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Wireless Multi-AntennaChannels

Wiley Series on Wireless Communications and Mobile Computing

Series Editors: Dr Xuemin (Sherman) Shen, University of Waterloo, CanadaDr Yi Pan, Georgia State University, USA

The “Wiley Series on Wireless Communications and Mobile Computing” is a seriesof comprehensive, practical and timely books on wireless communication andnetwork systems. The series focuses on topics ranging from wireless communi-cation and coding theory to wireless applications and pervasive computing. Thebooks provide engineers and other technical professionals, researchers, educators,and advanced students in these fields with invaluable insight into the latest devel-opments and cutting-edge research.

Other titles in the series:

Misic and Misic: Wireless Personal Area Networks: Performance, Interconnection,and Security with IEEE 802.15.4 , January 2008, 978-0-470-51847-2

Takagi and Walke: Spectrum Requirement Planning in Wireless Communications:Model and Methodology for IMT-Advanced , April 2008, 978-0-470-98647-9

Pérez-Fontán and Espiñeira: Modeling the Wireless Propagation Channel: A Simu-lation Approach with MATLAB, August 2008, 978-0-470-72785-0

Ippolito: Satellite Communications Systems Engineering: Atmospheric Effects,Satellite Link Design and System Performance, August 2008, 978-0-470-72527-6

Lin and Sou: Charging for Mobile All-IP Telecommunications , September 2008,978-0-470-77565-3

Myung and Goodman: Single Carrier FDMA: A New Air Interface for Long TermEvolution , October 2008, 978-0-470-72449-1

Wang, Kondi, Luthra and Ci: 4G Wireless Video Communications , April 2009,978-0-470-77307-9

Cai, Shen and Mark: Multimedia Services in Wireless Internet: Modeling and Anal-ysis , June 2009, 978-0-470-77065-8

Stojmenovic: Wireless Sensor and Actuator Networks: Algorithms and Protocolsfor Scalable Coordination and Data Communication, February 2010, 978-0-470-17082-3

Liu and Weiss, Wideband Beamforming: Concepts and Techniques , March 2010,978-0-470-71392-1

Riccharia and Westbrook, Satellite Systems for Personal Applications: Conceptsand Technology , July 2010, 978-0-470-71428-7

Zeng, Lou and Li, Multihop Wireless Networks: Opportunistic Routing , July 2011,978-0-470-66617-3

Qian, Muller and Chen: Security in Wireless Networks and Systems , February 2013,978-0-470-512128

Wireless Multi-AntennaChannels

Modeling and Simulation

Serguei L. Primak

The University of Western Ontario, Canada

Valeri Kontorovich

CINVESTAV, Mexico

A John Wiley & Sons, Ltd., Publication

This edition first published 2012 2012 John Wiley & Sons Ltd

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Library of Congress Cataloging-in-Publication Data

Primak, L. Serguei.aWireless multi-antenna channels : modeling and simulation / Serguei L. Primak, Valeri Kontorovich.aa p. cm. – (Wireless communications and mobile computing)a Includes bibliographical references and index., ISBN 978-0-470-69720-7 (hardback)1. Roaming (Telecommunication)–Mathematical models.aa 2. MIMO systems–Mathematical models.aa

3. Antenna radiation patterns–Mathematical models.aa 4. Antenna arrays–Mathematical models.aa5. Adaptive antennas–Mathematical models.aa I. Kontorovich, V. IA. (Valeri Kontorovich)aa II. Title.a TK5103.4874.P75 2011a 621.3845′6–dc23,aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa2011025940

A catalogue record for this book is available from the British Library.

ISBN: 978-0-470-69720-7 (H/B)ISBN: 978-1-119-95471-2 (ePDF)ISBN: 978-1-119-95472-9 (oBook)ISBN: 978-1-119-96086-7 (ePub)ISBN: 978-1-119-96087-4 (eMobi)

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Contents

About the Series Editors xi

1 Introduction 11.1 General remarks 11.2 Signals, interference, and types of parallel channels 3

2 Four-parametric model of a SISO channel 72.1 Multipath propagation 72.2 Random walk approach to modeling of scattering field 13

2.2.1 Random walk in two dimensions as a modelfor scattering field 13

2.2.2 Phase distribution and scattering strength 142.2.3 Distribution of intensity 142.2.4 Distribution of the random phase 17

2.3 Gaussian case 182.3.1 Four-parametric distribution family 182.3.2 Distribution of the magnitude 202.3.3 Distribution of the phase 272.3.4 Moment generating function, moments and cumulants

of four-parametric distribution 292.3.5 Some aspects of multiple scattering propagation 29

3 Models of MIMO channels 333.1 General classification of MIMO channel models 333.2 Physical models 33

3.2.1 Deterministic models 343.2.2 Geometry-based stochastic models 35

3.3 Analytical models 363.3.1 Channel matrix model 37

3.4 Geometrical phenomenological models 473.4.1 Scattering from rough surfaces 48

vi CONTENTS

3.5 On the role of trigonometric polynomials in analysisand simulation of MIMO channels 493.5.1 Measures of dependency 503.5.2 Non-negative trigonometric polynomials and their use

in estimation of AoD and AoA distribution 513.5.3 Approximation of marginal PDF using non-negative

polynomials 513.6 Canonical expansions of bivariate distributions and the structure

MIMO channel covariance matrix 523.6.1 Canonical variables and expansion 523.6.2 General structure of the full covariance matrix 543.6.3 Relationship to other models 54

