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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 59, NO. 5, JUNE 2010 2363

Performance Analysis of Scheduling Schemes forRate-Adaptive MIMO OSFBC-OFDM Systems

Mohammad Torabi, Member, IEEE, David Haccoun, Life Fellow, IEEE, and Wessam Ajib, Member, IEEE

Abstract—Dynamic channel-aware user-selection and resource-allocation schemes are attractive for providing high sys-tem performance for multiple-input–multiple-output orthogonalfrequency-division multiplexing (MIMO-OFDM) systems. In thispaper, we investigate the combination of different techniques,resulting in user scheduling schemes for multiuser MIMO-OFDMsystems employing orthogonal space–frequency block coding(OSFBC) over multipath frequency-selective fading channels. Ourcontribution is a performance analysis framework that evaluatesthe advantages of employing user scheduling in MIMO-OFDMsystems employing OSFBC in conjunction with adaptive modu-lation schemes. We derive analytical expressions for the averagespectral efficiency (ASE), the average bit error rate (BER), theoutage probability, and the average channel capacity for differentscheduling and adaptive modulation schemes. Discrete-rate andcontinuous-rate adaptive modulation schemes are employed toincrease the spectral efficiency of the system. We assume a signal-to-noise-ratio (SNR)-based user-selection scheme and the well-known proportional fair scheduling (PFS) scheme. Both full- andlimited-feedback channel information scenarios are considered.Using the results obtained from both mathematical expressionsand numerical simulations, we compare the presented schemesand show their significant advantages. Finally, the impact of spa-tial correlation on the performance of the system under study isanalyzed and evaluated.

Index Terms—Adaptive modulation, link adaptation, multiple-input–multiple-output orthogonal frequency-division multiplex-ing (MIMO-OFDM), resource allocation, user scheduling.

I. INTRODUCTION

MULTIPLE-input–multiple-output (MIMO) technologyhas been recognized as a key approach for improving the

system performance and the channel capacity of wireless com-munication systems. On the other hand, orthogonal frequency-division multiplexing (OFDM) has been considered as apromising technique in future broadband wireless communi-cations. In particular, the MIMO-OFDM system is consideredas an attractive solution for broadband wireless communica-tions [1]. Since OFDM converts the frequency-selective fadingchannel into several parallel flat-fading subchannels, it allowsthe implemention of MIMO-related algorithms on each sub-

Manuscript received November 25, 2008; revised June 3, 2009 andOctober 13, 2009; accepted November 17, 2009. Date of publication December28, 2009; date of current version June 16, 2010. The review of this paper wascoordinated by Prof. W. Su.

M. Torabi and D. Haccoun are with the Department of Electrical Engi-neering, École Polytechnique de Montréal, Montréal, QC H3C 3A7, Canada(e-mail: [email protected]; [email protected]).

W. Ajib is with the Department of Computer Science, Université du Québecà Montréal, Montréal, QC H2X 3Y7, Canada (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TVT.2009.2039235

carrier. OFDM can facilitate the employment of an adaptivemodulation scheme, leading to an improvement in both biterror rate (BER) performance and average spectral efficiency(ASE) [2]. A significant advantage of MIMO-OFDM systemsis that they allow rate and power allocation (through adaptivemodulation) and dynamic resource allocation to the system[3], [4]. In MIMO flat-fading channels, a multiuser scheduleris used to exploit the antenna and multiuser diversities at thesame time and to increase the system performance [5]–[7]. Anoverview of user-scheduling algorithms in MIMO systems ispresented in [8]. In the multiuser transmission scenario overfrequency-selective fading channels employing MIMO-OFDM,the so-called MIMO orthogonal frequency-division multiple-access (OFDMA) system is capable of exploiting differenttypes of diversities, namely, the frequency, space, and multiuserdiversities. An important issue in these systems is user schedul-ing, which is the efficient allocation of OFDM subcarriersamong users according to their channel conditions. In multiuserMIMO-OFDM, users can share the OFDM symbols so that Nsubcarriers of the OFDM symbol can be dedicated to someof the users. Both fixed and dynamic resource allocation canbe considered. Dynamic resource allocation provides higherspectral efficiency and, hence, better system performance, sinceaccording to the channel variations, it allocates the time or thefrequency slots to the users adaptively.

