pγs 2(γ) is the PDF of a scheduled user in case of Ka < Ks , the probability

of which is equal to

Pr [Ka < Ks ] =Ks −1∑Ka =0

(K

Ka

)[1 − Fγ (γT )]Ka [Fγ (γT )]K−Ka . (6.70)

In the case of Ka ≥ Ks , i.e. all scheduled users have acceptable SNR, the PDF ofthe received SNR at a scheduled user is the truncated version of standard fadingdistribution from below at γT , which is given by [22, eq. (5)]

pγs 1(γ) =

pγ (γ)1 − Fγ (γT )

U (γ − γT ) , (6.71)

where U (x) is the unit step function. For the case of Ka < Ks , we first notethat with the SBS scheme, Ka < Ks holds if and only if the Ksth largest user


SNR is below the threshold and, as such, some unacceptable users are scheduled.Therefore, the PDF of a scheduled user’s SNR is the PDF of any one of the Ks

best user SNRs, given the fact that the Ksth best user SNR is below threshold.Since the scheduled user has equal probability to be the first to the Ks best user,the PDF of a scheduled user can be written as

pγs 2(γ) =

1Ks

(Ks −1∑i=1

pγi :K |γK s :K <γT(γ) +

pγK s :K (γ)FγK s :K (γT )

(1 − U (γ − γT ))

),

(6.72)

where pγi :K |γK s :K < γ T(γ) is the conditional PDF of the ith (i < Ks) best user SNR

given the Ksth best user SNR is below the threshold γT , which can be calculatedusing the joint PDF of γi:K and γKs :K as

pγi :K |γK s :K <γT(x) =

1∫ γT

0 pγK s :K (γ )dγ

∫ min(γT ,x)

0pγi :K ,γK s :K (x, y) dy, (6.73)

where pγi :K (γ) is the PDF of the ith largest user SNR and pγi :K ,γK s :K (x, y) isthe joint PDF of the ithand Ksth largest user SNRs, both of which have beendiscussed in earlier chapters but reproduced here for the reader’s convenience.

pγi :K ,γK s :K (x, y) =K!

(K − Ks)!(Ks − i − 1)! (i − 1)!

× [Fγ (y)]K−Ks [Fγ (x) − Fγ (y)]KS −i−1

× [1 − Fγ (x)]i−1 pγ (x)pγ (y), (6.74)

x > y and 1 ≤ i < Ks ≤ K

pγj :K (γ) =(

K

j

)j · pγ (γ) [1 − Fγ (γ)]j−1 [Fγ (γ)]K−j . (6.75)

Note that the second term in the parenthesis of (6.72) corresponding to the caseof i = Ks , i.e. the PDF of the Ksth largest user SNR given that it is smallerthan the threshold.

Finally, after proper substitution, the PDF of the received SNR at a scheduleduser with the SBS scheme can be obtained as

pγs(γ) = Pr [Ka < Ks ]

1Ks

(Ks −1∑i=1

pγi :K s |γK s :K <γT(γ)

+pγK s :K (γ)FγK s :K (γT )

(1 − U (γ − γT ))

)

+ Pr [Ka ≥ Ks ]pγ (γ)

1 − Fγ (γT )U (γ − γT ) , (6.76)

which can be utilized to evalulate the ASE of the SBS scheme. Specifically, theASE of a scheduled user can be calculated, after applying (6.76) to the definition


in (6.57) as

ASEu =N∑

n=1

Rn

∫ γT n + 1

γT n

pγs(γ) dγ. (6.77)

Note that SBS scheme always schedules Ks users. Therefore, the total systemASE can be simply calculated by adding a factor of Ks , i.e. ASEt = KsASEu .


In this subsection, we discuss the trade-off involved in the three multiuser parallelscheduling schemes presented in previous sections through selected numericalexamples. Note that the number of scheduled users with the OOBS scheme israndom varying, whereas both GSMuS and SBS schemes schedule a fixed numberof Ks users. To allow a fair comparison between these scheduling schemes, wemust define an appropriate basis for this comparison. In particular, we select thefeedback threshold γT for the OOBS scheme such that the average number ofscheduled users with the OOBS scheme is equal to Ks . For the i.i.d. Rayleighfading scenario of interest, starting from (6.53), the threshold values satisfyingthis constraint can be calculated by solving the following equation for γ∗

T

Ks = K exp(−γ∗

T

γ

), (6.78)

which leads to

γ∗T = γ ln

(K

Ks

). (6.79)

Note that γ∗T is the equivalent SNR threshold of the OOBS scheme and is varying

with γ.In Fig. 6.2, we present the AFL of the OOBS scheme, the SBS scheme, and the

GSMuS scheme over i.i.d. Rayleigh fading conditions with K = 5 and Ks = 3.Specifically, we plot the AFL of three schemes with different threshold values asthe function of the average SNR γ. We first note that the AFL of the SBS schemeis decreasing from K and eventually converging to Ks as γ increases while thatof the GSC-based scheme is always equal to K. We can also see that the AFLof the SBS scheme increases as γT increases for a fixed γ. On the other hand,for fixed γT , the AFL of the OOBS scheme is increasing from 0 to K, which isthat of the GSC-based scheme, as γ increases. For fixed γ, the AFL of the OOBSscheme is decreasing as γT increases because the feedback load is the same as thenumber of scheduled users in the OOBS scheme and the number of scheduledusers is decreasing as γT increases. Finally, with the equivalent threshold γ∗

T , theAFL of the OOBS scheme remains constant and varies with γ.

Figure 6.3 presents the ASE of the OOBS scheme, the SBS scheme, and theGSMuS scheme with adaptive coded M-QAM modulation as a function of average


GSMuS scheme

SBS scheme with

29 dB

OOBS scheme with equivalent

T

Figure 6.2 Average feedback loads for the (i) OOBS scheme, (ii) SBS scheme, and(iii) GSMuS scheme (K = 5 and Ks = 3) [20]. c© 2009 IEEE.

SNR γ over the i.i.d. Rayleigh fading condition with K = 5 and Ks = 3. We cansee that, with the fixed γT , the ASE of the OOBS scheme is increasing andeventually converging to K · RN as γ increases because with fixed γT , (i) as γ

increases the number of scheduled users increases and eventually converges toK, (ii) the total data rate depends on the number of scheduled users and eachdata rate depends on its SNR. We also notice that the ASE of the OOBS schemewith small γT is converging faster than that with large γT . When the channelcondition is good, the ASE of the OOBS scheme is much larger than that of theGSMuS scheme because the number of scheduled users with the OOBS scheme isconverging to K while the number of scheduled users with the GSMuS scheme isalways Ks . By applying the equivalent threshold γ∗

T in the OOBS scheme, whenthe channel condition is poor, the OOBS scheme has almost the same ASE as theGSMuS scheme and the ASE of the OOBS scheme is slightly higher than thatof the GSMuS scheme from 5 to 35 dB. For high average SNR (above 35 dB),the ASE of the OOBS scheme is re-converging to that of the GSMuS scheme,Ks · RN . Comparing the ASE between the SBS scheme and the GSMuS scheme,we can see that for the fixed γT , when the channel condition is poor, the SBSscheme has almost the same ASE as the GSMuS scheme and as γ increases, theGSMuS scheme has a slightly better ASE and eventually these two schemes havethe same ASE again.


T*

GSMuS scheme

SBS scheme with T 29 dB

SBS scheme with

T 7.1 dB

OOBS scheme with T

T 29 dB

Figure 6.3 Average spectral efficiencies of adaptive coded M -QAM modulation for the(i) OOBS scheme, (ii) SBS scheme, and (iii) GSC-based scheduling scheme over i.i.d.Rayleigh fading conditions with K = 5 and Ks = 3 [20]. c© 2009 IEEE.

6.5 Power allocation for SBS

So far, we assume that the transmit powers are uniformly allocated to the sched-uled users, irrespective of their instantaneous channel conditions. The perfor-mance of the multiuser parallel user scheduling scheme studied in a previous sec-tion can be further improved with power allocation among scheduled users. Powerallocation for GSMuD scheme was investigated by Ma et al. in [24]. Specifically,optimal power allocation strategies based on one- or two-dimensional water-fillingsolution are derived and demonstrated to achieve a better sum-rate performancethan equal power allocation. In this section, we focus on the transmit powerallocation for selected users with the SBS scheme [25].

Based on the mode of operation of the SBS scheme, when there are not enoughacceptable users, at least one user will have an SNR smaller than the SNR thresh-old, which will limit the performance of the SBS scheme. One way to reduceor eliminate the number of scheduled unacceptable users is to redistribute thetransmission power to different scheduled users while satisfying the total trans-mitting power constraint. Note that because of the discrete nature of adaptivemodulation, such power redistribution among scheduled users may not affect theachieved ASE of those acceptable users, as the same modulation mode can beused for all values of the received SNRs in the same interval. With the above

Power allocation for SBS 183

motivation in mind, we propose in this section several threshold-based powerallocation algorithms for the SBS multiuser parallel scheduling scheme.

6.5.1 Power reallocation algorithms

The main purpose of the power reallocation algorithms for the SBS-basedscheduling scheme is to increase the number of the scheduled acceptable userswithout consuming any additional down-link transmit power. This is achieved byre-allocating the excess transmit power extracted from the scheduled acceptableusers to those scheduled unacceptable users. Based on this general principle, wepresent three specific power allocation algorithms. For later reference, γ

′i denotes

the SNR of the scheduled user after power allocation. Ka denotes the number ofacceptable users before power allocation, and γT denotes the preselected SNRthreshold.

Algorithm 1: ranking maintained power reallocationThis algorithm maintains the SNR ranking information of scheduled acceptableusers when extracting power from acceptable users. The algorithm extracts onlythe required transmitting power from the acceptable users and allocates it tothe unacceptable users sequentially from the strongest user to the weakest user.The operation of Algorithm 1 is summarized as follows. After SBS-based userscheduling, the system calculates the required additional transmission power for

the unacceptable users in dB scale as γreq =Ks∑

i=Ka +1(γT − γi). Then, the system

tries to extract the same amount of excess power from Ka acceptable users whileensuring the acceptance of user SNR, as

(i) If(γi − γr e q

Ka

)≥ γT , then we reduce the transmitting power for user i in

the amount of γreq /Ka . Then the resulting received SNR of user i becomesγ

′i = γi − (γreq /Ka).

(ii) If (γi − (γreq /Ka)) < γT P , then we reduce the transmitting power for user i

in the amount of (γi − γT P ). As such, the modified SNR of the acceptableuser is γ

′i = γT .

The total excess transmission power from the acceptable users is

γexcess =Ka∑i=1

(γi − γ

′

i

), (6.80)

which will be less or equal to γreq . The system will then allocate γexcess to theunacceptable users. More specifically, if γexcess = γreq , the received SNRs of allscheduled unacceptable users will be increased to γT and as such become accept-able with power reallocation. If γexcess < γreq , i.e. when case (ii) above occurs forat least one acceptable user, then γexcess is sequentially allocated to the unac-ceptable users, starting from that with the strongest SNR. With this approach,


the number of remaining unacceptable users will be the smallest after the powerreallocation process.

Algorithm 2: rate maintained power reallocationThis algorithm extracts transmit power from scheduled acceptable users whileensuring that their supported transmission rates remain unchanged. Similar toAlgorithm 1, the extracted power is allocated to unacceptable users sequentiallystarting from the strongest user. Specifically, excess SNR of the ith acceptableuser is calculated as γexcess,i = γi − γTj

, where γTjis the lower threshold for

modulation mode that ith user’s SNR can support, i.e. γTj≤ γi ≤ γTj + 1 . The

total excess transmission power from all acceptable users becomes

γexcess =Ka∑i=1

γexcess,i, (6.81)

which will be reallocated to the users, such that the number of scheduled unac-ceptable users is minimized. In particular, these excess transmission powers arethen allocated to the unacceptable users in a similar fashion as in Algorithm1. The only difference is that in this case, some excess power may remain afterpower allocation to all unacceptable users, in which case, the system equally allo-cates it to all the scheduled users. Note that Algorithm 2 has lower complexityin comparison with Algorithm 1 as the system does not need to calculate γreq .

Algorithm 3: equal SNR power reallocationWith Algorithm 3, the received SNR of all acceptable users after power realloca-tion will be the same. Specifically, the transmission power will be redistributedamong the scheduled users with the best channel conditions such that theirreceived SNRs are equal and above the threshold γT . The mode of operationof Algorithm 3 is as follows. When some unacceptable users are scheduled, i.e.Ka < Ks , the base station will first determine the number of acceptable users

after power reallocation, denoted by K ′a . In particular, if

i∑k=1

γk :K ≥ iγT buti+1∑k=1

γk :K < (i + 1)γT , then we conclude that K ′a = i and the received SNR of all

K ′a acceptable users after power reallocation is equal to 1

i

i∑k=1

γk :K . Note that

if Ka ≤ K ′a < Ks , then there will still remain Ks − K ′

a unacceptable users afterpower reallocation, whose SNR will not be changed with power reallocation. Thisalgorithm has the lowest implementation complexity among the three proposedalgorithms.

6.5.2 Performance analysis

In this section, we analyze the performance of the third power allocation algo-rithm for SBS scheduling through accurate analysis based on order statistics.


We first calculate the average numbers of scheduled acceptable users before andafter power reallocation based on Algorithm 3. The average number of scheduledacceptable users before power reallocation can be calculated as

Ka =Ks −1∑i=1

iPr[i acceptable users in total] + Ks Pr[at least Ks acceptable users].

(6.82)Using the joint PDF and marginal PDF of ordered statistics, the probabilitiesinvolved can be calculated as

Pr[i acceptable users in total] =∫ ∞

γT

∫ γT

0pγi :K ,γi + 1 :K (x, y)dydx (6.83)

and

Pr[at least Ks acceptable users] =∫ ∞

γT

pγK s :K (z)dz, (6.84)

where pγi :K ,γi + 1 :K (x, y) is the joint PDF of the ith largest and the i + 1 largestreceived SNRs, which is given by

pγi :K ,γj :K (x, y) =K!

(K − j)! (j − i − 1)! (i − 1)!

× [Fγ (y)]K−j [Fγ (x) − Fγ (y)]j−i−1 [1 − Fγ (x)]i−1

×pγ (x)pγ (y), (6.85)

x > y & 1 ≤ i < j ≤ K,

and pγK s :K (z) is the PDF of the Ksth largest received SNR, given by

pγ(K s :K ) (γ) =(

K

Ks

)Ks · pγ (γ) [1 − Fγ (γ)]Ks −1 [Fγ (γ)]K−Ks . (6.86)

Based on the mode of operation of Algorithm 3, the average number of sched-uled acceptable users after power reallocation can be calculated as

K′a =

Ks −1∑i=1

iPr

[i∑

k=1

γk :K ≥ iγT &i+1∑k=1

γk :K < (i + 1)γT

](6.87)

+Ks Pr

[Ks∑k=1

γk :K ≥ KsγT

].

In this case, the probabilities involved in the about summation can be calculated

using the joint PDF ofi∑

k=1γi:K and γi+1:K and the marginal PDF of

Ks∑k=1

γi:K ,


respectively, as

Pr

[i∑

k=1

γk :K ≥ iγT &i+1∑k=1

γk :K < (i + 1)γT

](6.88)

=∫ γT

0

∫ (i+1)γT −x

iγT

p i∑k = 1

γk :K ,γi + 1 :K

(x, y)dydx,

and

Pr

[Ks∑k=1

γk :K ≥ KsγT

]=∫ ∞

Ks γT

p K s∑k = 1

γk :K

(z)dz. (6.89)

These joint statistics of the partial sums of order statistics have been investigated

in a previous chapter. For example, the joint PDF ofj∑

i=1γi:K and γj+1:K can be

shown to be given by

pγj + 1 :K ,

j∑i = 1

γi :K

(x, y) =K!