3.7 Bivariate von Mises distribution with correlated transmitand receive sides 563.7.1 Single cluster scenario 563.7.2 Multiple clusters scenario 58

3.8 Bivariate uniform distributions 583.8.1 Harmonic coupling 583.8.2 Markov-type bivariate density 61

3.9 Analytical expression for the diversity measureof an antenna array 623.9.1 Relation of the shape of the spatial covariance

function to trigonometric moments 623.9.2 Approximation of the diversity measure for a large

number of antennas 643.9.3 Examples 663.9.4 Leading term analysis of degrees of freedom 70

3.10 Effect of AoA/AoD dependency on the SDoF 723.11 Space-time covariance function 72

3.11.1 Basic equation 723.11.2 Approximations 73

3.12 Examples: synthetic data and uniform linear array 753.13 Approximation of a matrix by a Toeplitz matrix 773.14 Asymptotic expansions of diversity measure 783.15 Distributed scattering model 79

4 Modeling of wideband multiple channels 814.1 Standard models of channels 82

4.1.1 COST 259/273 824.1.2 3GPP SCM 834.1.3 WINNER channel models 84

4.2 MDPSS based wideband channel simulator 844.2.1 Geometry of the problem 844.2.2 Statistical description 854.2.3 Multi-cluster environment 874.2.4 Simulation of dynamically changing environment 88

CONTENTS vii

4.3 Measurement based simulator 894.4 Examples 91

4.4.1 Two cluster model 924.4.2 Environment specified by joint AoA/AoD/ToA

distribution 934.4.3 Measurement based simulator 95

4.5 Appendix A: simulation parameters 96

5 Capacity of communication channels 995.1 Introduction 995.2 Ergodic capacity of MIMO channel 100

5.2.1 Capacity of a constant (static) MIMO channel 1005.2.2 Alternative normalization 1025.2.3 Capacity of a static MIMO channel under different

operation modes 1035.2.4 Ergodic capacity of a random channel 1045.2.5 Ergodic capacity of MIMO channels 1065.2.6 Asymptotic analysis of capacity and outage capacity 106

5.3 Effects of MIMO models and their parameterson the predicted capacity of MIMO channels 1095.3.1 Channel estimation and effective SNR 1105.3.2 Achievable rates in Rayleigh channels with partial CSI 1135.3.3 Examples 116

5.4 Time evolution of capacity 1195.4.1 Time evolution of capacity in SISO channels 1195.4.2 SISO channel capacity evolution 120

5.5 Sparse MIMO channel model 1225.6 Statistical properties of capacity 124

5.6.1 Some mathematical expressions 1245.7 Time-varying statistics 125

5.7.1 Unordered eigenvalues 1255.7.2 Single cluster capacity LCR and AFD 1265.7.3 Approximation of multi-cluster capacity

LCR and AFD 1265.7.4 Statistical simulation results 129

6 Estimation and prediction of communication channels 1316.1 General remarks on estimation of time-varying channels 1316.2 Velocity estimation 131

6.2.1 Velocity estimation based on the covariance functionapproximation 131

6.2.2 Estimation based on reflection coefficients 1326.3 K-factor estimation 133

6.3.1 Moment matching estimation 1336.3.2 I/Q based methods 134

6.4 Estimation of four-parametric distributions 135

viii CONTENTS

6.5 Estimation of narrowband MIMO channels 1386.5.1 Superimposed pilot estimation scheme 1386.5.2 LS estimation 1406.5.3 Scaled least-square (SLS) estimation 1426.5.4 Minimum MSE 1446.5.5 Relaxed MMSE estimators 146

6.6 Using frames for channel state estimation 1486.6.1 Properties of the spectrum of a mobile channel 1496.6.2 Frames based on DPSS 1506.6.3 Discrete prolate spheroidal sequences 1506.6.4 Numerical simulation 154

7 Effects of prediction and estimation errors on performanceof communication systems 1577.1 Kolmogorov–Szegö-Krein formula 1607.2 Prediction error for different antennas and scattering

characteristics 1627.2.1 SISO channel 1627.2.2 SIMO channel 1657.2.3 MISO channel 1677.2.4 MIMO channel 170

7.3 Summary of infinite horizon prediction results 1747.4 Eigenstructure of two cluster correlation matrix 1757.5 Preliminary comments on finite horizon prediction 1767.6 SISO channel prediction 178

7.6.1 Wiener filter 1787.6.2 Single pilot prediction in a two cluster environment 1797.6.3 Single cluster prediction with multiple past samples 1817.6.4 Two cluster prediction with multiple past samples 1827.6.5 Role of oversampling 187

7.7 What is the narrowband signal for a rectangular array? 1887.8 Prediction using the UIU model 190

7.8.1 Separable covariance matrix 1917.8.2 1 × 2 unseparable example 1927.8.3 Large number of antennas: no noise 1937.8.4 Large number of antennas: estimation in noise 1947.8.5 Effects of the number of antennas, scattering geometry,

and observation time on the quality of prediction 1957.9 Numerical simulations 198