Although the issue of adaptive modulation with multi-user scheduling has been studied in the past, particularly forsingle-input–single-output (SISO) systems, most of the workhas been focused on flat-fading channels [9], [10]. However,for frequency-selective fading channels, adaptive modulation inMIMO-OFDMA has attracted research interest and continuesto do so as an open research area. In our case, transmissionpower and rate (i.e., modulation modes), along with user allo-cation, can be adapted for every subcarrier. Most of the relatedwork that has been done for variable-rate and variable-powerallocation in these systems introduces high system complexity,particularly with using the well-known water-filling technique[11]–[17]. For conventional OFDM systems, it is shown that theoptimum power adaptation provides a small spectral efficiencygain on the order of 1 dB compared with the constant-powervariable-rate system [18]. Therefore, it was recommended touse a constant-power spectrum to save computational com-plexity in the adaptive modulation. In our analysis of thegeneral performance of user-scheduling schemes for MIMO-OFDM system employing orthogonal space–frequency blockcoding (OSFBC) in conjunction with rate-adaptive modula-tion, we obtained closed-form expressions for the ASE, outageprobability, average channel capacity, and average BER. To

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2364 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 59, NO. 5, JUNE 2010

the best of our knowledge, these results have been previouslyobtained under neither full nor limited-feedback scenarioconsideration.

In this paper, we consider two rate-adaptive modulationschemes (i.e., discrete and continuous rates) for increasing thespectral efficiency of MIMO-OFDMA systems constrained tomaintain a given target BER. Our objective is to apply the rate-adaptive modulation strategy to the MIMO-OFDMA systemand to present a mathematical performance analysis to studythe resulting improvement in system performance. We studythe effects of the number of users and the effects of the numberof receive and transmit antennas on the system performance. Inthis paper, we consider the popular orthogonal space–frequencyblock-coded OFDM (OSFBC-OFDM), where the orthogonalstructure leads to a low-complexity receiver. On the other hand,the fairness in user scheduling is a key parameter and shouldbe taken into consideration. Proportional fair scheduling (PFS)has been proposed for exploiting multiuser diversity whilemaintaining fairness among the users over flat-fading channels[5]. We use a similar proportional fairness scheduling schemeand extend it to frequency-selective fading channels. In thispaper, we derive the expressions for the probability densityfunction (pdf) and the cumulative distribution function (cdf) ofthe SNR for each OFDM subchannel for user scheduling ona MIMO channel. Then, we express the pdf and cdf for theSNR-based user scheduling scheme, enabling us to establisha mathematical analysis and formulation for the ASE and theaverage BER of the system under study. We consider two sce-narios: 1) full feedback and 2) limited feedback of the channelinformation to the transmitter. We derive analytical expressionsfor the ASE, the average BER, the outage probability, andthe average channel capacity of the multiuser MIMO-OFDMsystem under study. Finally, the impact of spatial correlationon the performance of the system under study is analyzed andevaluated, and analytical expressions of the ASE are derivedfor both continuous-rate and discrete-rate adaptive modulationschemes under spatial correlation.

This paper is organized as follows: Section II presentsthe MIMO OSFBC-OFDMA system model and the assump-tions used. The proposed scheduling schemes are presented inSection III. In Section IV, adaptive modulation schemes forthe OSFBC-OFDMA system are presented, and the averageBER, outage probability, ASE, and average channel capacityare analytically evaluated. The impact of spatial correlationon the performance of the system under study is presentedin Section V. Simulation and numerical results showing thesignificant advantages of the presented scheme are given inSection VI, and finally, Section VII concludes this paper.

II. SYSTEM MODEL

The system model is illustrated in Fig. 1(a). We considera multiuser MIMO OSFBC-OFDMA employing nT transmitantennas at the base station and K users, each with nR receiveantennas. A more detailed block diagram of the MIMO-OFDMtransceiver under study is shown in Fig. 1(b). We assumeOFDM with N subcarriers so that the MIMO channel betweenthe kth user and the base station on the nth frequency slot

can be expressed by the H[k, n] matrix of size nR × nT , withelements Hj,i[k, n] corresponding to the discrete Fourier trans-form of element hj,i(k, t), which corresponds to the complexchannel response between the ith transmit and jth receiveantennas. We assume that the fading process for each userremains unchanged during each OFDM block (one time slot)but is allowed to vary from one block to another. However,for each OFDM block, in the frequency domain, the amplitudeof the subchannel coefficients, i.e., |Hj,i[k, n]|, can vary asn varies from a value to the next. For each user, the fadingprocesses associated with different transmit–receive antennapairs are assumed to be uncorrelated. The fading process im-pulse response of the link between the ith transmit antennaof the base station and the jth receive antenna of user k (i =1, 2, . . . , nR, j = 1, 2, . . . , nT , k = 1, 2, . . . ,K) can be ex-pressed as hj,i(k, t) =

∑P−1m=0 dm,j,i(k, t)δ(t − τm(t)), where

dm,j,i(k, t) is a tap weight, τm(t) is the time delay of themth path, and P is the total number of resolvable paths. Thedm,j,i(k, t)s are complex Gaussian random processes with zeromean and variance 1/P (equal power). With this model, weassume that the path delays τm(t) are multiples of the symbolduration.