(K − j − 1)!(j)![Fγ (x)]K−j−1 [1 − Fγ (x)]j

× pγ (x)p j∑i = 1

γ +i

(y),

x ≥ 0, y ≥ jx, (6.90)

where p j∑i = 1

γ +i

(γ) denote the PDF of the sum of j i.i.d. random variables γ+i ,

whose PDF is a truncated version of the PDF of γi on the left at x.


The average number of the scheduled acceptable users, the ASE, and the averageBER with our three proposed power allocation algorithms are investigated usingMonte Carlo computer simulations. In our simulation, we assume the SBS schemethe coded adaptive modulation of [26] with N = 8 modes and i.i.d. Rayleighfading conditions.

Figure 6.4 presents the average number of scheduled acceptable users withour proposed power allocation algorithms over i.i.d. Rayleigh fading conditionswith K = 5, Ks = 3, and γT = 7.1 dB. As we can see, after power allocationwith our proposed power allocation algorithms, the average number of scheduledacceptable users increases significantly for all three algorithms. Among the threealgorithms, the third has the best performance among our proposed algorithmsand the second has a better performance than the first.

Figure 6.5 presents the ASE of SBS with our proposed power allocation algo-rithms over i.i.d. Rayleigh fading conditions with K = 5, Ks = 3, and γT = 7.1dB. Based on their mode of operation, when the average SNR is close to theSNR threshold for power allocation, Algorithm 2 acts like Algorithm 1 and in


0 2 4 6 8 10 12 14 160

0.5

1

1.5

2

2.5

3

Average SNR [dB]

Ave

rage

Num

ber

of E

ffec

tive

Acc

epta

ble

Use

rs

After power allocation with Algorithm 3



Before power allocation

Figure 6.4 Average number of effective acceptable users with SBS over i.i.d. Rayleighfading conditions with K = 5, Ks = 3, and γT = 7.1 dB [25]. c© 2009 IEEE.

the region of average SNR, γ, higher than the SNR threshold, Algorithm 2 actslike Algorithm 3. As expected, after power allocation with Algorithm 2, the ASEwas increased because the acceptable users still have the same ASE and the ASEof the power allocated to unacceptable users was increased. On the other hand,contrary to our original expectations, after power allocation with Algorithms 1and 3, the ASE was also increased. This can be explained by the fact that basedon the statistical property of the scheduled users with SBS, at least one of theacceptable users has very large SNR compared with others among the sched-uled users and then after power allocation, most of the acceptable users can stillmaintain their SNR regions after power allocation. The ASE with Algorithm1 was increased only when the average SNR is close to the SNR threshold forpower allocation because in the region of high γ, the required SNR is decreasingand the number of acceptable users is increasing and for that reason, most ofthe acceptable users can maintain their SNR regions. The ASE with Algorithm1 was increasing only in the region of low γ. Based on the mode of operation ofSBS, all the scheduled users with SBS may have acceptable SNRs in the regionof high γ. Therefore, based on the mode of operation of Algorithm 1, the system


0 5 10 15 20 25 30 350

5

10

15

20

25

Average SNR [dB]

ASE

ofth

eSch

edule

dU

sers





Figure 6.5 ASE [bits/s/Hz] with SBS over i.i.d. Rayleigh fading conditions withK = 5, Ks = 3, and γT = 7.1 dB [25]. c© 2009 IEEE.

did not need the power allocation process in the region of high γ. The ASE withAlgorithm 2 was increasing when γ is slightly lower than the SNR threshold andfinally approaching to the ASE prior to the application of the power allocation.Similar to Algorithm 2, the ASE with Algorithm 3 is also increasing when γ isslightly lower than the SNR threshold and finally it approaches to the ASE priorto the application of power allocation.

Figure 6.6 presents the average BER of SBS with our proposed power alloca-tion algorithms over i.i.d. Rayleigh fading conditions with K = 5, Ks = 3, andγT = 7.1 dB. After power allocation with algorithm 1, the average BER perfor-mance is degraded only when the average SNR is close to the SNR thresholdfor power allocation because the excess SNRs are extracted from the acceptableusers. In the region of high γ, the extracted SNR from the acceptable users isdecreased and then the average BER is approaching the average BER prior tothe application of the power allocation. Similar to the ASE results, the averageBER of Algorithm 2 acts like the ASE of Algorithm 2 when the average SNRis close to the SNR threshold for power allocation. Considering the three algo-rithms together, it is clear that Algorithm 3 provides the best performance andthe least complexity among the three proposed algorithms.


0 5 10 15 20 25 30 3510

−6

10−5

10−4

10−3

Average SNR [dB]

Ave

rage

BE

Rofth

eSch

edule

dU

sers





Figure 6.6 Average BER with SBS over i.i.d. Rayleigh fading conditions with K = 5,Ks = 3, and γT = 7.1 dB. c© 2009 IEEE.

6.6 Summary

In this chapter, we addressed the design and analysis of user scheduling schemesfor wireless communication systems. We focused on those schemes that havelow implementation complexity. Both single-user scheduling and multiple userscheduling were considered. For single-user schemes, we quantified their perfor-mance, not only from the conventional capacity benefit perspective but also interms of new metrics, such as channel access time and rate. The multiple userschemes assumed that users were competing for one of many channel resourceunits with common instantaneous quality to the same user. The performanceand complexity of each scheme were accurately quantified. Finally, we presentedand analyzed several practical power allocation strategies for one of the repre-sentative multiuser scheduling scheme.


A multiuser scheduling scheme based on switched diversity was studied in [27–29]. The performance of multiuser scheduling in a multi-antenna scenario with


different diversity combining scheme was analyzed in [30]. Feedback load reduc-tion strategies for multiuser diversity systems were also considered in [31–34]among many others. Ma and Tepedelenlioglu quantified the effect of outdatedfeedback on the performance of multiuser diversity systems in [35]. The jointdesign of a multiuser scheduling scheme with adaptive diversity combining strat-egy is considered in [36,37].

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7 Multiuser MIMO systems

7.1 Introduction

Multiple-antenna transmission and reception (i.e. MIMO) techniques can con-siderably improve the performance and/or efficiency of wireless communicationsystems. First introduced in mid 1990s [1, 2], MIMO technique has been anarea of active research and has found applications in various emerging wire-less systems. Most of early MIMO designs focus on point-to-point link whereboth transmitter and receiver have multiple antennas and demonstrate the hugepotential of MIMO techniques in terms of providing array gain, spatial diver-sity gain, spatial multiplexing gain and interference reduction capability, amongmany others [3–5]. Meanwhile, most mobile receivers will possess less antennasthan the base stations in the near future due to their size/cost constraints. Insuch scenarios, the capacity of a point-to-point link between the base station anda mobile will be limited by the number of antennas at the mobile. On the otherhand, if we consider the antennas of different receivers together, a virtual MIMOsystem is formed with huge capacity potential [6–10]. The design and analysisof efficient transmission strategy for resulting multiuser MIMO systems is thesubject of this chapter.

A rich literature on MIMO wireless communications already exists. There hasalready been a rich literature on MIMO wireless communications. Several bookshave been published on this general subject (see for example [11,12]). This chap-ter complements existing literature on MIMO wireless communications by focus-ing on the different multiuser scheduling schemes for multiuser MIMO systems.In general, the downlink transmission from the base station to mobile receivers ismore challenging in the multiuser MIMO system, as mobile users are randomlydistributed and cannot perform joint detection. In this context, the key idea tofully exploit the potential spatial multiplexing gain, offered by multiple antennaeat the base station and at different users, is to transmit simultaneously to multi-ple properly selected mobile receivers. We will examine several practical designsand evaluate their particular performance versus complexity trade-off throughaccurate statistical analysis. Note that most previous works on multiuser MIMOsystems have relied on asymptotic analysis, which has limited applicability inreal-world systems. Here, we address the exact sum-rate performance of differ-ent multiuser MIMO systems with user scheduling with the help of some orderstatistics results.

194 Chapter 7. Multiuser MIMO systems

In this chapter, we first review the basics of MIMO wireless communicationsand introduce multiuser MIMO systems. After that, we consider the zero-forcingbeamforming (ZFBF) transmission scheme for multiuser MIMO systems andanalyze the resulting sum-rate performance with two different user schedulingschemes. We then move onto the random unitary beamforming (RUB) scheme.For the RUB transmission scheme, we first consider the case of the single antennaper mobile user case, for which we present and analyze several low-complexityuser scheduling strategies and quantify their respective trade-off of sum-rateperformance versus feedback load. For the multiple antennae per user case, wefocus on the schemes where receivers apply linear combining to the receivedsignal on different antennas.

7.2 Basics of MIMO wireless communications

In this section, we review some basic results of MIMO wireless systems and thenintroduce the multiuser MIMO system model that will be used in a later section.

7.2.1 MIMO channel capacity

Let us consider a point-to-point link where the transmitter has Mt antennaeand the receiver has Mr antennae. We assume that the channel for the jthtransmit antenna to the ith receiver antenna experiences frequency flat fading,with complex channel gain hij , usually modeled as i.i.d. zero-mean unit-variancecomplex Gaussian random variables. As such, the received signal on the ithreceive antenna is given by

yi = hi1x1 + hi2x2 + · · · + hiMtxMt

+ ni, (7.1)

where xj is the transmitted symbol from the jth transmit antenna and ni isthe independent Gaussian noise with zero mean and variance σ2 = N0/2. Thediscrete time MIMO channel model can be rewritten in matrix form as⎡

⎢⎢⎢⎣y1

y2...

yMr

⎤⎥⎥⎥⎦ =

⎡⎢⎢⎢⎣

h11 h12 · · · h1Mt

h21 h22 · · · h2Mt

......

. . ....

hMr 1 hMr 2 · · · hMr Mt

⎤⎥⎥⎥⎦⎡⎢⎢⎢⎣

x1

x2...

xMt

⎤⎥⎥⎥⎦+

⎡⎢⎢⎢⎣

n1

n2...

nMr

⎤⎥⎥⎥⎦

or in matrix notation

y = Hx + n. (7.2)

The inherent spatial degree of freedom of the above MIMO system can beexploited for both spatial multiplexing gain and diversity gain.

The most convenient way to demonstrate the capacity benefit of a MIMOsystem is parallel decomposition [11,13]. Specifically, the MIMO channel can be

Basics of MIMO wireless communications 195

decomposed into multiple parallel channels as follows. After applying singularvalue decomposition to H, we have

HMr ×Mt= UMr ×Mr

ΣMr ×MtVMt ×Mt

H ,

where U and V are unitary matrices and Σ is the diagonal matrix of the sin-gular values σi of H. It is not difficult to see that the matrix will have RH

nonzero singular values, where RH ∈ [1,minMt,Mr]. Therefore, the MIMOchannel can be transformed into RH parallel independent channels if the chan-nel input is precoded by V, i.e. x = V x, and the channel output is preprocessedby U, i.e. y = UH y. The ith channel will have input xi , output yi , and channelgain σi with additive noise. Effectively, the data rate can be increased by RH

times. Note that the transmitter needs to have the complete knowledge of V orequivalently H.

Based on the parallel decomposition, the capacity of the MIMO wireless chan-nel can be determined as the sum of individual decomposed channels with opti-mal transmit power allocation. Let Pi denote the power allocation to the ithdecomposed channel, which satisfies a total power constraint of

∑i Pi ≤ PT .

The MIMO capacity with perfect CSIT can be calculated by solving the follow-ing optimization problem: maximizing over PiRH

i=1

RH∑i=1

B log2

(1 +

σ2i Pi

σ2

),

under the constraint∑

i Pi ≤ PT . The maximum is achieved with the well-knownwater-filling solution, given by

Pi =

1γ0

− 1γi

, γi ≥ γ0 ,

0, γi < γ0 ,(7.3)

where γi = σ2i /σ2 and γ0 is the cutoff value based on the total power constraint∑

i:γi ≥γ0Pi = PT . The resulting instantaneous channel capacity is equal to

C =∑

i:γi ≥γ0

B log2

(γi

γ0

). (7.4)

As the result of multipath fading, hij varies over time. The ergodic capacityof the MIMO channel over fading channels can be obtained by averaging theinstantaneous capacity over the fading distribution of channel matrix H.

Cf = EH [C(H)], (7.5)

or equivalently, over the distribution of the singular values of H, σ2i as

Cf = Eσ 2i

[max

Pi :∑

i Pi ≤P

∑i

B log2 (1 + Piγi)

]. (7.6)

The expectation can be solved in some cases while noting that σ2i is the ith

largest eigenvalue of HHH , which is of the Wishart type.


BS

user 1

user 2

user K

· · ·

Figure 7.1 Multiuser MIMO system.

7.2.2 Multiuser MIMO systems

While MIMO techniques can bring huge capacity gains to wireless systems, itis in general difficult to implement the same number of antennae at the mobilereceivers as at the base stations. Most mobile stations will still have only asingle antennae due to their size and cost constraints. In such a scenario, we canstill explore the potential spatial multiplexing gain offered by multiple antennaeat the base station through the simultaneous transmission to multiple single-antenna receivers. The resulting multiuser MIMO system, as illustrated in Fig.7.1, has received a significant amount of research interest. In particular, thedirty paper coding (DPC) technique [14] has been shown to be the optimaltransmission scheme from the sum-capacity perspective for MIMO broadcastchannels [6,8,15]. The sum-rate capacity of these channels with DPC scales at arate of M log log K, where M is the number of transmit antennae and K is thetotal number of receive antennae at different users. On the other hand, DPC hasprohibitively high computational complexity due to the associated successiveencoding process and requires the complete channel state information at thetransmitter (CSIT) side. Therefore, it is of great practical interest to designmultiuser MIMO transmission schemes that can achieve high sum-rate capacitywith low complexity and a minimum CSIT requirement.

The basic approach is to apply some linear precoding schemes, such as zero-forcing beamforming (ZFBF) [16, 17] or random unitary beamforming (RUB)[18–22], to the signals targeted at multiple selected users. With ZFBF, the pre-coding matrix is simply designed to be the pseudo-inverse of the channel matrixto the selected users. It has been shown that ZFBF is asymptotically optimalwhen the number of users approaches infinity [17]. With RUB, the precodingmatrix consists of M random orthogonal beamforming vectors and users areselected based on their channel quality on different beamforming direction. RUBis also shown to be able to achieve the same sum-rate scaling law as DPC,even if each user just feeds back its largest beam signal to interference plusnoise ratio (SINR) value and the index of the corresponding beam [20]. In the

Basics of MIMO wireless communications 197

following sections, we will present and analyze some practical user selectionstrategies for these linear precoding-based multiuser MIMO systems and focusspecifically on their exact sum-rate performance evaluation. To establish a com-mon context for the remaining discussion, we now present the signal model forthe multiuser MIMO system under consideration.

We consider the downlink transmission of a single-cell wireless system. Thebase station is equipped with M antennae and as such can serve as many as M

different selected users with a linear beamforming approach. The transmittedsignal vector from M antennae over one symbol period can be written as

x =M∑

m=1

bm sm , (7.7)

where sm is the information symbol, targeted to the mth selected user, and bm

is the beamforming vector for the mth selected user. The transmitted signalvector x has an average power constraint of trE[xxH ] ≤ P , where P is themaximum average transmitting power, E[·] denotes the statistical expectationand (·)H the Hermitian transpose. We assume that there are a total of K activeusers in the cell, where K ≥ M . Each user is equipped with a single antenna.We further assume that with a certain slow power control mechanism, the usersexperience homogeneous flat Rayleigh fading. In particular, the channel gainfrom the ith antenna to the kth mobile users, denoted by hik , is assumed tobe an independent zero mean complex Gaussian random variable with unitaryvariance, i.e. hik ∼ CN(0, 1). As such, the instantaneous multiple-input single-output (MISO) channel from the base station to the kth mobile user can becharacterized by a zero-mean complex Gaussian channel vector, denoted by hk =h1k , h2k , · · · , hM kT .