7.9.1 SISO channel single cluster 1987.9.2 Two cluster prediction 198

7.10 Wiener estimator 1997.11 Approximation of the Wiener filter 201

7.11.1 Zero order approximation 2027.11.2 Perturbation solution 202

CONTENTS ix

7.12 Element-wise prediction of separable process 2037.13 Effect of prediction and estimation errors on capacity

calculations 2047.14 Channel estimation and effective SNR 205

7.14.1 System model 2057.14.2 Estimation error 2057.14.3 Effective SNR 207

7.15 Achievable rates in Rayleigh channels with partial CSI 2087.15.1 No CSI at the transmitter 2087.15.2 Partial CSI at the transmitter 2097.15.3 Optimization of the frame length 211

7.16 Examples 2117.16.1 P(0, 0) Estimation 2117.16.2 Effect of non-uniform scattering 213

7.17 Conclusions 2147.18 Appendix A: Szegö summation formula 2157.19 Appendix B: matrix inversion lemma 216

8 Coding, modulation, and signaling over multiple channels 2198.1 Signal constellations and their characteristics 2198.2 Performance of OSTBC in generalized Gaussian channels

and hardening effect 2248.2.1 Introduction 2248.2.2 Channel representation 2258.2.3 Probability of error 2278.2.4 Hardening effect 229

8.3 Differential time-space modulation (DTSM) and an effectivesolution for the non-coherent MIMO channel 2338.3.1 Introduction to DTSM 2338.3.2 Performance of autocorrelation receiver of DSTM

in generalized Gaussian channels 2348.3.3 Comments on MIMO channel model 2358.3.4 Differential space-time modulation 2358.3.5 Performance of DTSM 2378.3.6 Numerical results and discussions 2438.3.7 Some comments 243

Bibliography 245

Index 257

About the Series Editors

Xuemin (Sherman) Shen (M′97-SM′02) receivedthe B.Sc degree in electrical engineering fromDalian Maritime University, China in 1982, andthe M.Sc. and Ph.D. degrees (both in electricalengineering) from Rutgers University, New Jersey,USA, in 1987 and 1990 respectively. He is aProfessor and University Research Chair, and theAssociate Chair for Graduate Studies, Departmentof Electrical and Computer Engineering, Univer-sity of Waterloo, Canada. His research focuses onmobility and resource management in interconnectedwireless/wired networks, UWB wireless communi-cations systems, wireless security, and ad hoc and

sensor networks. He is a co-author of three books, and has published morethan 300 papers and book chapters in wireless communications and networks,control and filtering. Dr. Shen serves as a Founding Area Editor for IEEETransactions on Wireless Communications ; Editor-in-Chief for Peer-to-PeerNetworking and Application; Associate Editor for IEEE Transactions on Vehicu-lar Technology; KICS/IEEE Journal of Communications and Networks, ComputerNetworks; ACM/Wireless Networks; and Wireless Communications and MobileComputing (Wiley), etc. He has also served as Guest Editor for IEEE JSAC,IEEE Wireless Communications , and IEEE Communications Magazine. Dr. Shenreceived the Excellent Graduate Supervision Award in 2006, and the OutstandingPerformance Award in 2004 from the University of Waterloo, the Premier’sResearch Excellence Award (PREA) in 2003 from the Province of Ontario, Canada,and the Distinguished Performance Award in 2002 from the Faculty of Engineering,University of Waterloo. Dr. Shen is a registered Professional Engineer ofOntario, Canada.

xii ABOUT THE SERIES EDITORS

Dr. Yi Pan is the Chair and a Professor in the Depart-ment of Computer Science at Georgia State Univer-sity, USA. Dr. Pan received his B.Eng. and M.Eng.degrees in computer engineering from TsinghuaUniversity, China, in 1982 and 1984, respectively, andhis Ph.D. degree in computer science from the Univer-sity of Pittsburgh, USA, in 1991. Dr. Pan’s researchinterests include parallel and distributed computing,optical networks, wireless networks, and bioinformat-ics. Dr. Pan has published more than 100 journalpapers with over 30 papers published in variousIEEE journals. In addition, he has published over130 papers in refereed conferences (including IPDPS,

ICPP, ICDCS, INFOCOM, and GLOBECOM). He has also co-edited over 30books. Dr. Pan has served as an editor-in-chief or an editorial board memberfor 15 journals including five IEEE Transactions and has organized many inter-national conferences and workshops. Dr. Pan has delivered over 10 keynotespeeches at many international conferences. Dr. Pan is an IEEE DistinguishedSpeaker (2000–2002), a Yamacraw Distinguished Speaker (2002), and a Shell OilColloquium Speaker (2002). He is listed in Men of Achievement, Who’s Whoin America, Who’s Who in American Education, Who’s Who in ComputationalScience and Engineering, and Who’s Who of Asian Americans.

1

Introduction

1.1 General remarks

The explosion in demand for wireless services experienced over the past 20 yearshas put significant pressure on system designers to increase the capacity of thesystems being deployed. While the spectral resource is very scarce and practicallyexhausted, the biggest possibilities are predicted to be in the areas of spectral reuseby unlicensed users or in exploiting the spatial dimension of the wireless channels.The former approach is now under intense development and is known as the cogni-tive radio approach (Haykin 2005). The latter approach is as old as communicationsystems themselves and is known mostly through the receive diversity techniques,well studied in both Western (Middleton 1960; Simon and Alouini 2000) and for-mer USSR literature (Fink 1970; Klovski 1982a). These techniques are mainlyused to improve the signal to noise ratio in the receiver in fading environments.In order to exploit the additional (spatial) dimension of the wireless channel, anumber of technologies were suggested in early 1990s, including smart antennas.The development of this antenna technology mainly focused on the developmentof the estimation of the angle of arrival, optimal beamforming, and space-timesignal processing. However, these techniques offered only a limited increase in thechannel capacity.