In every time slot, all users can have access to the Nsubcarriers of an OFDM symbol, where each frequency slotcan be dedicated to only one user (selected user). The multiuserscheduler at the base station can select the best user based onthe channel quality of the users in each frequency slot accordingto either full-feedback or limited-feedback channel informa-tion. Then, the rate-adaptive modulator chooses the highestmodulation mode for that user. The procedure is illustrated inFig. 2, which shows the user selection (out of two users) andthe bit allocation for that user (modulation mode selection) inthe frequency slots, whereas the amplitude of the bars indi-cates the data rate that the best user can transmit in each fre-quency slot.

Similar to the expression in [19] for the received signal inspace–time block code (STBC), the received signal for OSFBC-OFDM after OSFBC decoding can be written as

s̃ [k, n] = c ‖H[k, n]‖2F s[k, n] + η[k, n] (1)

where s̃ [k, n] is the output signal of the OSFBC decoder, s[k, n]is the transmitted symbol, ‖ · ‖2

F is the squared Frobeniusnorm of a matrix, ‖H[k, n]‖2

F =∑nR

j=1

∑nT

i=1 |Hj,i[k, n]|2,and η[k, n] is a complex Gaussian noise with distributionN (0, (cN0/2)‖H[k, n]‖2

F ) per dimension, where c is a con-stant that depends on the orthogonal STBC (OSTBC); forexample, c = 1 for codes G2, H3, and H4, in [20], and c = 2for G3 and G4 [20]. The total energy of the symbol transmittedthrough the nT antennas can be normalized to nT , and there-fore, as in [19], we can express the instantaneous SNR persymbol at the receiver of the kth user as

γ[k, n] =γ

nT Rc‖H[k, n]‖2

F (2)

where γ = Es/N0 is the average receive SNR per antenna andwhere Rc is the OSTBC code rate [20]. For example, Rc = 1for Alamouti code G2, Rc = 3/4 for codes H3 and H4, andRc = 1/2 for codes G3 and G4. Let us consider q as the period

TORABI et al.: ANALYSIS OF SCHEDULING SCHEMES FOR RATE-ADAPTIVE MIMO OSFBC-OFDM SYSTEMS 2365

Fig. 1. (a) Multiuser diversity with MIMO OSFBC-OFDM and scheduling. (b) MIMO OSFBC-OFDM transceiver.

Fig. 2. User scheduling and bit allocation.

of OSTBC, where q is equal to 2, 4, and 8 for codes G2, H3 (andH4), and G3 (and G4), respectively [20]. Considering the OFDMblock with N subcarriers and assuming that each frequency slotcontains q subcarriers, there are N/q frequency slots for eachOFDM block. This is a reasonable assumption when the OFDMblock size N is large enough. An example for the employedOSFBC-OFDM is given in [21].

III. MULTIUSER SCHEDULING

In this section, we express the pdf and the cdf for the SNR-based user scheduling scheme for the multiuser OSFBC-OFDMsystem, which will enable us to establish a mathematical analy-sis of the system under study. For the feedback channel, we

consider two scenarios: full-feedback and limited-feedback ofthe channel information to the transmitter. For both scenarios,we also express the pdf and the cdf for the SNR of the best user.

A. PDF and CDF for OSFBC-OFDM

In MIMO-OFDM systems, OFDM converts the frequency-selective fading channel into several parallel flat-fading sub-channels; therefore, the subchannel fading, i.e., |Hj,i[k, n]|, canbe assumed to be Rayleigh flat fading, and hence, |Hj,i[k, n]|2for each user in each subchannel will be a Chi-squared dis-tributed random variable. We omit the index [k, n] in γ forsimplicity. Since ‖H[k, n]‖2

F is the sum of nT nR independentidentically distributed (i.i.d.) |Hj,i[k, n]|2 random variables,then ‖H[k, n]‖2

F is distributed with 2nT nR degrees of freedom.Thus, using a change in variables, similar to the approach in[22, eq. (2.1-110)], we can show that the pdf and cdf of thereceived SNR for each subchannel of each user in OSFBC-OFDM, which is given in (2), can be expressed as [23]

fγ(γ)=γnT nR−1

(nT nR − 1)!

(nT Rc

γ

)nT nR

exp(−nT Rcγ

γ

)(3)

Fγ(γ)= 1 − exp(−nT Rcγ

γ

)nT nR−1∑v=0

1v!

(nT Rcγ

γ

)v. (4)


Note that we use the term i.i.d. for |Hj,i[k, n]|2 with respectto the users k ∈ {1, 2, . . . ,K} and antennas but not necessarilyfor the frequency-slots; n ∈ {0, 1, . . . , N − 1}. In other words,from one user to another, the subchannel amplitudes are i.i.d.,but for a given user, the subchannels can be correlated withrespect to the frequency slot index. This is due to fact thatthe user channels [i.e., hj,i(k, t)] are assumed to be i.i.d. fork ∈ {1, 2, . . . ,K}.