During each symbol period, the base station transmits simultaneously to asubset of M different chosen users using all M available beams. The receivedsymbol of the ith chosen user, when the jth beam is assigned to it, can bewritten as

yi = hTi x + ni = hT

i bj sj +M∑

m=1,m =j

hTi bm sm + ni, (7.8)

where ni are independent zero-mean additive Gaussian noise with unit variance.It is easy to see that the received signal will experience interuser interferenceas well as additive noise. The metric that will determine the quality of detec-tion becomes the SINR, which is given for the ith user receiving on the jthbeamforming direction, assuming uniform power allocation to different users, by

γi,j =PM |hT

i bj |2PM

∑Mm=1,m =j |hT

i nm |2 + 1, j = 1, 2, · · · ,M. (7.9)

The interuser interference will be controlled through beamforming vector designand/or user selection, as will be detailed in the following subsections.


Finally, the total sum rate of the multiuser MIMO system with linear precodingis given, with the assumption of uniform power allocation among selected users,by

R = E

[M∑

m=1

log2(1 + γBm )

], (7.10)

where γBm is the received SINR on the mth beamforming direction.

7.3 ZFBF-based system with user selection

ZFBF is a practical multiuser transmission strategy for multiuser MIMO sys-tems [16, 17]. By designing one user’s beamforming vector to be orthogonal toother selected users’ channel vectors, ZFBF can completely eliminate multiuserinterference. A proper user selection scheme is essential for the ZFBF-basedmultiuser MIMO system to fully benefit from multiuser diversity gain [23].The desired properties of selected user channels are (i) large channel powergain, and (ii) near orthogonal to one another. The optimum ZFBF user selec-tion strategy involves the exhaustive search of all possible user subsets, whichbecomes prohibitive to implement when the number of users is large. Severallow-complexity user selection strategies have been proposed and studied in theliterature [17, 23–26]. Among them, two greedy search algorithms, the succes-sive projection ZFBF (SUP-ZFBF) [17, 24] and the greedy weight clique ZFBF(GWC-ZFBF) [23], are attractive for simplicity of implementation and their abil-ity to achieve the same scaling rate of double log as the DPC scheme. We willcarry out accurate statistical analysis on these two schemes for the importanttwo-transmit antenna special case. These analytical results will not only facil-itate the trade-off study between them, but will also apply to the parameteroptimization for each scheme.

7.3.1 Zeroforcing beamforming transmission

We consider the multiuser MIMO system as shown in Fig. 7.1. Specifically,the base station is equipped with two transmit antennae. There are in totalK users in the system, where K ≥ 2, and each user has one receive antenna. Weassume a flat homogeneous Rayleigh fading channel model. The channel gainfrom the ith transmit antenna to the kth user hki is assumed to be i.i.d. zero-mean complex Gaussian random variable with unitary variance. As such, theinstantaneous MISO channel from the base station to the kth mobile user canbe characterized by a zero-mean complex Gaussian channel vector, denoted byhk = [hk1 hk2 ]T , k = 1, 2, · · · ,K. As such, the norm square of the kth user’schannel vector ‖hk‖2 (termed channel power gain) is a Chi-square distributed

ZFBF-based system with user selection 199

hπ(1)

hπ(2)

bπ(1)

bπ(2)

θπ

Figure 7.2 Zero-forcing beamforming for the two-user case [48].

random variable with four degrees of freedom, with PDF and CDF given by

p‖h‖2 (x) = xe−x , x≥0, (7.11)

and

F‖h‖2 (x) = γ(2, x), x≥0, (7.12)

respectively, where γ(2, x) =∫ x

0 te−tdt is the lower incomplete gamma function.We assume that the base station knows the downlink channel vector for everyuser either through a feedback mechanism or the reciprocity of the channel intime division duplexing (TDD) mode.

Let π(1) and π(2) denote the indexes of the selected users and bπ (1) and bπ (2)

denote the beamforming vectors for the two selected users. As shown in Fig. 7.2,the beamforming vector of one user is chosen to be orthogonal to the channelvector of the other user, resulting in so-called zero-forcing beamforming. Thetransmitted signal vector from two antennae over one symbol period can thenbe written as

x =2∑

i=1

√Pπ (i)bπ (i)sπ (i) , (7.13)

where sπ (i) and Pπ (i) are the data symbol and transmit power scaling factor,respectively, for user π(i). The power constraint imposed on the transmittedsignal is E[‖x‖2 ]≤P . Because of the orthogonality between beamforming vectorsand channel vectors, the received symbol at user π(i) is consequently given by

rπ (i) = hTπ (i)x + nπ (i) =

√Pπ (i)h

Tπ (i)bπ (i)sπ (i) + nπ (i) , (7.14)

where nπ (i) is the additive zero-mean Gaussian noise with variance N0 . Note thatthe interference for the two-user transmission is completely eliminated, whereasthe effective channel power gain for user π(i) becomes the norm square of theprojection of its channel vector hπ (i) onto the corresponding beamforming vectorbπ (i) , which is termed projection power in this chapter. Let θπ , as illustrated in


Fig. 7.2, denote the angle between hπ (1) and hπ (2) . The effective power gain γπ (i)

can be written as

γπ (i) = ‖hπ (i)‖2 | sin (θπ )|2 . (7.15)

Note that | sin (θπ )|2 can be viewed as a projection power loss factor due toZFBF. Assuming the total transmit power P is equally divided between twoselected users, then the instantaneous sum rate of the system is given by

R =2∑

i=1

log(1 +ρ

2γπ (i)) (7.16)

where ρ = P/N0 denotes the total transmit SNR.

7.3.2 User selection strategies

The base station will select two users to serve simultaneously using either theSUP-ZFBF [17] or GWC-ZFBF [23] scheme. Basically, both schemes sequentiallyperform user selection from a user subset, which contains users whose channelvectors are near orthogonal to the channel vectors of already selected users. Here,two channel vectors are claimed to be near orthogonal if θ, the angle between thetwo channel vectors, satisfies | sin (θ)|2≥λd , where 0 < λd < 1 is a constant value.With GWC-ZFBF, the next selected user is the one with the largest channelpower gain, whereas with SUP-ZFBF, the next user is the one with the largestprojection power of its channel vector onto the complement space of the spacespanned by the channel vectors of already selected users.

GWC-ZFBF1. Order all K users based on their channel power gain ‖hk‖2 as ‖h(1)‖2 >

‖h(2)‖2 > · · · > ‖h(K )‖2 .2. Select the one with the largest channel power gain as the first user, i.e. π(1) =

(1).3. Examine sequentially the angle between hπ (1) and h(k) , k = 2, 3, . . . ,K and

select the first user whose channel vector is near orthogonal to hπ (1) . Equiva-lently, the second selected user will be the one with the largest channel powergain among all users in the set

U = i|| sin(θi)|2 = 1 −| < hi ,hπ (1) > |2‖hi‖2‖hπ (1)‖2 ≥λd, (7.17)

where θi is the angle between hi and hπ (1) .

SUP-ZFBF1. Select the user with the largest channel power gain as the first user.2. Calculate and rank the projection power onto the complementary space of

hπ (1) for all users in U , which is defined in (7.17).


3. Select the user with the largest projection power as the second user, i.e.

π(2) = maxi∈U

‖hi‖2 | sin(θi)|2 (7.18)

As SUP-ZFBF needs to calculate the projection power of all users in U whileGWC-ZFBF stops searching when it encounters the first user that is near orthog-onal to hπ (1) , SUP-ZFBF exhibits higher computational complexity than GWC-ZFBF.

7.3.3 Sum-rate analysis

In this section, we analyze the ergodic sum rate of a ZFBF-based multiuserMIMO system with SUP-ZFBF and GWC-ZFBF schemes. For that purpose, wefirst derive the statistics of channel power gains of selected users ‖hπ (i)‖2 , i = 1, 2and projection power loss factor | sin (θπ )|2 .

Common analysis for both schemesFor both user selection schemes, the first selected user has the largest channelpower gain among K users. Since the power gains of K users are i.i.d. randomvariables, the PDF of ‖hπ (1)‖2 for both schemes is expressed as [27]

p‖hπ ( 1 ) ‖2 (x) = Kp‖h‖2 (x)(F‖h‖2 (x))K−1 . (7.19)

If no user’s channel vector is near orthogonal to the channel vector of the firstselected user, i.e. U = ∅, the base station transmits to the first selected userusing traditional beamforming approach. In this case, the sum rate for both userselection schemes is expressed as

E[R|U = ∅] =∫ ∞

0log(1 +

ρ

2x)p‖hπ ( 1 ) ‖2 (x)dx. (7.20)

It is shown in [17] that | sin(θ)|2 , where θ is the angle between a random channelvector and hπ (1) , follows the standard uniform distribution over the interval [0, 1].Therefore, after the selection of the first user, the probability that a remaininguser belongs to U is Pr[| sin(θ)|2 ≥ λd ] = 1 − λd. It follows that

Pr[U = ∅] = λK−1d and Pr[U = ∅] = 1 − λK−1

d . (7.21)

where Pr[U = ∅] is the probability of U = ∅ and as such the second user can beselected. Considering these two mutually exclusive cases of U = ∅ and U = ∅, theergodic sum rate of both schemes can be calculated as

E[R] = Pr[U = ∅]E[R|U = ∅] + Pr[U = ∅]E[R|U = ∅]. (7.22)

We now need to determine the average sum rate when U is not empty. Thisanalysis requires the PDF of ‖hπ (2)‖2 and | sin (θπ )|2 , which will be derivedseparately for GWC-ZFBF and SUP-ZFBF in the following.


GWC-ZFBF specific analysisWith GWC-ZFBF, the second selected user is the one with the ith largest channelpower gain, i.e. π(2) = (i), if and only if user (i) belongs to U and none of theusers from (2) to (i − 1) do. Therefore, the probability of π(2) = (i), given thatU is not empty, can be calculated as

Pr[π(2) = (i)|U = ∅

]=

(1 − λd)1 − λK−1

d

λi−2d , i = 2, 3, . . . ,K. (7.23)

By applying the total probability theorem, the PDF of the channel power gainof the second selected user is obtained as

p‖hπ ( 2 ) ‖2 (x|U = ∅) =K∑

i=2

Pr[π(2) = (i)|U = ∅

]· p‖h( i ) ‖2 (x), (7.24)

where p‖h( i ) ‖2 (·) is the PDF of the ith largest channel power gain among K users,given by [27]

p‖h( i ) ‖2 (x) = K

(K − 1i − 1

)p‖h‖2 (x)

(1 − F‖h‖2 (x)

)i−1FK−i‖h‖2 (x). (7.25)

After proper substitution and some mathematical manipulations, (7.24) can berewritten as

p‖hπ ( 2 ) ‖2 (x|U = ∅) = (7.26)

K(1 − λd)λd(1 − λK−1

d )p‖h‖2 (x)

((λd + (1 − λd)F‖h‖2 (x)

)K−1 − FK−1‖h‖2 (x)

).

As the second user is always selected from set U , the projection power loss factor,| sin (θπ )|2 , follows a uniform distribution over the interval [λd, 1]. Therefore,the ergodic sum rate for the GWC-ZFBF scheme conditioned on U = ∅ can becalculated, by averaging (7.16) over the distribution of ‖hπ (i)‖2 , i = 1, 2 and| sin (θπ )|2 as

E[R|U = ∅] =2∑

i=1

∫ ∞

0

∫ 1

λd

log(1 +ρ

2xy) (7.27)

·p| sin(θπ )|2 (y|U = ∅)dyp‖hπ ( i ) ‖2 (x|U = ∅)dx,

which can be further simplified to

E[R|U = ∅] =2∑

i=1

∫ ∞

0

g(x)1 − λd

p‖hπ ( i ) ‖2 (x|U = ∅)dx, (7.28)

where

g(x) =2ρx

[(1 +ρx

2) log(1 +

ρx

2) − (1 +

λdρx

2) log(1 +

λdρx

2)] − 1

ln(2), (7.29)

and p‖hπ ( i ) ‖2 (·) for i = 1 and 2 are given in (7.19) and (7.26), respectively.Finally, the final expression of sum-rate capacity of the multiuser MIMO sys-

tem with GWC-ZEBF strategy can be obtained by substituting (7.20), (7.21),


and (7.28) into (7.22) as

E[R] = λK−1d

∫ ∞

0log(1 +

ρ

2x)p‖hπ ( 1 ) ‖2 (x)dx (7.30)

+1 − λK−1

d

1 − λd

2∑i=1

∫ ∞

0g(x)p‖hπ ( i ) ‖2 (x|U = ∅)dx.

SUP-ZFBF specific analysisWith SUP-ZFBF, the second selected user is the user in U that has the largestprojection power onto the complementary space of the first selected user’s chan-nel vector. From (7.15), the second selected user’s projection power is also itseffective power gain. As such, the ergodic capacity of the second selected usercan be obtained using the PDF of the largest projection power among all usersin U .

We first need to find the PDF of the channel power gain for a user in U , whichis denoted as ‖h‖2 . Noting that users in U cannot have the largest channel powergain, the PDF of ‖h‖2 can be obtained as

p‖h‖2 (x) =1

K − 1

K∑i=2

f‖h( i ) ‖2 (x) (7.31)

=K

K − 1p‖h‖2 (x)(1 − FK−1

‖h‖2 (x)),

where p‖h( i ) ‖2 (x) is the PDF of the ith largest channel power gain. Letting ‖hp‖2

represent the projection power of a user in U , we have ‖hp‖2 = ‖h‖2 | sin(θ)|2 ,where θ is the angle between channel vectors of that user and the first selecteduser. Since | sin(θ)|2 follows the uniform distribution over the interval [λd, 1], theCDF of ‖hp‖2 can be obtained as

F‖hp ‖2 (z) = Pr[ ˜‖h‖2 | sin(θ)|2 < z] =1

1 − λd

∫ 1

λd

∫ zy

0p ˜‖h‖2 (x)dxdy. (7.32)

After taking the derivative with respect to z, the corresponding PDF is given by

p‖hp ‖2 (z) =1

1 − λd

∫ 1

λd

1yp‖h‖2 (

z

y)dy. (7.33)

Denote the cardinality of the set U , i.e. the number of users in U , as |U|. As thechannel directions of users are independent, the distribution of |U| conditionedon U = ∅ can be shown to be given by

Pr[|U| = k|U = ∅] = (7.34)1

1 − λK−1d

(K − 1

k

)(1 − λd)kλK−1−k

d , k = 1, · · · ,K − 1.

When |U| = k, the effective channel gain of the second selected user, γπ (2) , isequal to the largest projection power among k users and its PDF is thus given


by

pγπ ( 2 ) (g∣∣|U| = k) = kp‖hp ‖2 (g)Fk−1

‖hp ‖2 (g). (7.35)

Combining (7.34) and (7.35) and averaging over |U|, the PDF of γπ (2) given thatU = ∅ can be obtained as

pγπ ( 2 ) (g|U = ∅) =K−1∑k=1

Pr(|U | = k|U = ∅)pγπ ( 2 ) (g||U | = k) (7.36)

=(K − 1)(1 − λd)λK−2

d

1 − λK−1d

f‖hp ‖2 (g)(1 +

1 − λd

λdF‖hp ‖2 (g)

)K−2.

To derive the distribution of | sin(θπ )|2 , we start with the joint PDF of aremaining user’s channel power, ‖h‖2 , and its projection power loss factor,| sin(θ)|2 ,

p‖h‖2 ,| sin(θ)|2 (x, y) =1

1 − λdp‖h‖2 (x), x≥0, λd≤y≤1. (7.37)

With a change of variables, the joint PDF of projection power ‖hp‖2 and | sin(θ)|2of a remaining user is obtained from (7.37) as

p‖hp ‖2 ,| sin(θ)|2 (z, y) =1

(1 − λd)yp ˜‖h‖2 (

z

y), z≥0, λd≤y≤1. (7.38)

It follows that the conditional PDF of | sin(θ)|2 given that ‖hp‖2 = z is given by

p| sin(θ)|2 (y∣∣‖hp‖2 = z) =

1(1 − λd)y

p ˜‖h‖2 (z

y)/p‖hp ‖2 (z), z≥0, λd≤y≤1. (7.39)

Since the projection power of the second selected user is given by (7.36), wecan obtain the PDF of | sin(θπ )|2 conditioned on U = ∅ by averaging (7.39) over(7.36) as

p| sin(θπ )|2 (y|U = ∅) =∫ ∞

0p| sin(θ)|2 (y

∣∣‖hp‖2 = z)pγπ ( 2 ) (z|U = ∅)dz (7.40)

=(K − 1)λK−2

d

(1 − λK−1d )y

∫ ∞

0p‖h‖2

(z

y

)(1 +

1 − λd

λdF‖hp ‖2 (z)

)K−2dz, λd≤y≤1.