In recent years, however, development of the multiple-input multiple-output(MIMO) system has emerged as the most potent technique for increasing the capac-ity of wireless channels. This technique exploits sampling in the spatial dimensionon both sides of the communication links, combined in such a way that they eithercreate virtual, multiple, parallel spatial data pipes to increase capacity linearly withthe number of pipes and/or to add diversity to improve the quality of the links.A large number of research articles and monographs have treated different aspectsof this subject (Correia, Ed. 2007; Paulraj et al. 2003; Tse and Viswanath 2005;

Wireless Multi-Antenna Channels: Modeling and Simulation, First Edition.Serguei L. Primak and Valeri Kontorovich. 2012 John Wiley & Sons, Ltd. Published 2012 by John Wiley & Sons, Ltd.

2 WIRELESS MULTI-ANTENNA CHANNELS

Verdu 1998), including channel modeling, modulation, diversity-multiplexing trade-off, and so on. Since initial papers by Teletar (Teletar 1999), MIMO technologieshave been included in many existing standards of 4G communications. TodayMIMO technology appears to be the natural candidate for most large-scale com-mercial wireless products.

Most of the researchers are focusing on investigation of MIMO systems underthe important assumption that fading is Rayleigh, channel state information isperfectly known, and the scattering is reach. Such results provide good limitingestimates for capacity, performance, and delay. However, it often provides overoptimistic results. The main contribution of this book lies in addressing the follow-ing issues:

• Suggestion of the generalized Gaussian model of MIMO wireless channels.• Investigation of performance of different coding schemes in generalized

Gaussian channels.

• Suggestion of an efficient simulator of MIMO wireless channels based on ageometric-based modeling paradigm.

• In-depth studying of the effect of channel estimation on performance inMIMO systems.

• Introduction of the multitaper approach to channel estimation.• Investigation of the second-order statistics of MIMO channel capacity.The book is organized as follows. In this chapter we briefly discuss models

for signals used in this text. Chapter 2 describes a novel, four-parametric modelof a SISO wireless channel and extends the concept to MIMO configuration. Wealso consider channels with a fluctuating number of scatterers and other deviationsfrom Gaussian models. Chapter 3 expands on the modeling of MIMO channelsbased on scattering geometry and explores different geometry characteristics thateffect the channel model. It also describes narrowband MIMO channel modelswhile Chapter 4 is dedicated to wideband models. Chapter 5 treats topics relatedto the investigation of the capacity of the MIMO channel under different geo-metrical conditions, treats time variation of the capacity, and capacity of sparsechannels. Chapter 6 deals with the methodology of MIMO channel prediction,while Chapter 7 deals with effects of errors on different aspects of communicationsystem performance. Finally, Chapter 8 deals with the investigation of space-timecode performance in generalized Gaussian MIMO channels.

Finally we would like to express our gratitude to a number of people andagencies that were instrumental in supporting the research that has resulted in mostof the content of this book. First of all, we would like to express our admirationfor our late teacher and colleague Prod. Daniil (Dani) Klovski. His investigationof diversity in the time-space communication channel in the late 1960s to the early1980s laid a solid foundation of smart antenna techniques in the former USSRwhile also inspiring us to offer a generalized Gaussian model of MIMO channels.

WIRELESS MULTI-ANTENNA CHANNELS 3

We are also deeply indebted to our graduate students, now independent researchersthemselves. Our deepest thanks to Drs. Vanja Subotic, Khaled Almustafa,Dan Dechene, and A. F. Ramos-Alarcón for their diligent work in developingmost of the ideas presented in this manuscript. We would like to thank Drs. TriciaWillink and Karim Baddour from the Communications Research Centre Canada(CRC Canada) for numerous discussions, especially on topics related to MIMOchannel estimation. Our research and graduate students have been financiallysupported through a number of research grants, provided by NSERC Canada,CONACYT Mexico, and CRC Canada. We are grateful to the University ofWestern Ontario and CINVESTAV-IPN, Mexico for creating excellent workingconditions and the opportunity to spend a few months both in Mexico and Canadajointly working on the manuscript. We also would like to thank our colleaguesat the University of Agder, Prof. Matthias Patzold, Drs. Gulzaib Rafiq, DmitriUmanski, and others for a number of useful discussions and suggestions. And last,but not least, we would like to thank wonderful staff at Wiley for their patienceand indispensable help in preparing the manuscript.

1.2 Signals, interference, and types of parallelchannels

Here we consider the problem of transmission of digital information over a set ofparallel channels. The discrete message is chosen from an alphabet A = {xk} of sizem = dim(A). Once a symbol, say x from the alphabet A is chosen it is encodedto a signal waveform zk(t). A unique waveform corresponds to each symbol ofthe alphabet. In general one would select n symbols from the alphabet at a singlemoment of time to transmit them over n channels simultaneously. This can beaccomplished by using mn different signals zrk(t), r = 1, · · · , m, and k = 1, · · · , n.We consider coherent systems where the duration of the symbols at every channelis fixed to be T and the start of the symbols in each channel coincide.

The received signals z′k(t) at the output of each of the parallel channels have astatistical relation to the transmitted signal zk(t), albeit one that does not coincidewith it due to noise and interference. The received signals are processed by adecision-making block. We will focus on synchronous detection. This means thatthe decision-making algorithm observes a symbol over time period T and thendecides which symbol has been transmitted.