B. User Scheduling With Full-Feedback Load Scenario

Considering K users and assuming that full channel infor-mation is sent to the base station, for each frequency-slot n, thescheduler selects the best user (user m) such that

m(n) = arg maxk∈K

{γ[k, n]} (5)

where K = {1, 2, . . . ,K}, and γ[k, n] is the instantaneous SNRfor user k in the frequency slot n. This means that the user withthe largest channel SNR will be scheduled for transmission inthat frequency slot. Therefore, assuming that the users’ SNRare i.i.d., using the theory of order statistics [24], the cdf of thebest user (with SNR γm) selected from K available users canbe obtained from

Fγm(γ) = Pr (γm ≤ γ)

= Pr (γm ≤ γ; γm ≥ all other γu (u �= m))

= Pr (γ1 ≤ γ; . . . ; γm ≤ γ; . . . ; γK ≤ γ)

= [Fγ(γ)]K (6)

and therefore, the pdf of γm can readily be obtained by takingthe derivative of the corresponding cdf in (6) with respectto γ as

fγm(γ) = Kfγ(γ) [Fγ(γ)]K−1 (7)

where fγ(γ) and Fγ(γ) are defined in (3) and (4), respectively.By substituting (4) into (6) and using the binomial expansiondefined by (1 − x)K =

∑Ki=0

(Ki

)(−1)ixi after some manipu-

lations, we obtain

Fγm(γ) =

[1 − exp

(−nT Rcγ

γ

)nT nR−1∑v=0

1v!

(nT Rcγ

γ

)v]K

=K∑

i=0

(K

i

)(−1)i exp

(−inT Rcγ

γ

)

×i(nT nR−1)∑

t=0

at,i

(nT Rcγ

γ

)t

(8)

where at,i denotes the coefficient of ((nT Rc/γ)γ)t in theexpansion [

∑nT nR−1v=0 (1/v!)((nT Rc/γ) γ)v]i, which is defined

by a0,i =1, a1,i = i, at,i =(1/t)∑min(t,nT nR−1)

k=1 ((k(i + 1) −t)/k!)at−k,i, for 2 ≤ t < i(nT nR − 1), and at,i = [(nT nR −1)!]−i, for t = i(nT nR − 1) [25]. Similarly, by substituting

(3) and (4) into (7) and after some manipulations, the pdf isobtained as

fγm(γ) =

K

(nT nR − 1)!

(nT Rc

γ

)·

K−1∑i=0

(K − 1

i

)(−1)i exp

(−(i + 1)nT Rcγ

γ

)

·i(nT nR−1)∑

t=0

at,i

(nT Rc

γγ

)t+nT nR−1

. (9)

C. User Scheduling With Limited-Feedback Load Scenario

In full-feedback communications, we assume that all theusers should send their channel SNR information to the basestation. Although the base station only requires feedback fromthe user with the best channel quality, the users are not awareof the channel conditions of the other users. Therefore, forthe purpose of objective scheduling and resource allocation,the base station needs feedback from all the active users. Toreduce the feedback load, in [9], a scheme has been presentedfor the multiuser scheduling of SISO channels. Here, we use asimilar algorithm and extend it to the MIMO-OFDMA systemfor the MIMO channels. In this scheme, in each frequency slot,only a set of active users whose channel SNR is greater than apredefined threshold (γ > γth) should feedback their channelinformation to the base station; other users remain silent. Ifnone of the users has an SNR above the threshold (γ ≤ γth), thescheduler in the base station selects a random user. Assumingthat the users’ SNR are i.i.d., we can express the cdf of theselected user SNR as [9]

Fγm(γ) =

⎧⎪⎨⎪⎩Fγ(γ) [Fγ(γth)]K−1 , γ ≤ γth∑K

k=1

(Kk

)[Fγ(γth)]K−k

× [Fγ(γ) − Fγ(γth)]k , γ > γth.

(10)

Therefore, the pdf fγm(γ) can be obtained by taking the deriv-

ative of the cdf Fγm(γ) in (10) with respect to γ. Then, it can

be written as

fγm(γ) =

⎧⎪⎨⎪⎩fγ(γ) [Fγ(γth)]K−1 , γ ≤ γth∑K

k=1

(Kk

)kfγ(γ) [Fγ(γth)]K−k

× [Fγ(γ) − Fγ(γth)]k−1 , γ > γth.(11)

Substituting the expression for fγ(.) and Fγ(.) given in (3) and(4) into (10) and (11) yields new closed-form expressions forthe pdf and cdf of the selected user’s SNR under the limited-feedback scenario. These closed-form expressions can also bewritten using binomial series expansions followed by similarsteps, as used in (8). However, for the sake of brevity, theseexpressions are omitted.