Finally, E[R|U = ∅] for the SUP-ZFBF scheme as

E[R|U = ∅] =∫ ∞

0log(1 +

ρ

2g)fγπ ( 2 ) (g|U = ∅)dg + (7.41)∫ ∞

0

∫ 1

λd

log(1 +ρ

2xy)p‖hπ ( 1 ) ‖2 (x)p| sin(θπ )|2 (y|U = ∅)dydx

Consequently, the expression of ergodic sum rate of the multiuser MIMO systemwith SUP-ZEBF strategy can be obtained by substituting (7.20), (7.21) and(7.41) into (7.22).


0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

|sin(θπ)|2

GWC, λ

d=0

SUP, K=20, λd=0

SUP, K=100, λd=0

GWC, λd=0.7

SUP, K=20, λd=0.7

SUP, K=100, λd=0.7

Figure 7.3 The probability density function of | sin(θπ )|2 for GWC-ZFBF andSUP-ZFBF (ρ = 10 dB) [48]. c© 2009 IEEE.


Figure 7.3 illustrates the PDF of projection power loss factor | sin(θπ )|2 for bothGWC-ZFBF and SUP-ZFBF schemes with different λd and/or K values. Aswe can see, | sin(θπ )|2 with the GWC-ZFBF scheme always follows a uniformdistribution over the interval [λd, 1]. On the other hand, | sin(θπ )|2 with the SUP-ZFBF scheme is not uniformly distributed. This is because its user selection isbased on projection power, which is correlated to channel direction. We canalso observe from Fig. 7.3 that the PDF of | sin(θπ )|2 with SUP-ZFBF is amonotonically increasing function and the mass of the PDF shifts to the rightas K increases. We can conclude that SUP-ZFBF can explore more multiuserdirectional diversity gain than GWC-ZFBF by selecting “more orthogonal” users.

In Fig. 7.4, we plot the ergodic sum rate of both GWC-ZFBF and SUP-ZFBFschemes as a function of channel direction constraint λd . As we can see, whilethe ergodic sum rate with GWC-ZFBF is a unimodal function of λd , that ofSUP-ZFBF remains roughly constant for the small to medium values of λd anddecreases when λd becomes very close to 1. Note that when λd is small, theGWC-ZFBF may select fewer orthogonal users, which incurs larger projectionpower loss, whereas the SUP-ZFBF scheme tends to select “more orthogonal”users. On the other hand, when λd is very large, the number of candidate usersfor selection decreases, which leads to sum-rate performance degradation forboth schemes. With the analytical sum-rate expression derived in this chapter,


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.95

5.5

6

6.5

7

7.5

8

8.5

9

9.5

10

λd

Erg

odic

Sum

Rat

e (b

it/s/

Hz)

K=20, SUPK=50, SUPK=100, SUPK=20, GWCK=50, GWCK=100, GWC

0.85

0.8

0.7

Figure 7.4 Effects of λd on the ergodic sum rate of GWC-ZFBF and SUP-ZFBF(ρ = 10 dB) [48]. c© 2009 IEEE.

we can easily find the optimal value of λd by using one-dimensional optimizationalgorithms. We can also observe that the optimal value of λd increases with thenumber of users K, which indicates that when more users are in the system weshould trade channel power diversity for directional diversity gain.

As an additional example, Fig. 7.5 plots the ergodic sum rate with both GWC-ZFBF and SUP-ZFBF schemes as a function of the number of users K. The sum-rate performance of the more complex scheme proposed in [26], dubbed the D-Sscheme, is also plotted for comparison purposes. As shown in this figure, theperformance of the GWC-ZFBF scheme with optimal λd values for different K

approaches that of SUP-ZFBF, especially for a larger value of K. In addition,the performance gap between SUP-ZFBF and the D-S scheme is very small if λd

for SUP-ZFBF is properly chosen. Note that both GWC-ZFBF and SUP-ZFBFenjoy much lower computational complexity than the D-S scheme.

7.4 RUB-based system with user selection

RUB is a low-complexity solution for exploring spatial multiplexing gain overmultiuser MIMO broadcast channels [18–22]. With RUB, the base station gener-ates M random orthogonal beams to communicate with as many as M selectedusers simultaneously. To explore multiuser diversity gain, each user will feed

RUB-based system with user selection 207

10 20 30 40 50 60 70 80 90 1006.5

7

7.5

8

8.5

9

9.5

10

Number of Users

Erg

odic

Sum

Rat

e (b

it/s/

Hz)

D−S SchemeSUP, λ

d=0.5

GWC with optimal λd

GWC, λd

=0.6

GWC, λd

=0.85

Figure 7.5 Ergodic sum rate of GWC-ZFBF and SUP-ZFBF schemes as the functionof the number of users (ρ = 10 dB) [48]. c© 2009 IEEE.

back some quality information, usually in terms of SINR, of each beam, basedon which the base station carries out proper user selection. The major complex-ity saving of RUB over the ZFBF system is the lower feedback load, as userswill no longer need to feed back their channel vectors. It has been proven in [20]that if each user just feeds back its maximum beam SINR and the index of thecorresponding beam, RUB can achieve the same sum-rate scaling law as that ofthe DPC-based scheme (i.e. M log log K) when K approaches infinity. Note thatthe resulting feedback load is only a real number and an integer per user.

We attack the exact sum-rate analysis of RUB schemes over MIMO broadcastchannels with an arbitrary number of users. We focus on the RUB schemes withlow feedback load. In particular, we investigate the best beam SINR and index(BBSI)-based RUB scheme [20] and the best index (BBI)-based RUB [28,29], inwhich each user only feeds back the index of its best beam. The major difficultyin the sum-rate analysis for these schemes resides in the determination of thestatistics of ordered beam SINRs for a particular user. Note that while the beamSINRs of different users can be assumed to be independent random variables [20],those corresponding to the same users are correlated to each other as they involvethe same channel vector. As such, the SINR of the best beam of a user is thelargest one among some correlated random variables. With the help of the orderstatistics results discussed in a previous chapter [30], we derive the accuratestatistical characterization of ordered-beam SINRs for a user, in terms of the PDF


and the CDF, which are then applied to obtain the exact analytical expressionsof the ergodic sum rate of different RUB schemes. These accurate analyticalresults greatly facilitate the investigation of the trade-off analysis on differentRUB schemes in a practical scenario where the number of users is finite.

7.4.1 User selection strategies

We consider the downlink transmission of a single-cell wireless system. The M -antenna base station employs M random orthonormal beams, generated froman isotropic distribution [18], to serve M selected users simultaneously over thesame frequency channel. Let U = u1 ,u2 , · · · ,uM denote the set of beamform-ing vectors, assumed to be known to both the base station and the mobile users.The transmitted signal vector from M antennae over one symbol period can bewritten as

x =M∑

m=1

um sm , (7.42)

where sm is the information symbol, targeted to the mth selected user.We assume that there are a total of K active users in the cell, where K ≥ M .

Each user is equipped with a single antenna. We further assume that with a cer-tain slow power control mechanism, the users experience homogeneous Rayleighfading. In particular, the channel gain from the ith antenna to the kth mobileusers, denoted by hik , is assumed to be an independent zero-mean complex Gaus-sian random variable with unitary variance, i.e. hik ∼ CN(0, 1). Consequently,the SINR of user i while using the jth beam is given by

γi,j =PM |hT

i uj |2PM

∑Mm=1,m =j |hT

i um |2 + 1, j = 1, 2, · · · ,M. (7.43)

We assume that the users can accurately estimate their own channel vector anddetermine the instantaneous SINR corresponding to different beams using Eq.(7.43) with the knowledge of beamforming vector set U .

Best-beam SINR and index strategyWith the BBSI strategy, each active user in the system feeds back its best beamindex and the corresponding SINR value to the base station. For example, if themth beam leads to the largest SINR for the kth user, i.e. γk,m = γ∗

k , then the kthuser will feedback the beam index m and the corresponding SINR value γk,m .Note that with the BBSI strategy, each user needs to feed back a real numberfor the SINR value and a finite integer for the index of the best beam.

Based on the feedback information, the base station assigns a beam to theuser with the largest SINR value among all users who feed back the index ofthat beam. Specifically, the base station ranks all K feedback best beam SINRs.If γk,m is the largest among all K SINRs, then the base station selects the kthuser for the mth beam. After that, the base station will rank the feedback SINRs


for the remaining beams. If now γn,l is the largest, where l = m and n = k, thenthe base station assigns the lth beam to the nth user. This process is continueduntil either all beams have been assigned to selected users or there are someunrequested beams remaining. In the latter case, the base station will randomlyselect users for the remaining beams.

Quantized best-beam SINR and index strategyIn practical systems, only a quantized value of the best beam SINR can be fedback, which leads to the quantized best beam SINR and index (QBBSI) strat-egy. Similar to the BBSI strategy, the users with the QBBSI strategy will firstcalculate the instantaneous SINRs for all M beams using its channel vector anddetermine the largest one after some comparison. Then, the users feed back thequantized SINR value of their best beam in the following fashion. In particular,the value range of SINR is divided into N intervals, with boundary values givenas 0 = α0 < α1 < α2 < · · · < αN = ∞. If the SINR value of the best beam forthe kth user, γk,1:M , is in the ith interval, i = 1, 2, · · ·N , i.e. αi−1 < γk,1:M < αi ,then the kth user will feedback the index of that interval, i, together with theindex of its best beam. The base station allocates the beams to selected usersbased on the feedback information. Specifically, a beam will be assigned to auser who has the largest quantized SINR value, i.e. who feeds back the largestSINR interval index, among all the users requesting that beam. Note that a tiemay exist in this case, which will be resolved using random selection. For thosebeams that no user requests for, the base station will assign them to randomlyselected users. Based on the model of operation of QBBSI strategy, the amountof feedback per user is equal to log2 MN bits.

Best-beam index strategyWith the best beam index (BBI) strategy, each user only feeds back the index ofits best beam. After receiving the best beam information of all users, the basestation allocates a beam to a randomly selected user among all users requestingthat beam, i.e. all users who feed back the corresponding beam index. When nouser requests one or several beams, the base station will assign that beam to arandomly selected user. Note that with the BBI strategy, each user only needsto feedback log2(M) bits of information.

7.4.2 Asymptotic analysis for BBSI strategy

In this subsection, we investigate the asymptotic behavior of the beam SINRwhen the number of users in the system K approaches infinity. In particular, asK is large, we can ignore the cases that a user is the best user on two differentbeams and that a beam is not requested by any users. As a result, the SINR of abeam will be the largest of all user SINRs on that beam. Mathematically, we haveγB

j∼= maxγ1,j , γ2,j , · · · , γK,j .= γ1:K,j . To investigate the asymptotic behavior

of beam SINR γ1:K,j , we first obtain the statistics of an unordered beam SINR


of a user γi,j , as defined in (7.43). We first rewrite γi,j as

γi,j =|hT

i uj |2∑Mm=1,m =j |hT

i um |2 + ρ, (7.44)

where ρ = M/P . Since uj s are orthonormal beamforming vectors, it can beshown that |hT

i uj |2 , j = 1, 2, · · · ,M , are i.i.d. Chi-square random variables withtwo degrees of freedom. It follows that the summation term in the denominatorfollows a chi-square distribution with 2M − 2 degrees of freedom and is inde-pendent of the numerator. As such, the PDF of unordered SINR can be writtenas

pγi , j(x) =

e−ρx

(1 + x)M(ρ(1 + x) + M − 1) . (7.45)

The CDF can be obtained after carrying out integration as

Fγi , j(x) = 1 − e−ρx

(1 + x)M −1 . (7.46)

Since the user channels are assumed to be independent, γi,j , i =1, 2, · · · ,K are i.i.d. random variables. As such, the CDF of their maximummaxγ1,j , γ2,j , · · · , γK,j is simply (Fγi , j

(x))M , which can be applied to calcu-late the approximate sum-rate performance of the BBSI strategy.

To obtain some insights into the nature of the sum-rate behavior as K

approaches infinity, we examine the limiting distribution of γ1:K,j in what follows.It can be shown that

limx→∞

1 − Fγi , j(x)

pγi , j(x)

= limx→∞

(1ρ− (M − 1)/ρ

ρ(1 + x) + M − 1

)(7.47)

=1ρ.

Therefore, the limiting distribution of γ1:K,j is of the Gumbel type. Solving theequation 1 − Fγi , j

(bK ) = 1/K for bK , we can show that bK is approximatelygiven by [20]

bK =1ρ

log K − M − 1ρ

log log K. (7.48)

The limiting CDF of γ1:K,j as K approaches infinity is then given by

limK→+∞

Fγ1 :K −bK(x) = exp

(−e−x/aK

). (7.49)

We can conclude that γ1:K,j behaves like 1ρ log K when K approaches infinity.

7.4.3 Statistics of ordered-beam SINRs

We now derive the statistics of the ordered-beam SINRs for user i. Note that thebeam SINRs for the same user, i.e. γi,j in (7.43) with the same i but different j,are correlated random variables as they involve the same channel vector hi .


For notational conciseness, we use αj to denote the norm square of the pro-jection of user i’s channel vector onto the jth beam direction, i.e. αj = |hT

i uj |2 ,j = 1, 2, · · · ,M . The M beam SINRs for user i can be rewritten as

γi,j =αj∑M

m=1,m =j αm + ρ, j = 1, 2, · · · ,M. (7.50)

We are interested in the statistics of the ordered version of these M correlatedrandom variables, i.e. γi,1:M ≥ γi,2:M ≥ · · · ≥ γi,M :M . We first note that all M

beam SINRs for the ith user are calculated using the same set of projectionpowers αjM

j=1 and, as such, they are correlated random variables. On the otherhand, we also note from (7.50) that the interference terms for signal on a particu-lar beam are always the project power of the channel vector on to the remainingM − 1 beams. Therefore, we conclude that γi,j is the largest beam SINR foruser i if and only if αj is the largest one among M projection norm squares forthis user, i.e. αj = α1:M . Based on this observation, the largest SINR of user i,denoted by γ∗

i , can be written in terms of the ordered norm squares αm :M Mm=1

as

γ∗i =

α1:M

z1 + ρ, (7.51)

where z1 is the sum of the remaining M − 1 norm squares, i.e. z1 =∑M

m=1 αm −α1:M . Consequently, the CDF of γi,1:M can be calculated in terms of the jointPDF of α1:M and z1 , denoted by fα1 :M ,z1 (y, z), as

Fγi , 1 :M (x) =∫ ∞

0

∫ x(ρ+z )

0pα1 :M ,z1 (y, z)dydz. (7.52)

It follows that the PDF of γ∗i can be calculated using the joint PDF of α1:M and

z1 as

pγ ∗i(x) =

∫ ∞

0(z + ρ) · pα1 :M ,z1 ((z + ρ)x, z)dz, (7.53)

From (7.50), we can also conclude that γi,j is the lth largest SINR for user i ifand only if αj is the lth largest one among all M norm squares, i.e. γi,l:M = γi,j

if and only if αj = αl:M . Consequently, γi,l:M can be rewritten as

γi,l:M =αl:M

(wl + zl) + ρ, (7.54)

where wl =∑l−1

m=1 αm :M and zl =∑M

m= l+1 αm :M . It can be shown that the PDFof γi,l:M can then be calculated in terms of the joint PDF of wl , αl:M , and zl ,denoted by fwl ,αl :M ,zl

(w, y, z), as

pγi , l :M (x) =∫ ∞

0

∫ (M −l )wl−1

0(ρ + z + w) · pwl ,αl :M ,zl

(w, x(ρ + z + w), z)dzdw, (7.55)

where the integration limits is set using the fact w/(l − 1) > αl:M > z/(M − l).The PDFs fα1 :M ,z1 (·, ·) and fwl ,αl :M ,zl

(·, ·, ·) are joint PDFs of partial sums ofordered random variables, the general expressions of which have been obtained


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

1.2M = 4, P = 30 dB

x

Pro

babi

lity

Den

sity

Fun

ctio

n (P

DF

) of

the

larg

est S

INR

per

use

r

Simulation (1000 000 samples)Analysis

Figure 7.6 Simulation verification of the PDF expression given in (7.118). c© 2008IEEE.

in a previous chapter. As such, the statistical results of ordered SINR are quitegeneral and independent of the distribution αj . For the specific Rayleigh chan-nel model under consideration, it can be shown, after proper substitution andcarrying out integration, that we can obtain the following closed-form expressionfor the PDF of the largest beam SINR of a user as

pγ ∗i(x) =

M −1∑j=0

(−1)jM !(M− j − 1)!j!

exp(− (1 + j)ρx

jx − 1

)(7.56)

(M + ρ − 1 + (ρ − jM + j)x)(1 − jx)M −3

(1 + x)M.