Parallel channels are assumed to be linear and the signals zrk(t) narrowband.This means that most of the energy of the signals zrk(t) is concentrated in afrequency band Frk that is much smaller than the carrier frequency frk. In thiscase the following representation is valid

zrk(t) = Erk cos[2πfrkt + �rk(t)

], 0 ≤ t < T (1.1)

In this case the envelope

Erk(t) =√z2rk + z̃2rk (1.2)


and the initial phase

�rk(t) = atanzrkz̃rk

− 2πfrkt (1.3)

are slowly varying factions with respect to cos(2πfrkt). Here z̃rk is the Hilberttransform (Proakis 1997).

zrk(t) = Erk sin[2πfrkt + �rk(t)

], 0 ≤ t < T (1.4)

The low pass equivalent of this signal is thus

srk = Erk exp(j�rk) (1.5)The received signal can then be written as

z′rk = µk(t)zrk [t − τrk(t)] + ξk(t) (1.6)where µk(t) is a coefficient describing the attenuation of the signal transmittedthrough the k-th channel, τk is the delay associated with this channel, and ξk(t)is the associated additive noise component. For the majority of realistic channelswith variable parameters the time delay can be written as a sum

τk(t) = τk av − �τk(t) (1.7)where τk av = E {τk(t)} is the average delay and �τk(t) is the random fluctuationof the delay. The former can be associated with the overall propagation delaydue to the finite distance between the transmit and the receive antennas, while thelatter can be associated with variation in the channel and mutual position betweenthe receiver and the transmitter. In order to have an effect on the performanceof the system the fluctuating component should have a root-mean-square valuecomparable to the duration of the bit interval.

Let θk(t) = 2πfrk�τk(t) be a random excessive phase associated with the k-thchannel and r-th frequency. Then Equation (1.6) can be rewritten as

z′rk = µk(t)zrk [t − τk av, θk(t)] + ξk(t) (1.8)where z(t − τ, θ) represents signal z(t) delayed by τ and the phase of its carriershifted by θ .

During a one-bit detection the receiver processes segments of the signal z′k(t)of duration T one by one. It is quite clear that without loss of generality a con-stant delay τk av can be eliminated from consideration. Thus, the receiver observesa signal

z′k(t) = µk(t)zrk[t, θk(t)] + ξk(t), (l − 1)T ≤ t < lT (1.9)where l indicates the order of the transmitted symbols.

For the narrowband process z′k we can write its expansion in terms of in-phaseand quadrature components (equation 2.44)

z′k(t) = µk(t) cos θk(t)zrk(t) + µk(t) sin θk(t)z̃rk(t) + ξk(t), (l − 1)T ≤ t < lT(1.10)


Here µk(t) and θk(t) represent the magnitude and the phase of the transmissioncoefficient of the k-th channel.

In practice it is more efficient to represent fading either in terms of in-phaseand quadrature components

µck(t) = µk(t) cos θk(t) (1.11)µsk(t) = µk(t) sin θk(t) (1.12)

or in its phasor (complex low-pass equivalent) form

µk(t) = µk(t) exp[−jθk(t)] (1.13)

In this case

z′k = µck(t)zrk(t) + µsk(t)z̃rk(t) + ξk(t), (l − 1)T ≤ t < lT (1.14)

If µk(t) and θk(t) are random functions of time, they can be considered asmultiplicative noise.

Without going into detail about the statistical properties of the random channeltransmission coefficient it is important to provide a general classification of thedigital communication systems utilizing such channels. The main purpose of asystem with multiple channels is to increase the capacity of the channels. This canbe achieved when the additive noise and the multiplicative noise are decorrelated.

Digital systems with multiple channels can be categorized as follows

• Channels with distinct media of propagation corresponding to differentchannels;

• Channels that share a media with the same physical properties.The former category may include channels where the information is simultaneouslytransmitted over wired and wireless links. Such systems would mostly be used toprovide redundancy and reliability of communication systems since the amount ofinformation is the same regardless of number of channels used.

Systems belonging to the latter category are widely used in the diversity recep-tion and recently in so-called MIMO systems. The improvement in the capacityof such systems is to a large degree defined by the number of channels used, thephysical location of the antennas, and the signal processing techniques.

Depending on the method of forming multiple channels one can distinguish thefollowing groups of parallel channels

1. systems with parallel channels formed on the transmitting side;

2. systems with parallel channels formed on the receiving side;

3. systems with parallel channels formed on both the receiving and transmit-ting side.


In the systems belonging to the first group the information about each bit/symbolis transmitted into the channel by means of multiple signals that are formed by thetransmitter during the modulation stage. In this case the total number of differ-ent signals is mn where n is the number of parallel channels. Each bit/symbolcan be transmitted simultaneously over all channels or sequentially over the sametime interval T . Systems with frequency and time diversity are two well-knownrepresentatives of this class. In the case of frequency diversity the same signalis transmitted over different frequencies. The total signal in the receive antennais thus

z′(t) =n∑

k=1

[µck(t)zrk(t) + µsk(t)z̃rk(t)

]+ ξ(t) (1.15)where µck(t) + j ∗ µsk is a complex transmission coefficient for the signal zrk(t).