D. Determining the SNR Threshold for Limited-FeedbackLoad Scenario

In the following, we determine the SNR threshold γth interms of a normalized average feedback load. The normalizedaverage feedback load F is defined as the ratio of the average


load per time slot over the total number of active users K. Itis equivalent to the probability of γ > γth; therefore, it can beexpressed in terms of the cdf as

F = Pr(γ > γth) = 1 − Fγ(γth) (12)

where 0 ≤ F ≤ 1, and F = 1 corresponds to the full feedbackload. Therefore, for a given feedback load, the correspondingSNR threshold γth depends on the MIMO-OFDMA systemsettings, such as the number of transmit and receive antennas.This threshold can be obtained from

γth = F−1γ (1 − F ) (13)

where F−1γ (.) is the inverse function of the given cdf in (4) and

can be calculated numerically.

IV. ADAPTIVE MODULATION

As earlier stated, for each frequency slot, after selectingthe best user by a user scheduler, a rate-adaptive modulatorshould select the highest modulation mode for that user. Thegeneral idea of adaptive modulation is to choose a set ofsuitable modulation parameters based on the full or partial(limited) feedback channel-state information and thus achievehigh spectral efficiency. We assume the use of a constant-power spectrum to save computational complexity since theoptimum power adaptation provides a small spectral efficiencygain on the order of 1 dB, compared with the constant-powervariable-rate system [18]. Throughout this paper, we consideran adaptive modulation with constant-power and adaptive-ratetransmission. For the rate adaptation, we consider two schemes:1) discrete-rate and 2) continuous-rate adaptive modulations.The term discrete rate means that the number of bits per symbolis restricted to integer values.

In the following, we present the adaptive modulation andprovide a mathematical analysis for the proposed system.We derive closed-form expressions for the average spectralefficiency, which is denoted by ASE, and the average BER isdenoted by BER. Although they are general and applicable forboth full-feedback and limited-feedback scenarios, the resultswill somehow be different by considering their associated pdfand cdf for each scenario.

A. Discrete-Rate Adaptive Modulation

In this section, we consider a discrete-rate adaptive modu-lation for the system under study. Consequently, the numbersof bits/symbol (bl[m,n]) are integer values, and the constel-lation sizes can be determined by Ml = 2bl[m,n] for the nthOFDM frequency slot, where l is a positive integer, and m isthe best user index. In this case, L-mode modulation (M -arymodulation) will be considered. The modulation modes will beselected according to the instantaneous channel conditions perfrequency slot. Discrete-rate adaptive modulation is performedby splitting the SNR regions into L + 1 fading regions definedby the SNR levels 0 < α1 < · · · < αL < αL+1 = ∞. For thefading region αl ≤ γ < αl+1, bl bits will be allocated to thecorresponding channel/user. The lower part of each SNR regionis the smallest SNR required to achieve the target BER. For the

signals whose SNR is lower than the SNR level α1, there will beno transmission (outage), whereas for the signals whose SNR ishigher than αL, the highest modulation mode will be dedicated.Note that the index (n) is omitted from αl(n) for simplicity.

1) Average Spectral Efficiency (ASE): The ASE for the nthOFDM frequency slot and for the user m (best user) expressedin terms of bits per second per hertz for the base station can bedefined as [26], [27]

ASE(n) =L∑

l=1

bl[m,n]Pl[m,n] (14)

where Pl[m,n] is the probability of modulation mode selec-tion, Pl[m,n] =

∫ αl+1

αlfγm

(γ)dγ = [Fγm(αl+1) − Fγm

(αl)],and fγm

(γ) is defined in (6) for the full-feedback load scenarioand in (11) for the limited-feedback load scenario. Fγm

(γ) is

the cdf defined as Fγm(ξ) =

∫ ξ

0 fγm(γ)dγ and is given in (8)

for the full-feedback load scenario and in (10) for the limited-feedback load scenario. Note that in the region αL ≤ γ <αL+1 = ∞, we set Fγm

(αL+1) = 1.Then, by averaging (14) over N subcarriers and taking the

OSFBC code rate (Rc) into account, the overall ASE can beexpressed as

ASE =Rc

N

N−1∑n=0

L∑l=1

bl[m,n] [Fγm(αl+1) − Fγm

(αl)] . (15)

We obtain new closed-form expression for the ASE by insertingthe cdf expression (8) into (15) as

ASE =Rc

N

N−1∑n=0

L∑l=1

bl[m,n]K∑

i=0

(K

i

)(−1)i

×i(nT nR−1)∑

t=0

at,i

[(nT Rc

γαl+1

)texp(−inT Rc

γαl+1

)

−(

nT Rc

γαl

)texp(−inT Rc

γαl

)](16)

where, for simplicity, the index (n) is omitted from αl and αl+1.To calculate the ASE of our system under study for the full-

feedback load scenario, we use the closed-form expression ofASE given by (16). As for the limited-feedback load scenario,we could use the closed-form expression in (15) and substituteinto it the expressions in (10) for Fγm

(.). For brevity, theresulting expression is not reported, but we have used it for thesimulation and numerical evaluations.