Figure 7.6 plots the PDF of the largest SINR for a particular user as given in(7.56) with M = 4 and P = 30 dB in comparison with the simulation results. Aswe can see, the analytical result matches the simulation result perfectly.


BBSI strategyBased on the mode of operation of the BBSI strategy with ordered beam assign-ment, we can see that the SINR of the first beam is always the largest of all K

best-beam SINRs, i.e. γB1 = γ∗

1:K , where γ∗i:K denotes the ith largest best-beam


SINRs among all K ones. Noting that the largest SINRs of different users areindependent, the PDF of γB

1 can be obtained easily as

pγ B1

(x) = KFγ ∗i(x)K−1pγ ∗

i(x), (7.57)

where Fγ ∗i(·) and pγ ∗

i(·) are the CDF and PDF of the best beam SINR for a

particular user, given in (7.52) and (7.53), respectively.We now consider the SINR of the second beam. Based on the mode of operation

of the BBSI strategy, the SINR of the second beam may be the ith largest oneamong all K best-beam SINRs, i = 2, 3, · · · ,K, if the previous i − 2 largest onesare for the first beam. In particular, if, for example, the second and third largestbest-beam SINRs are also for the first beam, then the SINR of the second beamwill be equal to the fourth largest best-beam SINR. Since the user SINRs fordifferent beams are i.i.d., each beam has the same probability to lead to thelargest SINR for a particular user. It can be shown that the probability that thesecond beam is assigned to a user with the ith largest best-beam SINR amongall K ones, i.e. γB

2 = γ∗i:K , i = 2, 3, · · · ,K, is given by

P 2i =

(1M

)i−2 (1 − 1

M

), (7.58)

where 1/M gives the probability that the best SINR of a user is for a particularbeam. In the worst case, where all users request the same beam, the SINR ofthe second beam is equal to the SINR of a randomly chosen user.1 Note thatthe SINR of a randomly assigned beam has the same probability to be the jthlargest one, j = 2, 3, · · · ,M , among M beam SINRs. Correspondingly, the PDFof the SINR of a randomly assigned beam can be obtained as

pγ R (x) =1

M − 1

M∑j=2

pγi , j :M (x), (7.59)

where fγi , j :M (·) is the PDF of the jth largest beam SINR for a user, given in(7.55). The probability that the second beam is assigned to a random user isequal to (1/M)K−1 . Consequently, the PDF of the SINR for the second beamγB

2 can be shown to be given by

pγ B2

(x) =K∑

i=2

P 2i pγ ∗

i :K(x) +

(1M

)K−1

pγ R (x), (7.60)

where fγ ∗i :K

(x) is the PDF of the ith largest best beam SINR among all K ones,given by [27]

pγ ∗i :K

(x) =K!

(K − i)!(i − 1)!Fγ ∗

i(x)(K−i) (7.61)

×[1 − Fγ ∗i(x)]i−1pγ ∗

i(x).

1 This approach is to maintain the validity of (7.43). Alternatively, we may turn off thosebeams to reduce mutual interference.


Table 7.1 Sample scenario for the event that the mth beam is assigned to the user withthe ith largest best-beam SINR among all K users, i.e. γB

m = γ∗i :K .

Indexes of ordered best-beam SINRs Associated beams No. of users1 First beam 1

2 to n1 + 1 First beam n1

n1+2 Second beam 1n1+3 to n1+n2+2 First two beams n2

......

...∑j−1k=1 nk + j jth beam 1∑j−1

k=1 nk + j + 1 to∑j−1

k=1 nk+ nj + j First j beams nj

......

...∑m−2j=1 nj + m to

∑m−2j=1 nj+ nm−1 + m − 1 First m − 1 beams nm−1

i =∑m−1

j=1 nj + l mth beam 1

Similarly, the SINR of the third beam may be the ith largest best-beam SINRsamong all K ones, i = 3, 4, · · · ,K, if one of the previous i − 2 largest best-beamSINR is assigned to the second beam and the remaining i − 3 largest ones arefor the first and second beams. It can be shown that the probability that thethird beam is assigned to a user with the ith largest best-beam SINRs amongall K users, i.e. γB

3 = γ∗i:K , i = 3, 4, · · · ,K, is given by

P 3i =

i−3∑j=0

(1M

)j (1 − 1

M

)(2M

)i−3−j (1 − 2

M

), (7.62)

where 2/M gives the probability that the largest SINR for a particular useris for the first and second assigned beams. Note that in the worst case whereall best-beam SINRs belong to the first and second beam, whose probabilityis (2/M)K−1 , the SINR of the third assigned beam is equal to the SINRs of arandomly chosen user, whose PDF was given in (7.59). Consequently, the PDFof the SINR for the third beam with BBSI strategy can be shown to be given by

pγ B3

(x) =K∑

i=3

P 3i pγ ∗

i :K(x) +

(2M

)K−1

pγ R (x). (7.63)

In general, the SINR of the mth beam may be the ith largest best-beam SINRs,where i = m,m + 1, · · · ,K. More specifically, γB

m = γ∗i:K when a scenario shown

in Table 7.1 occurs, where nj denotes the number of users whose best beam is oneof the first j assigned beams. After determining the probability of the scenario inTable 7.1 for a particular vector njm−1

j=1 and summing over all possible vectorsnjm−1

j=1 , while noting that njm−1j=1 needs to satisfy nj ∈ 0, 1, · · · , i − m and∑m−1

j=1 nj = i − m, we obtain the probability that the mth beam is assigned to theuser with the ith largest best-beam SINR, i.e. γB

m = γ∗i:K , i = m,m + 1, · · · , K


as

Pmi =

∑∑m −1

j = 1n j = i−m

n j ∈0 , 1 , ··· , i−m

⎡⎣m−1∏

j=1

(j

M

)nj(

1 − j

M

)⎤⎦ (7.64)

i = m,m + 1, · · · ,K.

Consequently, after noting that it is with a probability of(

m−1M

)K−1 that themth beam is assigned to a random selected user, we can obtain the PDF of theSINR on the mth beam γB

m as

pγ Bm

(x) =K∑

i=m

Pmi pγ ∗

i :K(x) +

(m − 1

M

)K−1

pγ R (x). (7.65)

Note that additional SINR feedback from the randomly selected users is requiredto achieve the data rate. When K is not too small, the extra feedback load canbe neglected since the probability of beams not being requested is small.

Finally, after applying (7.57) and (7.65) in (7.10), the exact sum-rate expres-sion for the BBSI strategy can be calculated as

R =∫ ∞

0log2(1 + x)KFγ ∗

i(x)K−1pγ ∗

i(x)dx (7.66)

+M∑

m=2

(K∑

i=m

Pmi

∫ ∞

0log2(1 + x) pγ ∗

i :K(x)dx

+(

m − 1M

)K−1 ∫ ∞

0log2(1 + x)pγ R (x)dx

),

where Pmi and pγ ∗

i :K(x) are given in (7.64) and (7.61), respectively.

Quantized best-beam SINR and index strategyLet us consider one of the M available beams. It can be shown that the proba-bility that exact j users among the total K users request this particular beam isequal to

Pr[Nr = j] =K!

(K − j)!j!

(1M

)j (M − 1

M

)K−j

, (7.67)

j = 0, 1, · · · ,K.

If Nr = 0, then that beam will be assigned to a randomly selected user whosebest beam is a different one. In this case, the PDF of the SINR of this beam is thesame as that given in (7.59). If Nr = j > 0, then that beam will be assigned toa user for whom this beam is the best one. It can be shown that the probabilitythat the beam is assigned to a user with a best-beam SINR in the ith interval


to be given by

Pr[γB ∈ (αi−1 , αi)|Nr = j > 0] = (7.68)

Fγi , 1 :M (αi)j − Fγi , 1 :M (αi−1)j , 1 ≤ i ≤ N,

where Fγi , 1 :M (·) is the CDF of the SINR of the best beam of a user, as given in(7.52). The corresponding SINR PDF of the beam when it is assigned to a userwith best beam SINR in the ith interval can then be obtained as

pγ B |γ B ∈(αi−1 ,αi )(x) = (7.69)1

Fγi , 1 :M (αi) − Fγi , 1 :M (αi−1)pγi , 1 :M (x), αi−1 ≤ x ≤ αi.

After successively removing the conditionings, the SINR PDF for the beam canbe obtained as

pγ B (x) =K∑

j=1

K!(K − j)!j!

(1M

)j (M − 1

M

)K−j

(7.70)

×N∑

i=1

Fγi , 1 :M (αi)j − Fγi , 1 :M (αi−1)j

Fγi , 1 :M (αi) − Fγi , 1 :M (αi−1)pγi , 1 :M (x)

×(U(x − αi−1) − U(x − αi)) +(

M − 1M

)K

pγ R (x),

where pγ R (·) was given in (7.59).Finally, the sum rate of a RUB-based multiuser MIMO system with QBBSI-

based user selection can be calculated as

RQBBSI = M

∫ ∞

0log2(1 + x)pγ B

m(x)dx (7.71)

= M

⎛⎝ K∑

j=1

(K)!(K − j)!j!

(1M

)j (M − 1

M

)K−j

×( N∑

i=1

Fγi , 1 :M (αi)j − Fγi , 1 :M (αi−1)j

Fγi , 1 :M (αi) − Fγi , 1 :M (αi−1)

∫ αi

αi−1

log2(1 + x)pγi , 1 :M (x)dx

+(

M − 1M

)k 1M − 1

M∑j=2

∫ ∞

0log2(1 + x)pγi , j :M (x)dx

⎞⎠ .

BBI strategyBased on the mode of operation of the BBI strategy, the SINR of the user assignedto the mth beam, γB

m in (7.10), may follow two different types of distribution.More specifically, if there is at least one user requesting the mth beam, thenthis beam will be the best beam of the selected user. Therefore, γB

m is the largestamong M beam SINRs for a user, i.e. γB

m = γ∗i , whose PDF is given in (7.53). On

the other hand, when no user requests the mth beam and that beam is assigned


to a randomly chosen user, γm can be any one of M beam SINRs, except forthe largest one, for the selected user. Therefore, the PDF of γB

m for this case wasgiven in (7.59), in terms of the PDF of the jth largest beam SINR for a userfγi , j :M (·).

Let Na denote the number of beams that are requested by at least one user.Clearly, Na is a discrete random variable, taking values 1, 2, · · · ,M . We caneasily see that the probability that only one beam is active, i.e. Na = 1 is equalto the probability that all k users feed back the same beam index, which can becalculated as

Pr[Na = 1] =(

M

1

)(1M

)K

. (7.72)

The probability of exactly two beams being active can be calculated by subtract-ing from the probability that at most two beams are active, which is given by(M2

) ( 2M

)K , the probability of only one of those two beams is active, which isequal to

(M2

)(21

) ( 1M

)K . Therefore, we have

Pr[Na = 2] =(

M

2

)[(2M

)K

−(

21

)(1M

)K]

. (7.73)

Similarly, the probability that exactly three beams are active can be calculatedby subtracting from the probability that at most three beams are active, theprobabilities that not all three beams are active. Hence, the probability of Na = 3is given by

Pr[Na = 3] =(

M

3

)(3M

)K

−(

32

)· (7.74)

[(2M

)K

−(

21

)(1M

)K]−(

31

)(1M

)K

.

In general, the probability that exact m beams out of M available beams areactive given that k users feed back their best beam indexes can be recursivelycalculated as

Pr[Na = m] =(

M

m

)(m

M

)K

(7.75)

−m−1∑j=1

(m

j

)Pr[Na = j](

Mj

)

, m = 1, 2, · · · ,M.

We also can prove by induction that (7.75) can be simplified to the followingcompact expression

Pr[Na = m] =1

MK

(M

m

)·

m∑q=1

(−1)m−q

(m

q

)qK , m = 1, 2, · · · ,M. (7.76)


It is easy to verify that (7.76) holds for the i = 1 case. Assume that (7.76) holdsfor all 1≤ i≤m − 1, then we have

Pr[Na = i] =1

MK

(M

i

)· (7.77)

i∑q=1

(−1)i−q

(i

q

)qK , i = 1, 2, · · · ,m − 1.

We now consider the case of i = m. From (7.75), we have

Pr[Na = m] =(

M

m

)(m

M

)k

− (7.78)

m−1∑j=1

(m

j

)Pr[Ma = j|Nf = k](

Mj

)

.

Substituting (7.77) into (7.78), we have

Pr[Ma = m|Nf = k]

=1

Mk

(M

m

)mk −

m−1∑j=1

(m

j

) j∑q=1

(−1)j−q

(j

q

)qk

(a)=

1Mk

(M

m

)mk −

m−1∑q=1

m−1∑j=q

(−1)j

(m

j

)(j

q

)(−1)q qk

(b)=

1Mk

(M

m

)mk −

m−1∑q=1

m−1∑j=q

(−1)j

(m − q

j − q

)(−1)q

(m

q

)qk

(c)=

1Mk

(M

m

)mk −

m−1∑q=1

m−q−1∑

j=0

(−1)j

(m − q

j

)(m

q

)qk

(d)=

1Mk

(M

m

)mk +

m−1∑q=1

(−1)m−q

(m

q

)qk

=1

Mk

(M

m

) m∑q=1

(−1)m−q

(m

q

)qk

where(a) follows by change order of summation;(b) follows since

(mj

)(jq

)=(mq

)(m−qj−q

), which can be proven by direct calcula-

tion;(c) follows by the change of variable;(d) follows from the formula

∑mj=0(−1)j

(mj

)= 0, which results from the bino-

mial expansion of (1 − 1)m . Therefore, (7.76) holds in general.Finally, the sum rate of a RUB-based multiuser MIMO system with BBI strat-

egy can be obtained, by conditioning on the number of beams that have been


5 10 15 20 25 300.5

1

1.5

2

2.5

3

3.5

4

4.5

5

K

Sum

Rat

e (b

it/s/

Hz)

Upper BoundExactLower Bound

Figure 7.7 Sum-rate capacity in comparison with upper and lower bounds (M = 4,P = 5 dB).

requested by at least one user, as

RBBI =M∑i=1

Pr[Na = i](i

∫ ∞

0log2(1 + x)pγ ∗

i(x)dx

+M − i

M − 1

M∑j=2

∫ ∞

0log2(1 + x)pγi , j :M (x)dx

), (7.79)

where Pr[Na = i] is given in (7.76).In Fig. 7.7, we plot the exact sum rate of the MIMO broadcast channel with

BBSI-based RUB as given in (7.66) as the function of the number of users underconsideration. On the same figure, we also plot the upper and lower boundderived in [20] for the asymptotic sum-rate analysis. As we can see, the upperbounds are much closer to the exact sum rate, especially when the number ofusers is large. We also notice that when the number of users is small, there isalso a noticeable gap between the upper bound and the exact sum rate. As such,the analytical sum-rate expression provides a valuable tool to predict the systemperformance when the number of users in the system is limited.