If the signals zrk(t) and zrp(t) are orthogonal for k �= p in the strong sense,then instead of a single received signal in Equation (1.15) one can consider a set ofparallel channels with output signals defined by Equation (1.14) where µck(t) andµsk(t) are in-phase and quadrature components of the transmission coefficient forthe signal zrk(t) and ξk(t) is the additive noise acting within the bandwidth allocatedto the signal zrk(t). In general, quantities µck(t), µsk(t), ξk(t), and µcp(t), µsp(t),ξp(t) are represented by correlated processes for all k �= p. An optimal choice ofsignals zrk(t) will be such as to provide nearly complete decorrelation betweenthe fading components and the additive noise of signals transmitted over differentchannels. For example, if all zrk(t) have non-overlapping spectra, than the additivenoise ξk(t) and ξp(t) will be uncorrelated for k �= p. Furthermore, if the propagationmedium has a selectivity property,1 one can choose carrier frequencies fk in such away that µck(t), µsk(t) and µcp(t), µsp(t) are also almost completely decorrelated.

The simplest method of formation of two parallel channels is frequency shiftkeying (FSK). Indeed, FSK can be considered as two inverse amplitude modulationswith different carriers.

1 The majority of situations which involve the reflection of the transmitted signal from multipleobstacles will have this property.

2

Four-parametric modelof a SISO channel

2.1 Multipath propagation

It has been recognized from the onset of communication theory that it is importantto accurately represent communication channels, taking into account the specificsof electromagnetic wave propagation (Middleton 1960). Here we confine ourselfto systems that use propagation in free (but perhaps non-homogeneous) media.In most situations such channels are random (stochastic) with randomly placedand varying in time inhomogeneities, such as reflecting surfaces, edges, absorbers,and so on. In this case an accurate representation of all inhomogeneities is ratherdifficult and even counterproductive due to the computational burden and lack ofdetailed information. Therefore, it seems only natural to deploy a phenomenologicalapproach, that is to model only a selective set of (statistical) characteristics ofchannels, such as mean values, covariance functions, and so on, and disregard thespecific mechanism of wave propagation assuming, in general, that the waves inquestion could be modeled well as a set of individual rays, which represent a highproportion of the power intersected by receive antennas. Each such ray may bethe result of reflections from a relatively smooth surface,1 diffracted ray, or others(Bertoni 2000). Such “hot spots” could be formed by a few volumes significantlyseparated in space, as shown in Figure 2.1. If the scattering region (volume) slightlychanges in time, the “hot spots” start to move and change their intensity and newhot spots may appear. This is equivalent to variation of the phase and magnitudeof each ray. Such a picture chimes well with the principles of physical optics,

1 So called “hot spots”.

Wireless Multi-Antenna Channels: Modeling and Simulation, First Edition.Serguei L. Primak and Valeri Kontorovich. 2012 John Wiley & Sons, Ltd. Published 2012 by John Wiley & Sons, Ltd.


Figure 2.1 Physical optics model of scattering.

especially at the high propagation frequencies currently used in wireless systems.In the frame of this model each ray may be reflected a number of times beforereaching the receive antenna. Often, the Born approximation is evoked to neglectall secondary reflections (see e.g., (Abdi and Kaveh 2002; Yu and Ottersten 2002)).In this case each ray can be considered as propagating in parallel to all others: wewill call this mechanism a parallel single-bounce scattering . However, in manysituations it is important to take into account multiple reflections (Middleton 1960;Salo et al. 2006). In this case we still have multiple rays propagating in parallel toeach other, however each of them experience sequential reflections, attenuations,absorption, diffraction, and so on. This scenario will be referred to as multiplescattering or sequential-parallel multiple scattering .

If a single bounce is assumed, the received signal r(t, xr ) can be readily writtenas a sum of the random numbers of independent components

r(t, xr ) =N(t)∑n=1

rn(t, xr , pn) (2.1)

Here xr is the vector position of the receive antenna, pn is the vector of theparameters2 associated with the n-th ray. This model will be considered in moredetails in Section 2.2. Often, the central limit theorem can be invoked to justify thatthe distribution of the received signal is Gaussian for a fixed number of scatterers.

The case of multiple scattering often arises in situations where there is no line-of-sight and where there is a dense scattering environment between the receive andtransmit sites (Bliss et al. 2002; Dohler et al. 2007; Gesbert et al. 2002; Oestgeset al. 2003). The received signal can also be represented as a sum (2.1), however, allrays cannot be treated as independent. This could be explained by the fact that someof the components of the sum (2.1) are produced by the same “hot spots”. Generalanalysis of such a situation is quite complicated since the central limit theoremcannot be invoked without preconditions on the amount of correlation between theindividual terms. The general structure of such channels can be described by asequence of time-varying filters as shown in Figure 2.2. Among the many modelswith multiplicative scattering it is possible to select one in which rays are still

2 Such as intensity, phase, delay, etc.


S (t, r )

u(t, r )

L1

L2

L3

L4

LL

Figure 2.2 Mechanism of propagation with multiple scattering but indepen-dent rays.

S (t )

H11

H21

H31

HQ1 HQ2

H12

H22

H32

H1K H1N

H2N

H3N

HQNHQK

H3K

H2K

u(t, r )

Figure 2.3 Mechanism of propagation with multiple scattering but indepen-dent rays.

independent but where each ray experiences multiple scattering. This situation isshown in Figure 2.3.