2) Average BER: As stated earlier, the goal of this adaptivemodulation system is to maximize the ASE while respectinga target BER requirement. Therefore, the average BER of thesystem should not be larger than the target BER, i.e., BER ≤BERt. To calculate the average BER of the system, we firstcalculate the mode-specific average BER for the best user in thenth OFDM frequency slot BER l(n), which can be written as

BER l(n) =

αl+1∫αl

BERMl(γ)fγm

(γ)dγ

= Ψ(αl+1, gl) − Ψ(αl, gl) (17)


where Ψ(x, gl) =∫ x

0 BERMl(γ)fγm

(γ)dγ, and BERMl(γ) is

the instantaneous BER of the M quadratic-amplitude mod-ulation (M -QAM) mode over the additive white Gaussiannoise (AWGN) channel (the index (n) is omitted from gl, αl,and αl+1 for simplicity). Using the exponential approxima-tion given in [28] for OSFBC-OFDM, we have BERMl

(γ) =0.2 exp(−glγ), where gl = 1.6/(nT Rc(2bl − 1)) [21]. Thedistribution function fγm

(γ) is defined in (9) for the full-feedback load scenario and in (11) for the limited-feedback loadscenario. Therefore, we can write

Ψ(x, gl) = 0.2

x∫0

e−glγfγm(γ)dγ. (18)

The closed-form expression for Ψ(., .) in the full-feedback loadscenario can be obtained by substituting (9) into (18). Using theequation from [29, eq. 3.351.1], i.e.,

x∫0

zve−μzdz =v!

μv+1− exp(−μx)

v∑j=0

v!j!

xj

μv−j+1(19)

the analytical solution of the integral in (18) yields (20), shownat the bottom of the page. Thus, the closed-form expressionfor BER l(n) is obtained by substituting (20) into (17) as (21),shown at the bottom of the page, where the index (n) isomitted from gl, αl, and αl+1 for simplicity. The closed-formexpression for Ψ(., .) in the limited-feedback load scenariocan similarly be obtained by substituting (11) into (18) but isomitted here for brevity.

The overall average BER of this adaptive modulation on allN OFDM subcarriers can thus be represented as the average

of the sum of the mode-specific average BERs divided by theoverall ASE and can be written as

BER =1

N ASE

N−1∑n=0

L∑l=1

bl[m,n]BER l(n) (22)

where bl[m,n] is the number of bits per symbol assigned to thebest user (user m) in frequency slot n and for the lth modulationmode, and ASE and BER l(n) have been defined earlier.

3) Outage Probability: When the SNR of the best user fallsbellow the smallest threshold value α1, no data are transmitted.The probability of no transmission (outage probability) can beexpressed as

Pout = Pr(γm ≤ α1) =

α1∫0

fγm(γ)dγ = Fγm

(α1). (23)

Substituting the cdf expressions given in (8) and (10) into(23) yields new closed-form expressions for the outage prob-ability under the full- and limited-feedback load scenarios,respectively. In the full-feedback load scenario, the followingexpression is obtained:

Pout =K∑

i=0

(K

i

)(−1)i exp

(−inT Rcα1

γ

)

×i(nT nR−1)∑

t=0

at,i

(nT Rc

γα1

)t

. (24)

Clearly, the outage probability per frequency slot decreases aseither the lowest threshold value α1 becomes smaller or thenumber of users K increases. However, when the number ofusers increases, then clearly, some users may have less of achance to access the channel.

Ψ(x, gl) =0.2K

(nT nR − 1)!

K−1∑i=0

(K − 1

i

)(−1)i

i(nT nR−1)∑t=0

at,i

(nT Rc

γ

)t+nT nR

⎡⎢⎣ (t + nT nR − 1)!((i+1)nT Rc

γ + gl

)t+nT nR

− exp(−[(i + 1)nT Rc

γ+ gl

]x

) t+nT nR−1∑j=0

(t + nT nR − 1)!xj

j!(

(i+1)nT Rc

γ + gl

)t+nT nR−j

⎤⎥⎦ (20)

BER l(n) =0.2K

(nT nR − 1)!

K−1∑i=0

(K − 1

i

)(−1)i

i(nT nR−1)∑t=0

at,i

(nT Rc

γ

)t+nT nR

⎡⎢⎣exp(−[(i + 1)nT Rc

γ+ gl

]αl

)

×t+nT nR−1∑

j=0

(t + nT nR − 1)!αjl

j!(

(i+1)nT Rc

γ + gl

)t+nT nR−j− exp

(−[(i + 1)nT Rc

γ+ gl

]αl+1

)t+nT nR−1∑j=0

(t + nT nR − 1)!αjl+1

j!(

(i+1)nT Rc

γ + gl

)t+nT nR−j

⎤⎥⎦(21)