In Fig. 7.8, we plot the sum-rate performance of the MIMO broadcast channelwith BBSI-based RUB as given in (7.66) as a function of the average SNR fora different number of active users K and a different number of antennae (or,equivalently, the number of beams) M . First of all, we can observe a common


0 5 10 15 20 25 30 35 400

2

4

6

8

10

12

14

Average SNR (dB)

Sum

Rat

e (b

it/s/

Hz)

M=4, K=20M=4, K=50M=2, K=20M=2, K=50

Figure 7.8 Sum-rate performance for a different number of transmit antennae and/ornumber of users.

trend that, when SNR increases, the sum rate of RUB systems increases over thelow SNR region but saturate in the high SNR region. If we compare the curveswith the same K value but different M values, we can see that while using moreantennae, i.e. transmitting to more users simultaneously, helps slightly increasethe total sum rate over the low SNR region, activating fewer beams leads to amuch better sum-rate performance in the high SNR region. These behaviors canbe explained as follows. Over the high SNR region, the multiuser interferencebecomes a dominant factor in terms of limiting the system sum rate. Note that auser for the M = 2 case will only experience a third of the amount of multiuserinterference experienced by uses for the M = 4 cases. In addition, with the sametotal transmitting power constraint, the users in the M = 2 case enjoy twice thetransmitting power as the users in the M = 4 cases, which contributes furtherto the sum-rate advantage of the M = 2 case over the high SNR region. On theother hand, noise is the major limiting factor over the low SNR region whereasthe multiuser interference can be negligible. As such, transmitting to more userstranslates to approximately a linear increase in the system sum rate and canovercome the power loss due to the total power constraint.

In Fig. 7.9 we compare the sum-rate performance of different RUB schemesconsidered in this work for the case of four transmit antennae and a total of 20users. The sum rate of BBSI schemes with quantized SINR feedback (QBBSI),which leads to lower feedback load, is presented. Note that if N = 2r SINR


0 5 10 15 20 25 301

2

3

4

5

6

7

8

Average SNR (dB)

Sum

Rat

e (b

ps/H

z)

M = 4, K = 20

BBIQBBSI, N=2QBBSI, N=4QBBSI, N=8BBSI

Figure 7.9 Sum-rate performance of different RUB schemes as the function of averageSNR (SINR thresholds for QBBSI: 3 dB for N = 2; –5, 0 and 5 dB for N = 4; –6, –3,0, 3, 6, 9, and 12 dB for N = 8).

intervals are used, then each user needs to feed back r additional bits for itsbest-beam SINR feedback. As we can see, the sum-rate performance of all threeschemes improves as average SNR increases and saturate in the high SNR region,as expected. On the other hand, the sum rate of the BBI-based RUB schemeconverges to a much smaller value than other schemes. It is worth noting thatthe QBBSI-based scheme offers a much larger sum-rate capacity than the BBIscheme over high SNR, even with N = 2 SINR intervals, i.e. one additional bitfeedback for quantized best-beam SINR. We also notice that when the numberof quantization levels increases, the sum rate of the QBBSI scheme improves andquickly approaches that of the BBSI scheme. Note that the sum rate of QBBSIwith N = 8 intervals is within 0.3 bps/Hz to that of the BBSI scheme.

As a final numerical example, we plot the sum-rate of different RUB schemesas a function of the number of active users in the system in Fig. 7.10. We assumethat the total transmitting power budget is fixed to 10 dB with four transmitantennae. The SINR threshold values for QBBSI schemes are arbitrarily selectedto be the same as the ones used in Fig. 7.9. As we can clearly observe, whilethe sum-rate of BBSI and QBBSI-based RUB schemes increases as the numberof active users increases, that of BBI scheme quickly saturates at a small value(≈ 3.8 bps/Hz). This is because without any knowledge of SINR values, theBBI scheme cannot fully explore the multiuser diversity gain. We also see from


10 15 20 25 30 35 40 45 503

3.5

4

4.5

5

5.5

6

6.5

7

7.5M = 4, P = 10 dB

Number of Users, K

Sum

Rat

e (b

ps/H

z)

BBIQBBSI, N=2QBBSI, N=4QBBSI, N=8BBSI

Figure 7.10 Sum-rate performance of different RUB schemes as a function of thenumber of active users (SINR thresholds for QBBSI: 3 dB for N = 2; –5, 0 and 5 dBfor N = 4; –6, –3, 0, 3, 6, 9, and 12 dB for N = 8).

Fig. 7.10, the sum rate of QBBSI increases at a higher rate as K increases whenthe number of quantization levels becomes larger.

7.5 RUB with conditional best-beam index feedback

In this section, we present another low-feedback user scheduling strategy for aRUB-based multiuser MIMO system, termed adaptive beam activation basedon conditional best-beam index feedback (ABA-CBBI) [31]. In general, feedbackthresholding is a commonly adopted mechanism to reduce the feedback load inmultiuser systems [21, 28, 32]. For a multiuser MIMO system, it was suggestedin [21,28] to only allow those users with best-beam SINR greater than a thresholdto feed back their best beam information. With the ABA-CBBI scheme, similarto the strategy proposed in [28], only those users with their largest SINR greaterthan a threshold will feed back their best-beam indexes. When the thresholdis high and/or the number of users is moderate, it may be that a beam is notrequested by any user. When this happens, previous work [21, 28] suggests ran-domly selecting a user for those beams. Note that this random user selection foran unrequested beam cannot guarantee high SINR for a selected user and willintroduce more interuser interference. With the ABA-CBBI scheme, however,

RUB with conditional best-beam index feedback 223

only those beams that are requested by at least one user will be active for trans-mission. We study the sum-rate performance of the ABA-CBBI strategy throughaccurate statistical analysis. In particular, we derive the distribution of the num-ber of active beams and the distribution of their resulting SINR, which are thenapplied to obtain the exact analytical expression of the overall system sum rate.We also accurately quantify the average feedback load of the ABA-CBBI scheme.Based on these analytical results, we study the effects of the feedback thresholdon the sum-rate performance and its optimization. We show that the resultingABA-CBBI scheme can effectively explore multiuser diversity gain and controlinteruser interference for sum-rate performance benefit.

7.5.1 Mode of operation and feedback load analysis

At the beginning of the scheduling period, each user will determine the instan-taneous SINR on the available beams while assuming that all M beams will beactive. Specifically, with the knowledge of beamforming vector set U and the esti-mated channel vector hk , the SINR of user k on the jth beam can be calculatedas

γk,j =PM |hT

k uj |2PM

∑Mm=1,m =j |hT

k um |2 + 1, k = 1, 2, · · · ,K, j = 1, 2, · · · ,M. (7.80)

The users will then perform some comparison to determine their best beams j∗,which achieve the maximum SINR among all beams, namely j∗ = arg maxj (γk,j ).The corresponding SINR, denoted by γk,j ∗ , is called the best-beam SINR of userk. Each user compares its best-beam SINR with an SINR threshold, denoted byβ. Only when its best-beam SINR is greater than β will the user feed back itsbest-beam index, which results in a feedback of log2(M) bits per user. After col-lecting the best-beam indexes fed back from qualified users, the base station willassign a beam randomly to a user requesting it. If a beam is not requested by anyfeedback user, the base station will simply turn off that beam and redistributethe power to other beams. In this case, the number of active beams (or selectedusers) Ma will be less than the number of available beams M . Since ABA-CBBIrandomly assigns a beam to one of the users requesting it, this strategy has loweroperational complexity than the scheme proposed in [20], which ranks the SINRsof all users requesting that beam first and then assigns it to the user achievingthe maximum SINR on it.

The feedback load of the proposed ABA-CBBI strategy varies over time as thenumber of qualified users changes with the channel condition. We can quantifythe average feedback load of this strategy as

F = KPf log2 M, (7.81)

where Pf is the probability that a user is qualified to feed back. Based on themode of operation above, Pf is equal to the probability that the best-beamSINR of a user while assuming all M beams are active, γk,j ∗ , is greater than


the threshold β, i.e. Pf = Pr[γk,j ∗ < β]. With the homogeneous Rayleigh fad-ing assumption, it can be shown that γk,j ∗ are identically distributed with thecommon PDF given by

pγ

(M )j ∗

(x) =M

(M− 2)!

∫ ∞

0(M/P + z)e−x( M

P +z )−z× (7.82)

M −1∑j=0

(M − 1

j

)(−1)j

(z − jx(

M

P+ z)

)M−2U(z−jx(

M

P+ z)

)dz,

where U(·) denotes the unit step function. As such, the average feedback loadwith the proposed ABA-CBBI strategy is given by2

F = K log2 M

∫ ∞

β

pγ

(M )j ∗

(x)dx. (7.83)

Distribution of the number of active beamsA unique feature of the proposed ABA-CBBI strategy is no random user selectionfor unrequested beams. As a result, the number of selected users/active beams,Ma , becomes a discrete random variable taking values in [1, 2, · · · ,M ]. Note thatif every user feeds back, the probability of a beam not being requested by any userwill be small and even negligible, as long as the number of users is large enough.However, with conditional feedback strategies, this probability becomes larger asthe number of feedback users can be small due to the feedback thresholding. Inthe following, we derive the probability mass function (PMF) of Ma , which willbe applied to the sum-rate analysis of the proposed scheme in the next section.

With the ABA-CBBI strategy, the number of users that are qualified to feedback, denoted by Nf , is random. The probability of Nf = k is given by

Pr[Nf = k] =(

K

k

)Pk

f (1 − Pf )K−k , k = 0, 1, · · · ,K. (7.84)

Given that k users feed back their best-beam indexes, the probability that onlyone beam is active, i.e. Ma = m, can be calculated, by following the similarderivation in a previous section, as

Pr[Ma = m|Nf = k] =(

M

m

)(m

M

)k

(7.85)

−m−1∑j=1

(m

j

)Pr[Ma = j|Nf = k](

Mj

)

, m = 1, 2, · · · ,M.

which can be simplified to the following compact expression

Pr[Ma = m|Nf = k] =1

Mk

(M

m

)·

m∑q=1

(−1)m−q

(m

q

)qk , m = 1, 2, · · · ,M. (7.86)

2 This result corrects an error in [28, eq. (13)] as the SINR of a user for different beams arecorrelated random variables.


Finally, combining (7.84) and (7.86), the probability that exact m beams areactive can be obtained by applying the total probability theorem as

Pr[Ma = m] =K∑

k=1

Pr[Nf = k] Pr[Ma = m|Nf = k] (7.87)

=(

M

m

) m∑q=1

(−1)m+q

(m

q

)[(1 − M − q

MPf

)K

− (1 − Pf )K

],

where the feedback probability Pf is given in (7.83).


In this section, we investigate the performance of the proposed ABA-CBBI strat-egy by analytically deriving the exact sum-rate expression for the resulting mul-tiuser MIMO systems. Conditional on the number of active beams, the averagesum rate with ABA-CBBI strategy can be calculated as

E[R] =M∑

m=1

Pr[Ma = m]E

[m∑

i=1

log2(1 + γi)

], (7.88)

where Pr[Ma = m] was given in (7.87) and γi is the SINR of the ith selected user.Based on the mode of operation of the proposed strategy, the SINR of users ondifferent active beams are identically distributed. As such, we can rewrite thesum rate as

E[R] =M∑

m=1

Pr[Ma = m]m∫ ∞

0log2(1 + x)p

γ(m )B

(x)dx, (7.89)

where fγ

(m )B

(·) denotes the common PDF of the SINR of selected users given thatexactly m beams are active.

We now proceed to derive the PDF fγ

(m )B

(·). Based on the mode of operation

of the proposed ABA-CBBI strategy, γ(m )B is also the best-beam SINR of a

feedback user when m beams are active. Meanwhile, the best-beam SINR of sucha user when assuming all M beams are active, γ

(M )B , must be greater than the

threshold β. Therefore, the SINR distribution for active beams with the ABA-CBBI strategy is the conditional distribution of γ

(m )B given γ

(M )B > β. It follows

that the cumulative distribution function (CDF) of the SINR of the selectedusers when m beams are active can be determined as

Fγ

(m )B

(x) =Pr[γ(m )

B < x, γ(M )B > β]

Pr[γ(M )B > β]

. (7.90)

To proceed further, we note that a beam becomes the best beam for a particularuser if and only if the user’s channel vector has the largest projection norm


square (|hTk uj |2) on that beam direction. Therefore, we can rewrite γ

(m )B as

γ(m )B =

α1 :M

zm −1 +m/P , 1 < m ≤ M,

Pα1:M , m = 1,(7.91)

where α1:M = max |hTi uj |2 , j = 1, 2, · · · ,M is the largest projection norm

squares onto all M possible beam directions and zm−1 is the sum of arbitrarym − 1 projection norm squares out of the remaining M − 1 ones. As such, thejoint probability in (7.90) can be rewritten as

Pr[γ(m )B < x, γ

(M )B > β] (7.92)

=

⎧⎪⎪⎨⎪⎪⎩

Pr[Pα1:M < x, α1 :MzM −1 +M/P > β], m = 1;

Pr[ α1 :Mzm −1 +m/P < x, α1 :M

zm −1 +zM −m +M/P > β], 1 < m < M ;

Pr[β < α1 :MzM −1 +M/P < x], m = M,

where we define zM −m = zM −1 − zm−1 for the case of 1 < m < M . Consequently,the joint probability of the case of m = 1 can be calculated using the joint PDFof α1:M and zM −1 as

Pr[γ(m )B < x, γ

(M )B > β] (7.93)

=∫ x/P

0dy

∫ miny/β−M/P,(M −1)y

0dupα1 :M ,zM −1 (y, u),

where the joint PDF pα1 :M ,zM −1 (y, u) is given by

pα1 :M ,zM −1 (y, u) = pα1 :M (y)pzM −1 |α1 :M =y (u) (7.94)

= Me−y−uM −1∑j=0

(M − 1

j

)(−1)j (u − jy)M−2

(M− 2)!U(u−jy),

y > 0, u < (M − 1)y.

The joint probability of the cases of 1 < m < M can be calculated using the jointPDF of α1:M , zm−1 , and zM −m as

Pr[γ(m )B < x, γ

(M )B > β] =

∫ ∞

0dy

∫ y/β−M/P

y/x−m/P

du (7.95)

∫ y/β−u−M/P

0dvpα1 :M ,zm −1 ,zM −m

(y, u, v).

The joint PDF of α1:M , zm−1 and zM −m can be determined by following theBayesian approach. It has been shown that the unordered projection normsquares |hT

i uj |2 , j = 1, 2, · · · ,M , are i.i.d. χ2 random variables with two degreesof freedom [18]. Due to the ordering process, α1:M , zm−1 and zM −m are correlatedrandom variables. On the other hand, it can be shown that given α1:M = y, zm−1

and zM −m are the sum of m − 1 and M − m i.i.d. random variables with trun-cated distribution on the right at y, respectively, and as such are independent.