In this case, if the number of rays is large, the resulting received signalagain converges to a Gaussian one, regardless of the distribution of components.However, if the number of rays is relatively small3 the distribution of the signal inthe receive antenna is dominated by the distribution of individual rays.

3 N < 10 (Klovski 1982b).


Another important criteria of channel classification is related to the ability ofthe receiver to isolate different subrays (Rappaport 2002). This is defined by theaperture size on the receive and transmit sides as well as by the duration of thepulses used to transmit data symbols.

Let us assume that T is a duration of the transmitted symbol s(t), 0 ≤ t ≤ Tthat occupies effective bandwidth �fs , due to the fact that the received signal iscomposed of multiple copies of the transmitted signal s(t). The channel can beconsidered a single path channel if the maximum delay �τ experienced by anindividual ray is smaller than the resolution of the receiver

�τ � 1�fs

(2.2)

If the opposite is true, the channel should be considered multipath.Even in the case of a single path channel, the resulting signal is a superposition

of a large number of subpaths, mainly due to diffused scattering in the propagationenvironment.

In the case of a narrowband transmit signal on the interval [iT : (i + 1)T ]

si(t) = Si(t) exp(jω0t)u(t, iT ), u(t, iT ) ={

1 if iT ≤ t ≤ (i + 1)T0 otherwise

(2.3)

the received signal ri(t) is the sum of delayed replicas of the radiated signal, that is

ri(t) =L∑l=1

hlSi(t − �τl) exp[jω0(t − �τl − Td)

]u(t − �τl − Td, iT ) (2.4)

or, for its based-band equivalent

Ri(t) =L∑l=1

hlSi(t − �τl − Td) exp(−jω0�τl)u(t − �τl − Td) (2.5)

Here L is the number of distinguishable rays4 at the receiver, hl is the complexchannel coefficient for the l-th path, and Td is the average propagation delay.Due to the assumption about the narrowband nature of the transmitted signal, theparameters hl , �τl , and Td are the same for all frequency components of thetransmit signal.5

Due to changing environmental conditions and movement of the receiver (orthe transmitter, or both) parameters hl , �τl could be treated as random functions.However, over the duration of a single symbol T of a transmitted signal theseparameters can often be considered constants. These parameters, however, may bechange from symbol to symbol (or from a block of symbols to a block of sym-bols). In Equation (2.5) the transient process due to symbol transitions is neglectedbecause the signals considered are relatively narrowband.

4 Assumed to be a constant for the moment.5 These conditions are only for frequency independence of the scattering parameters on frequency,

and are not to be confused with the flat fading conditions.


As the next step let us assume that

tl = τ + �tl (2.6)where τ is the mean time of the ray propagation and �tl is the variation of prop-agation time from the mean value. Furthermore, let us assume that

|�tl | � 1�fi

= Tbi

(2.7)

|�tl | � T (2.8)In this case phase shifts of various frequency components of the signal are relativelysmall, the signal is not distorted in shape, and, therefore, the Equation (2.5) can bewritten as

r(t) = s(t − τ )L∑l=1

γl exp(−jφl) = s(t − τ )L∑l=1

hl = h · s(t − τ ) (2.9)

Here φl = ω0�tl . The complex transmission coefficient h = γ exp(−jφ),defined as

h = γ exp(−jφ) =L∑l=1

hl =L∑l=1

γl exp(−jφl) = x + jy (2.10)

is a complex random process with the in-phase and quadrature components x andy respectively:

x = γ cosφ =L∑l=1

xl =L∑l=1

γl cosφl (2.11)

y = γ sinφ =L∑l=1

xl =L∑l=1

γl sinφl (2.12)

where

γ =√x2 + y2, φ = arctan y

x(2.13)

γl =√x2l + y2l , φl = arctan

yl

xl(2.14)

When analyzing the channel, its transfer function

H(t, ω0) = γ exp(−jφ) exp(jω0t) (2.15)must be treated as a non-stationary complex random process. However, over a shortduration Tst the process could be treated as locally stationary and ergodic. Thechannel is considered to be locally stationary over time interval Tst in a frequencyband Fst , if its time-frequency covariance function Cω0,t (τ,) can be consideredindependent on t and ω0, that is

Cω0,t (τ,) = C(τ,) (2.16)


Furthermore, if transmitted signals are narrowband the dependency on frequencyin (2.16) vanishes

Cω0,t (τ,) = C(τ) (2.17)If the parameters of distribution of all random variables γ , φ, and τ are the same forall positions of the transmitted symbols the channel is considered to be symmetric.

Let τk be the correlation interval of the parameters of the channel. If

T � τk (2.18)that is, the channel coefficients vary slowly over the duration of many symbols,the channel is considered slow and the frequency flat. If the condition (2.18) is notsatisfied, the fading is called time-selective, or fast.