B. Switching Levels for Adaptive Modulation

The first attempt in finding the switching levels (αi; i =1, 2, . . . , L) in adaptive modulation schemes satisfying thetransmission requirement was made by Webb and Steele [30].They used the BER curves of different modulation modesover the AWGN channel, which are obtained by simulation,and simply found the SNR values achieving a target BER ineach modulation mode. This concept has widely been usedby many other researchers to find the switching levels [26].For example, for the target BER of 10−4 in square M -QAM,the SNR switching levels will be 8.4, 11.4, 18.2, and 24.3 dBfor BSPK (1 bit), 4-QAM (2 bits), 16-QAM (4 bits), and64-QAM (6 bits), respectively. Obviously, for an SNR valuebelow 8.4 dB should be “no transmission,” and for above24.3 dB, 64-QAM modulation will be employed. Alternatively,the switching levels can be obtained using the approximateBER of M -QAM over the AWGN channel given by [31]

BER l(γ) � 0.2 exp(− 1.5γ

2bl − 1

), l = 2, . . . , L. (25)

This approximation is good for bl ≥ 2 and 0 < γ ≤ 30 dB.The first switching level corresponding to binary phase-shiftkeying (BPSK) modulation (bl = 1) can more accurately beobtained [32] from α1 = [Q−1(BER t)]2/2, where Q−1(.) de-notes the inverse Gaussian Q-function. Therefore, the switchinglevels can simply be calculated for a certain target BER valueof BER t as [32]

αl =

⎧⎨⎩12

[Q−1(BER t)

]2, l = 1

− (2l−1)1.5 ln(5 BER t), l = 2, . . . , L

+∞, l = L + 1.

(26)

Note that the system considered here does not include an outererror correcting code. Obviously, using an additional outer errorcorrecting code in our model provides an extra diversity andimproves the system performance. For the system employing anouter error correcting code, the SNR switching levels requiredfor adaptive modulation can be obtained from the BER(γ)function for each modulation mode. Alternatively, from a set ofBER curves for the assumed M -QAM constellations obtainedfrom simulation or analysis for the coded system over AWGNchannels, one can simply find the SNR switching levels byreading the SNR points corresponding to a target BER. Then,the presented adaptive modulation and scheduling schemes canbe applied. In coded systems, one may expect a higher ASEcompared with the uncoded case, since in the coded system,the SNR switching levels achieving a target BER are lower, andhence, for a given SNR interval, a higher modulation mode willbe assigned. For example, in the uncoded system, the fadingregion 8.4 ≤ γ < 11.4 dB represents the BSPK modulation,whereas in the coded system, it represents higher modulationmodes, resulting in a higher ASE.

C. Continuous-Rate Adaptive Modulation

In continuous-rate adaptive modulation, we assume that thenumber of bits per symbol is not restricted to integer values.While continuous-rate adaptive modulation may conveniently

be considered from a theoretical point of view [33], discrete-rate adaptive modulation is practical. The goal is to find thenumber of bits per symbol (β[m,n]) to be assigned on eachfrequency slot corresponding to the best user as a functionof the target BER. We use an exponential approximation forthe BER in an “invertible” form as a function of the β[m,n]and the instantaneous SNR. It is shown that the expression forthe instantaneous BER of the nth subchannel and mth userof OSFBC-OFDM (square M -QAM with Gray bit mappingon each frequency-slot, where M = 2β[m,n]) over a frequency-selective fading channel can approximately be expressed as [21]

BER[m,n] � 0.2 exp

(− 1.6γ ‖H[m,n]‖2

F

nT Rc

(2β[m,n] − 1

)) . (27)

By inverting (27), the suitable modulation scheme and thecorresponding number of bits in OSFBC-OFDM to obtain thetarget BER (BER t) can be calculated from

β[m,n] = log2

⎛⎝1 +1.6γ ‖H[m,n]‖2

F

nT Rc ln(

0.2BER t

)⎞⎠ . (28)

Using (28), a continuous bit allocation can be performed oneach frequency slot. Taking the OSFBC code rate into account,the instantaneous spectral efficiency [modulation throughput(in bits per second per hertz)] for the nth OFDM frequency slotcan be obtained from SE[m,n] = Rcβ[m,n], and the ASE interms of bits per second per hertz is equal to

ASE(n) = RcE

⎧⎨⎩log2

⎛⎝1 +1.6γ ‖H[m,n]‖2

F

nT Rc ln(

0.2BER t

)⎞⎠⎫⎬⎭ (29)

where E{.} denotes the expectation operator. To obtain ananalytical expression for the ASE, we first express the instanta-neous spectral efficiency as a function of γm, i.e., the SNR ofthe best user in frequency slot n. We use SE[m,n] = Rcβ[m,n]and (28), which yields

SE[m,n] = Rc log2

⎛⎝1 +1.6γm

ln(

0.2BER t

)⎞⎠ (30)

where γm = γ[m,n] = γ‖H[m,n]‖2F /nT Rc is the SNR of the

best user in frequency slot n, and the index (n) is omitted fromγm for simplicity.