Specifically, the joint PDF of α1:M , zm−1 and zM −m can be written as

pα1 :M ,zm −1 ,zM −m(y, u, v) = pα1 :M (y)pzm −1 |α1 :M =y (u)pzM −m |α1 :M =y (v) (7.96)

= pα1 :M (y)p∑m −1j = 1 α−

j(u)p∑M

j = m + 1 α−j(v),

where α−j are i.i.d. random variables with PDF for the Rayleigh fading scenario

under consideration given by

pα−j(x) =

e−x

1 − e−y, 0 < x < y. (7.97)

After some mathematical manipulations, we can obtain the closed-form expres-sion for the joint PDF of α1:M , zm−1 and zM −m as

pα1 :M ,zm −1 ,zM −m(y, u, v) = Me−y−u−v

m−1∑j=0

(m − 1

j

)(−1)j (u − jy)m−2

(m− 2)!U(u−jy)

×M −m∑k=0

(M − m

k

)(−1)k (v − ky)M−m−1

(M−m− 1)!U(v−ky),

y > 0, u < (m − 1)y, v < (M − m)y. (7.98)

The joint probability for the case of m = M can be calculated using the PDF ofγ

(M )B given in (7.82) as

Pr[γ(m )B < x, γ

(M )B > β] =

∫ x

β

pγ

(M )B

(y)dy. (7.99)

After proper substitutions and taking derivatives, we obtain the PDF of activebeam SINR γ

(m )B as

pγ

(m )B

(x) = (7.100)

1Pf

⎧⎪⎪⎨⎪⎪⎩∫ x

P β −MP

01P pα1 :M ,zM −1 (

xP , u)du, m = 1;∫∞

0

∫ yβ − y

x −M −mP

0yx2 pα1 :M ,zm −1 ,zM −m

(y, yx − m

P , v)dvdy, 1 < m < M ;

fγ

(M )B

(x)U(x − β), m = M.

Finally, the exact sum-rate expression for RUB-based multiuser MIMO sys-tems with ABA-CBBI strategy can be obtained by substituting (7.87) and(7.100) into (7.89). The final expression can be readily evaluated using mathe-matical software such as Maple and Mathematica.

We now study the sum-rate performance of the proposed ABA-CBBI schedul-ing strategy through selected numerical examples. In particular, we compare itsperformance and complexity with some well-known user scheduling strategiesfor RUB-based multiuser MIMO systems. Table 7.2 summarizes the key featuresof the different user scheduling strategies under consideration. Figure 7.11 illus-trates the effects of random user selection for unrequested beams on the sum-rateperformance. In particular, we plot the sum rate of ABA-CBBI and conventionalCBBI as a function of SNR. As we can see, with the same threshold value, the


Table 7.2 Features of different strategies [31]. c© 2009 IEEE.

SINR thresholding SINR value feedback Random selection

BBSI No Yes YesCBBI Yes No YesABA-CBBI Yes No NoBBI No No Yes

5 10 15 20 25 30 35 400

1

2

3

4

5

6

7

8

9

10

SNR, P/M (dB)

Sum

Rat

e, R

(bi

t/s/H

z)

ABA−CBBI, β=2ABA−CBBI, β=1Conventional CBBI, β=1Conventional CBBI, β=2

Figure 7.11 Effect of random user selection on sum rate performance (K = 20) [31].c© 2009 IEEE.

sum rate of ABA-CBBI is always greater than that of conventional CBBI, whichshows that the RUB-based system is not benefiting from random user selection.We also notice from Fig. 7.11 that ABA-CBBI with different threshold values hasa better sum-rate performance over different SNR ranges. Therefore, to achievethe best sum-rate performance, we should use the optimal value of the feedbackthreshold β.

In Fig. 7.12, we plot the sum rate of the proposed ABA-CBBI strategy as afunction of the feedback threshold β. For a fixed SNR, there exists an optimalvalue of β maximizing the sum rate, which is marked in the figure. Intuitively,when β is very small, almost all users will feed back their best-beam indexes tothe base station. Since the base station randomly assigns a beam to one of theusers requesting it, that beam may be assigned to a user that does not achievehigh SINR, which leads to relatively poorer sum-rate performance. On the otherhand, when β is very large, few users will be qualified to feed back and some

RUB performance enhancement with linear combining 229

0 0.5 1 1.5 2 2.5 34

4.5

5

5.5

6

6.5

7

7.5

8

8.5

9

Threshold, β

Sum

Rat

e, R

(bi

t/s/H

z)

SNR=20dBSNR=40dB

1.9

2.2

Figure 7.12 Sum rate of ABA-CBBI as a function of SINR threshold for different SNRvalues (K =20) [31]. c© 2009 IEEE.

beams may not be requested. As such, not enough spatial multiplexing gain willexplored, which also causes poorer sum-rate performance.

In Fig. 7.13, we compare the sum-rate performance of the proposed ABA-CBBI strategy with two strategies without thresholding, BBSI and BBI. As wecan see, the sum rate of ABA-CBBI with optimal threshold values approachesthat of BBSI over low SNR region and as SNR increases, ABA-CBBI consid-erably outperforms BBSI. This somewhat surprising behavior can be explainedas follows. While the sum rate of the BBSI strategy saturates over high SNRregions due to multiuser interference, ABA-CBBI manages to reduce multiuserinterference by increasing the threshold value and then selecting fewer users toserve. As a result, the selected users will experience less multiuser interference,a bottleneck to achieving better sum-rate performance at high SNR for conven-tional RUB-based user scheduling schemes. By comparison, BBI has the worstperformance due to its complete lack of a mechanism to guarantee the high SINRof selected users.

7.6 RUB performance enhancement with linear combining

We investigate the performance enhancement of RUB-based multiuser MIMOsystems multiple receive antennae at each mobile user. Specifically, we assume


5 10 15 20 25 30 35 401

2

3

4

5

6

7

8

9

10

11

SNR (dB)

Sum

Rat

e, R

(bi

t/s/H

z)

ABA−CBBI with β*, K=50

ABA−CBBI with β*, K=20BBSI, K=50BBSI, K=20BBI, K=50BBI, K=20

Figure 7.13 Sum-rate comparison between ABA-CBBI, BBSI and BBI strategies [31].c© 2009 IEEE.

that each mobile user has N (N > 1) antennae. A conventional approach toexploit multiple receive antennae is to treat each antenna independently andlet each user feed back the best-beam indexes and corresponding SINR valuesfor all its receive antennae [20]. Intuitively, this approach would improve thesum-rate performance since it is equivalent to increasing the number of single-antenna users in the system to NK. The main disadvantage with this approachis, however, that the selected receiver will only use one designated antenna forreception and the received signals at other receive antennae are simply disgarded,which leads to inefficient utilization of the multiple receive antennae.

Alternatively, we can apply linear combining technique to fully explore themultiple receive antennae at each mobile [33,34]. In this section, we present andinvestigate two different feedback strategies for a RUB-based multiuser MIMOsystem with linear combining. In particular, we first present the system model,then describe the model of operation, and finally study their sum-rate perfor-mance through either asymptotic or exact statistic analysis. We consider theselection combining (SC) [35] and optimum combining (OC)3 [36–38] schemeshere.

3 We omit the maximum ratio combining (MRC) scheme because MRC is optimal only in thenoise-limited environment.


7.6.1 System and channel model

We consider the downlink transmission of a single-cell wireless system. The basestation is equipped with M antennae and each mobile terminal is equipped withN antennae, where N< M due to the size and cost limitations of the mobiles.We assume that there are a total of K active users in the cell where K M .With RUB, the base station utilizes M random orthonormal vectors generatedfrom an isotropic distribution [18] to transmit to as many as M selected userssimultaneously. We denote the vector set as U = u1 ,u2 , · · · ,uM . The trans-mitted signal vector from M antennae over one symbol period can be writtenas

x =M∑

m=1

um sm , (7.101)

where sm is the information symbol for the mth selected user. The transmittedsignal vector x has an average power constraint of E(xH x) ≤ P , where P is themaximum average transmitting power, E(·) denotes the statistical expectation,and (·)H stands for the Hermitian transpose.

We adopt a homogeneous flat fading channel model for analytical tractabil-ity. Specifically, the channel gains from each transmit antenna to each receiveantenna of users follow i.i.d. zero-mean complex Gaussian distribution with unitvariance. The channel gain from the mth transmit antenna to the nth receiveantenna of the kth mobile users is denoted by h

(k)nm . As such, the N×M channel

matrix for the kth user is constructed as

Hk =

⎛⎜⎜⎜⎜⎝

h(k)11 h

(k)12 · · · h

(k)1M

h(k)21 h

(k)22 · · · h

(k)2M

......

. . ....

h(k)N 1 h

(k)N 2 · · · h

(k)N M

⎞⎟⎟⎟⎟⎠ (7.102)

with the nth row, denoted by h(k)n , being the instantaneous MISO channel for

the nth receive antenna of the kth user.The received signal vector over a symbol period at the ith selected user, when

the jth beam is assigned to it, can be written as

y(i)j = Hix + ni = Hiuj sj +

M∑m=1,m =j

Hium sm + ni , (7.103)

where ni is the N × 1 additive Gaussian noise vector, whose entries follow inde-pendent complex Gaussian distribution with zero mean and variance σ2

n . Notethat Hiuj sj is the desired signal for user i while

∑Mm=1,m =j Hium sm acts as

interference.The mobile receivers apply linear diversity combining techniques, such as SC

and OC, to combine the signal received from N different receive antennae. Whenbeam j is assigned to the ith user, the combined signal at user i can be written


as

z(i)j = wH

ij y(i)j = wH

ij Hiuj sj + wHij

M∑m=1,m =j

Hium sm + wHij ni , (7.104)

where wij denotes the weighting vector of user i for the jth beam. Assumingthat the transmit power is allocated equally among M selected users, the SINRof the combined signal at the user i when the jth beam is assigned to it can beshown to be given by

γ(i)j =

|wHij Hiuj |2

M∑m=1,m =j

|wHij Hium |2 + ρ‖wij‖2

, (7.105)

where ρ = M σ 2n

P is the normalized transmit SNR. We assume that users havethe perfect knowledge of their downlink channel matrix through the channelestimation process. Applying (7.105), user i calculates the SINRs after beam-specific combining for all M available beams, γ

(i)j , j = 1, 2, · · · ,M .

7.6.2 M beam feedback strategy

With this strategy, each mobile evaluates the resulting SINR for each beam atthe beginning of each time slot, after linearly combining the received signals onall antennae with either SC or OC. Then, the mobile will return the informa-tion of M effective beam SINRs to the BS. The incurred feedback load withthis strategy is M real numbers per mobile. Note that OC performs active inter-ference suppression by exploiting the interference structure, whereas SC simplyselects the beam with the best SINR. It will be shown later that this charac-teristic interference-suppression feature enables OC to considerably outperformSC with higher implementation complexity. After receiving the effective SINRinformation from all mobiles, the BS schedules and starts data transmission tothe mobiles that reported the largest effective SINRs on different beams. Theselected mobile will combine the received signals from all antennae before datadetection.

We now apply an asymptotic approach to analyze the performance of thisstrategy based on the limiting distribution of the beam SINR [33]. It is worthnoting that the probability of awarding multiple beams to the same mobile israther small, when the number of mobiles K is large. As such, the SINR of the jthbeam after user selection is equal to γB

j = maxγ(1)j , γ

(2)j , · · · , γ(i)

j , where γ(i)j ,

i = 1, 2, · · · ,K, are the resulting SINR at user i for the beam j after applyingdiversity combining. With the homogeneous fading assumption, we can showthat the CDF of γB

j is given by

Fγ Bj

(x) = [Fγ

( i )j

(x)]K . (7.106)


In the following, we develop the limiting distribution of γBj and use it to exam-

ine the asymptotic throughput and sum-rate scaling laws of different combiningschemes. To make the analysis easy to follow, we focus on the case of four trans-mit antennae M = 4 and two receive antennae per mobile N = 2.

Selection combiningFor SC, we first obtain the CDF of the resulting SINR after beam-specific com-bining as

Fγ

( i )j

(x) =(

1 − e−ρx

(1 + x)3

)2

. (7.107)

It follows that the PDF of γi,j is given by

pγ

( i )j

(x) = 2(

1 − e−ρx

(1 + x)3

)e−ρx

(1 + x)4 (ρ(1 + x) + 3) . (7.108)

We can then verify easily that the following condition is satisfied

limx→+∞

1 − Fγ

( i )j

(x)

pγ

( i )j

(x)= 1/ρ > 0, (7.109)

which implies that the limiting distribution of γBj exists and follows the Gumbel

type. More specifically, the limiting distribution of γBj when K approach infinity

is given by

limK→+∞

Fγ Bj

(x) = exp(−e

− x −b Ka K

), (7.110)

where aK and bK are normalizing factors, which can be determined by solv-ing numerically the equations 1 − F

γ( i )j

(bK ) = 1/K and 1 − Fγ

( i )j

(aK + bK ) =

1/(eK).Starting from this limiting distribution, it can be shown that the corresponding

scaling law for SC scheme is given by [33]

limK→+∞

CSC

4 log bK= 1. (7.111)

When ρ = 1, we can approximate bK as

bK ≈ log 2K − 2 log(1 + log 2K). (7.112)

Optimal combiningThe resulting SINR after applying OC for a particular beam is equivalent to thecombined SINR with OC in the presence of M − 1 interfering sources, the CDFof which has been derived in [39]. For the four transmit antennae and two receiveantennae case under consideration, the CDF specializes to

Fγ

( i )j

(x) = 1 − e−ρx

(1 + x)3 (1 − 3x − ρx). (7.113)


It follows that the PDF is given by

pγ

( i )j

(x) =xe−ρx

(1 + x)4 [(3ρ − ρ2)x + (6 + 6ρ + ρ2)]. (7.114)

Again, based on these results, it can be shown that the limiting distribution ofγB

j is also following the Gumbel type. Following the similar procedure of theSC case, we can show that that the corresponding scaling law for OC scheme isgiven by [33]

limK→+∞

COC

4 log bK= 1. (7.115)

When ρ = 1, we can approximate bK as

bK ≈ log 4K − 2 log log K. (7.116)

7.6.3 Best-beam feedback strategy

In this section, we investigate the sum-rate performance of RUB systems withlinear combining when each user only feeds back the best-beam information.Specifically, the user first calculates the output SINR after beam-specific com-bining for each of the M available beams. Then, the user selects its best beamamong M beams, i.e. the one that achieves the largest combiner output SINR.Let bi denote the index of the best beam for user i. Mathematically, bi is given bybi = arg maxj γ

(i)j . The index of the best beam bi and its corresponding output

SINR γ(i)bi

are fed back to the base station. Note that the feedback load withreceiver combining techniques is one integer (for best-beam index) and one realnumber (for best-beam output SINR) per user, which is much lower than thatof the M beam feedback strategy discussed in the previous subsection.

The base station assigns a beam to the user that feeds back the largest SINRvalue among all users requesting for that beam. Specifically, let Km denote theuser index set containing all users who feed back beam index m, i.e. bk = m ifand only if k ∈ Km . If γ

(k)m is the largest among all γ

(k)m , k ∈ Km , then the mth

beam will be assigned to user k and the SINR of the mth beam γBm will be equal

to γ(k)m . If a particular beam is not requested by any user, i.e. the set Km is empty

for such a beam, then the base station knows that this beam is not the best onefor any user and will allocate it to a randomly chosen user. The selected userwill then apply proper weights to combine the signal from different antennae.

In what follows, we analyze the sum-rate performance of the resulting systemthrough accurate statistical analysis. Unlike the limiting distribution approachadopted in a previous section, which applies only when the number of users isvery large, this analysis applies to an arbitrary number of users. In particular,we derive the exact statistics of the feedback SINR for the SC scheme, based onwhich we arrive at the exact expression of its sum rate. By following a geometricapproach, we also develop the statistics of the OC output SIR over high SNR


region, which is then used to obtain an upper bound for the sum rate. The upperbound is shown to be tight for a moderate number of users, but the number ofusers must be at least five times the number of transmit antennae.