It can be seen from Equation (2.11), that the in-phase and quadrature compo-nents are the result of the summation of a large number L of subrays with limitedvariance, such that

liml→∞

σ 2lx

σ 2x= 0, lim

l→∞σ 2ly

σ 2y= 0 (2.19)

In this case, the central limit theorem could be applied, even for correlated subrays.Simple calculations allow us to evaluate the mean and the variance of the

in-phase and quadrature components based on the model (2.11). In this case

mI =L∑l=1

E{γl}E {cosφl} , mQ =L∑l=1

E{γl}E {sinφl} (2.20)

σ 2I =L∑l=1

[E{γ 2l }E {cos2 φl}− E2 {γl} E2 {cosφl}] (2.21)σ 2Q =

L∑l=1

[E{γ 2l }E {sin2 φl}− E2 {γl} E2 {sinφl}] (2.22)RIQ = 1

2

L∑l=1

E{γ 2l }E {sin 2φl}

+L∑

k=1

L∑l �=k

E{γl}E{γk} {sinφl} {sinφk} − mQmQ (2.23)

If all elementary scatterers have the same statistics, previous equations could befurther simplified to produce

mI = LE{γ }E {cosφ} , mQ = LE{γ }E {sinφ} (2.24)

σ 2I = LE{γ 2}E{cos2 φ

}− m2IL

(2.25)


σ 2Q = LE{γ 2}E{sin2 φ

}− m2QL

(2.26)

RIQ = L2E{γ 2}E {sin 2φ} − mImQ

L(2.27)

Some general conclusions could be reached based on Equations (2.20)–(2.27):

1. If the distribution of the phases of each partial scatter are random anddistributed uniformly on [−π, π ], then mI = mQ = 0, σ 2I = σ 2Q, r = 0.

2. If fluctuations of the unwrapped phase are much wider than the interval[−π, π ], the wrapped distribution approaches the uniform on [−π, π ].

3. If the fluctuations of the phases are symmetrical with respect to their meanvalues, are equal to zero but with a variance that is significantly smallerthan π , it can be easily seen that mQ = 0, mI �= 0, and σ 2I �= σ 2Q, r = 0.

4. Finally, if the distribution of the phases is not symmetrical and has a smallvariance, one obtains that mQ �= 0, mI �= 0, and σ 2I �= σ 2Q, r �= 0.

Therefore, the particular mechanism of scattering plays a significant role in thedistribution of the resultant field. The distribution of the phases is more importantin this case than particular distributions of magnitudes.

In the following we consider two cases of statistical description of the scat-tered field using representation (2.11): at first, we assume that L is large enoughso that the distributions could be assumed to be Gaussian. This case is treatedin Section 2.3 and is often used in communication applications. However, somedeviations from Gaussian statistics are also found in experiments. Correspondingstatistics are described in Section 2.2.1.

2.2 Random walk approach to modelingof scattering field

2.2.1 Random walk in two dimensions as a modelfor scattering field

Modeling the distribution of electromagnetic and optical fields in the presenceof multiple scatterers has been of great interest since the earliest times of com-munications theory. While classical Rayleigh models can be traced back as faras the end of the 19th century, more sophisticated models were developed inthe 1970s and 1980s (Barakat 1986, 1987; Jakeman and Pusey 1976; Jakemanand Tough n.d.). The content of this section follows mainly (Barakat 1986, 1987;Jakeman and Pusey 1976; Jakeman and Tough n.d.). First we consider the randomwalk in two dimensions as presented by R. Barakat in (Barakat 1986). In this case,the signal of interest ξ(t) at a given moment of time t is considered to be a sum


of complex phasors

ξ(t) =N(t)∑n=1

an(t) exp(jθn(t)) (2.28)

Magnitudes an(t), phases θn(t), and number of scatterers N(t) are independentrandom variables at any fixed moment of time. For simplicity it is assumed thatthe distribution of magnitudes and phases is identical for all scatterers, that ispa,n(A) = pa(A), pφ,n(θ) = pφ(θ) and the distribution of the number of scatterersis described by some known distribution P (N).

2.2.2 Phase distribution and scattering strength

It is common to distinguish two scattering regimes (Barakat 1986): the strongscattering regime and the weak scattering regime. In the strong scattering regimerms scatterer size σ >λ is larger than the wavelength λ associated with the carrierfrequency of the signal. In this case, no phase value is given preference and thephase could be described as uniformly distributed

pφ(θ) = 12π

, 0 ≤ θ ≤ 2π (2.29)

In contrast, in the weak scattering regime, the size of scatterers is smaller thanthe wavelength σ < λ, and, therefore, a predominant value of the phases exists.A convenient model for such a non-uniform phase could be in the form of thevon Mises–Tikhonov distribution (La Frieda and Lindsey 1973),6 often used incommunication applications (Abdi and Kaveh 2002; La Frieda and Lindsey 1973;Mardia and Jupp 2000)

pφ(θ) = exp [κ cos(θ − µ)]2πI0(κ)

, −π ≤ θ ≤ π (2.30)

Here κ is the shape parameter, with κ = 0 corresponding to the uniformly dis-tributed phase pφ = 1/2π . For large κ > 10, the von Mises-Tikhonov distribu-tion is well approximated by a Gaussian distribution with mean µ and varianceσ 2 = 1/κ (Barakat 1986; Mardia and Jupp 2000). To emphasize the effect of phasedistribution rather than the effects of the magnitude distribution, we assume, fol-lowing (Barakat 1986), that the amplitudes of each ray are deterministic constant a.

2.2.3 Distribution of intensity

First let us assume that the number of rays N(t) is deterministic and fixed butarbitrary: N(t) = N . In this case, the conditional density pγ (γ |N) of the processintensity γ (t) = |ξ(t)|2 can be represented as (Barakat 1986)

pγ (γ |N) = I−N0 (κ)I0(κ√γ /a) 1

2

∫ ∞0

JN0 (aω)J0(√γω)ωdω (2.31)

6 Also know as simply the von Mises distribution or Tikhonov distribution.

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