Selection combiningWith the SC scheme, the combined SINR at user i for the jth beam is equal tothe largest SINR among N ones corresponding to different receive antennae forthis beam. Specifically, the SINR on user i’s nth receive antenna, while assumingbeam j is assigned to it, can be written as

γ(i)n,j =

|(h(i)n )T uj |2∑M

m=1,m =j |(h(i)n )T um |2 + ρ

, 1≤n≤N, 1≤j≤M, (7.117)

where h(i)n is the M×1 channel vector from M transmit antennas to the nth

receive antenna of user i. The combined SINR for the jth beam is then givenmathematically by γ

(i)j = maxγ(i)

1,j , γ(i)2,j , · · · , γ

(i)N,j. If a user is selected for data

reception on a certain beam, the user will use only the best receive antenna forthis beam during reception, which constitutes the main complexity advantage ofthe SC scheme.

The sum-rate performance of the resulting multiuser MIMO system can beanalyzed following a similar approach as in a previous section while treating eachuser as a single antenna receiver. On the other hand, because of the antenna selec-tion process at the mobile users, the statistics of the feedback SINR is differentfrom the single receive antenna case. In particular, γ

(i)bi

is the largest of M i.i.d.

combined SINRs, i.e. γ(i)bi

= maxγ(i)1 , γ

(i)2 , · · · , γ(i)

M . Equivalently, the feedback

SINR γ(i)bi

can be viewed as the largest among N best-beam SINRs, i.e. γ(i)bi

=

maxγ(i)1,j ∗ , γ

(i)2,j ∗ , · · · , γ

(i)N,j ∗, where γ

(i)n,j ∗ = maxγ(i)

n,1 , γ(i)n,2 , · · · , γ

(i)n,M denotes the

best-beam SINR for the nth antenna at user i, whose PDF was obtained as

pγ

( i )n , j ∗

(x) =M −1∑j=0

(−1)jM(M − 1)(M− j − 1)!j!

∫ ∞

0(ρ + z) (7.118)

×((1 − jx)z − jxρ)M−2e−(1+x)z−xρU((1 − jx)z−jxρ)dz.

Therefore, the common PDF of the feedback SINRs γ(i)bi

can be obtained as

pγ

( i )b i

(x) = NFγ

( i )n , j ∗

(x)N −1pγ

( i )n , j ∗

(x), (7.119)

where Fγ

( i )n , j ∗

(x) =∫ x

0 pγ

( i )n , j ∗

(z)dz is the CDF of γ(i)n,j ∗ .

Based on the mode of operation for user selection, a certain beam will beassigned to a randomly selected user if no user feeds back the index of such abeam. In this case, the user will choose the best antenna, i.e. the receive antennathat leads to the largest SINR on this beam, for data reception. As such, thebeam SINR for such a beam will be equal to the largest SINR among N i.i.d.


ones, whose PDF is given by

fγ R (x) = N(F

γ( i )n , j

(x))N −1

pγ

( i )n , j

(x), (7.120)

where fγ

( i )n , j

(x) and Fγ

( i )n , j

(x) denote the PDF and CDF of γ(i)n,j in (7.117), respec-

tively. The closed-form expression of fγ

( i )n , j

(x) is given by [20, eq. (14)]

pγ

( i )n , j

(x) =e−x/ρ

(1 + x)M

(x + 1

ρ+ M − 1

). (7.121)

Finally, following the similar analytical approach of the previous section, whilenoting that the PDF of user feedback SINR is given in (7.118) and the PDF ofrandomly assigned beam SINR in (7.121), the sum rate of RUB system with SCat receivers can be calculated as

RSC =∫ ∞

0log2(1 + x)p

γ( 1 :K )b i

(x)dx (7.122)

+M∑

m=2

K∑i=m

(Pm

i

∫ ∞

0log2(1 + x) p

γ( i :K )b i

(x)dx

+(

m − 1M

)K−1 ∫ ∞

0log2(1 + x)pγ R (x)dx

),

where Pmi denotes the probability that the mth beam is assigned to the user

with the ith largest feedback SINR, i.e. γBm = γ

(i:K )bi

, i = m,m + 1, · · · ,K, whichis given by,

Pmi =

∑∑m −1

j = 1n j = i−m

n j ∈0 , 1 , ··· , i−m

⎡⎣m−1∏

j=1

(j

M

)nj(

1 − j

M

)⎤⎦ , i = m,m + 1, · · · ,K, (7.123)

and pγ

( i :K )b i

(x) represents the PDF of the ith user feedback SINR, given in terms

of fγ

( i )b i

(x) as

pγ

( i :K )b i

(x) =K!

(K − i)!(i − 1)!F

γ( i )b i

(x)K−i [1 − Fγ

( i )b i

(x)M ]i−1pγ

( i )b i

(x). (7.124)

Optimum combiningWith the OC scheme [35,36], it can be shown that the optimal weighting vectorof user i for the jth beam is given by

w∗ij =

⎛⎝ M∑

m=1,m =j

Hium (Hium )H + ρI

⎞⎠−1

Hiuj , (7.125)


and the corresponding maximum SINR of the combined signal is given by

γ(i)j = (Hiuj )H w∗

ij = (Hiuj )H

⎛⎝ M∑

m=1,m =j

Hium (Hium )H + ρI

⎞⎠−1

Hiuj ,

(7.126)

where I denotes the identity matrix. After proper comparison, user i will feedback the best-beam index bi and the corresponding combined SINR, γ

(i)bi

, to thebase station for user selection. If the user is selected for data reception on thebith beam, the user will combine the signal from different antennae using weightsw∗

ibicalculated using (7.125) for data detection.

After some mathematical manipulation, the optimal weight vector and thecorresponding output SINR are simplified to

w∗ij =

(HiHHi + 1

ρ I)−1Hiuj

1 − (Hiuj )H (HiHHi + 1

ρ I)−1Hiuj

, (7.127)

and

γ(i)j =

(Hiuj )H (HiHHi + 1

ρ I)−1Hiuj

1 − (Hiuj )H (HiHHi + 1

ρ I)−1Hiuj

, (7.128)

respectively. Note that each user needs only to calculate the inversion of a singlematrix (HiHH

i + 1ρ I) now. Although the statistics of the combined SINR for a

single beam can be obtained [39,40], because of the dependence between SINRsfor different beams, it is very difficult, if not impossible, to derive the statisticsof the feedback SINR with the OC scheme. Therefore, we focus on the sum rateupper bound analysis over a high SNR region in what follows.

When the SNR is very high and the system becomes interference-limited, theoptimal weighting vector can be approximately given by, after setting ρ in (7.127)to infinity,

w∗ij =

(HiHHi )−1Hiuj

1 − (Hiuj )H (HiHHi )−1Hiuj

. (7.129)

It follows that w∗ij is proportional to the least square solution of HH

i w = uj

[39, 41]. Therefore, with the optimal weighting vector, the angle between theeffective MISO channel for user i, HH

i w∗ij , and the jth beam, uj is minimized,

which is equal to the angle between uj , and the range space of HHi , denoted by

R(HHi ).

Lemma 1. Over a high SNR region, the maximum output SINR at user i afterperforming optimal combining with respect to the jth beam is given by

γ∗i,j = | cot(θij )|2 , (7.130)

where θij is the angle between uj and R(HHi ).


Proof. Let ρ in (7.128) go to infinity, the maximum output SINR at user i overhigh SNR region becomes

γ∗i,j =

(Hiuj )H (HiHHi )−1Hiuj

1 − (Hiuj )H (HiHHi )−1Hiuj

. (7.131)

Since HHi (HiHH

i )Hi is the projection matrix onto R(HHi ), we have

| cos(θij )|2 = (Hiuj )H (HiHHi )−1Hiuj . (7.132)

The lemma is proved after substituting (7.132) into (7.131).

Since cot(·) is a monotonically decreasing function, each user can feed back theindex and SINR value of the beam that forms the smallest angle with R(HH

i ).It can be shown that the random variable | cos(θij )|2 , where θij is the angle

between a random vector of length M and a space spanned by N independentrandom vectors of the same length, follows the beta distribution parameterizedby N and M − N , with PDF and CDF given by [17]

f| cos(θ)|2 (x) = (M − N)(

M − 1N − 1

)xN −1 ·(1 − x)M −N −1 , 0≤x≤1 (7.133)

and

F| cos(θ)|2 (x) = I(x;N,M), 0≤x≤1, (7.134)

respectively. To derive an upper bound of the system sum rate, we assume thatevery beam of the base station is requested by at least one user. In this case,since a user can only be assigned to one beam, the SINR of the jth assignedbeam can be shown to be equal to the largest among K − j + 1 combined SINRson that beam for the remaining K − j + 1 users (j − 1 users have been previ-ously assigned). Mathematically speaking, the SINR of the jth assigned beam isequal to | cot(θ∗j )|2 , where θ∗j is the smallest angle among K − j + 1 ones. Corre-spondingly, | cos(θ∗j )|2 will be the largest one among K − j + 1 independent Betadistributed random variables. The PDF of | cos(θ∗j )|2 is thus given by [27]

f| cos(θ∗j )|2 (x) = (K − j + 1)

(F| cos(θ)|2 (x)

)K−jf| cos(θ)|2 (x), 0≤x≤1. (7.135)

Noting that log2(1 + | cot(θ∗j )|2) = − log2(1 − | cos(θ∗j )|2), we obtain a sum rateupper bound for the proposed scheme over the high SNR region as

ROC =M∑

j=1

−(K − j + 1)∫ 1

0log2(1 − x)

(F| cos(θ)|2 (x)

)K−jf| cos(θ)|2 (x)dx.

(7.136)As this upper bound is derived based on the assumption that all beams arerequested, it will be tight as long as the probability of all beams being requestedis close to 1. In [31], we showed that the probability that each of M available


1020

3040

50

24

680

0.2

0.4

0.6

0.8

1

KM

prob

Figure 7.14 Probability of all beams being requested [34]. c© 2010 IEEE.

beam is requested by at least one user is given by

Pr[every beam requested] =M∑

q=1

(−1)M −q

(M

q

)( q

M

)K

. (7.137)

This probability is plotted as a function of the number of users K and the numberof beams M in Fig. 7.14. As we can see, when K is more than five times M , whichis usually the case in practice, every beam will be requested with probability veryclose to 1.

In Fig. 7.15, we plot the sum rate of SC-based systems as a function of SNRfor a different number of users. We set M = 4 and N = 2 here. The sum rate ofthe conventional approach, which treats a user’s receive antennae independently,is also plotted for comparison. Although requiring only 1/N of the feedback loadof the conventional approach, SC is shown to achieve nearly the same sum-rateperformance. In Fig. 7.15, we also compare our analytical results given by (7.123)to simulation results. The perfect match verifies the analytical approach adoptedin this work.

In Fig. 7.16, the sum rate of the OC-based system is plotted as a function ofSNR for a different number of users. The sum rate of the conventional approach,M beam feedback (the scheme considered in [33, 40]), and the analytical upperbound given by (7.136) are also plotted. As expected, the analytical upperbound developed for OC-based system is shown to be tight over the high SNRregion. Compared to the conventional approach, OC achieves noticeably better


0 5 10 15 20 25 30 35 401

2

3

4

5

6

7

8

9

10

SNR (dB)

Sum

Rat

e (b

it/s/

Hz)

Conventional, N=2,K=25SC, N=2,K=25, AnalysisSC, N=2,K=25, SimulationConventional, N=2,K=10SC, N=2,K=10, AnalysisSC, N=2,K=10, Simulation

Figure 7.15 Sum-rate performance of RUB systems with SC scheme (M = 4) [34].c© 2010 IEEE.

0 5 10 15 20 25 30 35 400

5

10

15

SNR (dB)

Sum

Rat

e (b

it/s/

Hz)

Bound, N=2, K=25M Beam Feedback, N=2,K =25Best Beam Feedback, N=2,K =25Bound, N=2,K =10M Beam Feedback, N=2,K =10Best Beam Feedback, N=2,K =10Conventional, N=2,K =25Conventional, N=2,K =10

Figure 7.16 Sum-rate performance of RUB systems with OC scheme (M = 4) [34].c© 2010 IEEE.

References 241

sum-rate performance while requiring less feedback load. We further observethat the proposed OC-based scheme offers nearly the same performance as theM beam feedback scheme in [33, 40], which mandates M real numbers per userfeedback load. It is worthwhile mentioning that the advantages of OC come atthe cost of additional processing complexity of channel matrix inversion whencalculating the combining weights and output SINR.

7.7 Summary

In this chapter, we focused on the low-complexity beamforming/user schedulingschemes for multiuser MIMO systems. While ZFBF can completely eliminatethe interuser interference when operating in multiuser MIMO channels, RUBschemes incur much lower feedback load, as the users need not feed back thecomplete channel gain vectors. As it is usually sufficient for the users to feedback their best-beam SINR information, we applied the order statistics resultsto obtain the exact statistics of users’ best-beam SINR, which found applicationsin the analysis of several user scheduling schemes for RUB systems. Finally, weshowed that the receiver combining approach can offer significant performancegain for the multiple antennae per user case.


For a more thorough discussion on point-to-point MIMO systems, including thetransceiver design and decoding algorithms, the reader may refer to [11,12]. Thecapacity of multiuser MIMO systems and its achieving with DPC was investi-gated in [6,8]. The optimality of the ZFBF scheme over multiuser MIMO channelis examined in [17, 42], whereas that for the RUB scheme can be found in [20].Refs [43–46] present some additional user scheduling scheme with limited feed-back for multiuser MIMO systems. A beamforming schemes with limited feed-back for multiple access channel was considered in [47]. Finally, more details foroptimal combining can be found in [35].

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Index

Adaptive modulation, 24–26Additive white Gaussian noise, 18

joint adaptive modulation and diversitycombining 145–159

Autocorrelation function, 16Average error rate, 2, 20–21

Best-beam SINR, 209-213Block fading model, 16

Cell coverage analysis, 9–10Channel correlation, 15–16

channel coherence bandwidth, 12, 172channel coherence time, 15–16, 172–175

Dirty paper coding, 197Diversity combining,

adaptive transmit diversity 127–135antenna reception, 26–27transmit diversity, 35–37

Exceedance distriubtion function, 55Equal gain combining, 26, 30

Fading,flat, 13–15general model, 10–11Nakagami, 15, 22Rayleigh, 14, 21, 35Rician, 14–15, 20selective, 11–12

Feedback load 167–169

Generalized extreme value, 62Generalized selection combining 3,

73–76absolute threshold, 76minimum estimation and combining

98,159minimum selection 105–116normalized threshold, 76output threshold 116–127threshold test per branch 76–79

Generalized switch and examine combining79–84

post-examining selection 84–94

Intersymbol interference, 11

Level crossing rate, 170Line of sight, 7–8,Linear bandpass modulation, 16–19

amplitude shift keying, 16phase shift keying, 16quadrature amplitude modulation,

17Log-normal, 9

Maximum ratio combining, 28–30output threshold 99–105

Mean square error, 8Moment generating function,

average error rate analysis 22definition and properties, 21–22joint distribution of order statistics

53–55Multiuser diversity 164–165Multiuser parallel scheduling 173–183

Optimal combining, 237–239Order statistics,

conditional distribution, 41–42distribution of extremes 61–64distribution of partial sum, 42–46joint distribution of partial sums,

46–53marginal distribution, 41

Orthogonal frequency division multiplexing(OFDM), 12

Orthogonal frequency division multipleaccess (OFDMA), 3, 5

Outage probability, 2, 20,

Path loss, 8log-distance model, 8–10

255

256 Index

Power allocation, 183–190Power spectrum density, 18

RAKE finger management 135–145Root mean square, 11Random unitary beamforming,

208–240

Selection combining, 27–28, 236–237Signal-to-noise ratio, 2Shadowing, 9Soft handover, 135–159Space-time block coding, 127–137

Threshold combiningswitch and examine combining

31–32

switch and examine combining withpost-examining selection, 32–35

switch and stay combining 30–31

User schedulingabsolute SNR-based 169–171normalized SNR-based 171–172on–off based, 175–283switched based, 177–190

Ultra wideband, 73–74, 135, 172

Wideband code division multiple access,73–74, 135

Zeroforcing beamforming, 195–241greedy weight clique, 199–206successive projection, 199–206

tele-0521199255

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