Then, by knowing the pdf of the best user’s SNR, the ASEon the frequency slot n in this case will be given by

ASE(n) = E {SE[m,n]}

= Rc

∞∫0

log2

⎛⎝1 +1.6γm

ln(

0.2BER t

)⎞⎠ fγm

(γm)dγm (31)

where the expression of fγm(γm) is given in (7) for full-

feedback scenario and in (11) for limited-feedback scenario. Weobtain the new closed-form expression for ASE(n) in the full-feedback load scenario by substituting (9) into (31), yielding


(32), shown at the bottom of the page, which is proved inthe Appendix, where ξ = 1.6/ ln(0.2/BER t), PM (.) is thePoisson distribution defined by PM (x) =

∑M−1v=0 (xv/v!)e−x,

and E1(.) is the exponential integral of first order defined byE1(x) =

∫∞x t−1e−tdt for x > 0. Finally, the ASE over all the

N subcarriers is given by

ASE =1N

N−1∑n=0

ASE(n). (33)

The overall ASE can be calculated from (33) and using theexpression [see (32)] for the full-feedback load scenario. For thelimited-feedback load scenario, the expression for ASE(n) cansimilarly be found by substituting (11) into (31) but is omittedfor brevity. In this case, the overall ASE can be calculated from(33), which requires a numerical integration of (31).

D. Average Channel Capacity Analysis

The channel capacity of the kth user in OSFBC-OFDM forthe nth frequency slot can be written as [23]

C[k, n] = Rc log2

(1 +

γ

nT Rc‖H[k, n]‖2

F

). (34)

Therefore, the capacity achieved by the best user can be ex-pressed as

C[m,n] = Rc log2(1 + γm) (35)

where γm is defined in the line after (30). The average capacitywill be given by

C(n) = E {C[m,n]}

=

∞∫0

Rc log2(1 + γm)fγm(γm)dγm. (36)

Finally, by averaging over N subchannels, the overall averagechannel capacity can be obtained from

Cavg =1N

N−1∑n=0

C(n). (37)

Similar to (32), we obtain the closed-form expression for Cavg

in the full-feedback load scenario by substituting (9) into (36)and (37), which yields (38), shown at the bottom of the page.For the limited-feedback load scenario, the average capacityC(n) can numerically be evaluated by substituting (11) forfγm

(γm) into (36). Then, the overall average channel capacityCavg can be obtained from (37).

E. Proportional Fair Scheduling (PFS)

From a practical point of view, user fairness is an importantissue that should be considered by the scheduling. In the idealcase, when the statistics of users are the same, the schedul-ing technique maximizes the total throughput as well as thethroughput of individual users. In reality, the statistics of theusers are not the same.

The scheduler presented in the previous section always se-lects the user with the highest SNR and therefore the highestthroughput at each frequency slot. Since the scheduler selectsfor each frequency slot the user having the best quality channel(where the channel quality is defined by the user SNR), andhence the largest throughput, users suffering from bad channelconditions may never transmit their information, yielding anunfair resource allocation among the users. To overcome thislimitation, a PFS technique has been proposed to provide agood compromise between fairness and throughput [5]. PFSallows some of the users with bad channel conditions tobe transmitted. Instead of selecting a user with the largestthroughput, PFS selects a user whose ratio of instantaneousthroughput Rt[k, n] to its own average throughput Rt[k, n] overthe past window of length τ is the largest. In other words, for

ASE(n) =KRc log2(e)(nT nR − 1)!

K−1∑i=0

(K − 1

i

)(−1)i

i(nT nR−1)∑t=0

(t + nT nR − 1)!at,i

(i + 1)t+nT nR

⎧⎨⎩Pt+nT nR

(− (i + 1)nT Rc

ξγ

)

×E1

((i + 1)nT Rc

ξγ

)+

t+nT nR−1∑j=1

1jPj

((i + 1)nT Rc

ξγ

)Pt+nT nR−j

(− (i + 1)nT Rc

ξγ

)⎫⎬⎭ (32)

Cavg =KRc log2(e)

N(nT nR − 1)!

N−1∑n=0

K−1∑i=0

(K − 1

i

)(−1)i

i(nT nR−1)∑t=0

(t + nT nR − 1)!at,i

(i + 1)t+nT nR

⎧⎨⎩Pt+nT nR

(−(i + 1)nT Rc

γ

)

×E1

((i + 1)nT Rc

γ

) t+nT nR−1∑j=1

1jPj

((i + 1)nT Rc

γ

)+ Pt+nT nR−j

(− (i + 1)nT Rc

γ

)⎫⎬⎭ (38)


a given OFDM frequency slot n, the PFS selects the user msuch that

m(n) = arg maxk

{Rt[k, n]Rt[k, n]

}. (39)

The value of Rt[k, n] is then updated as

Rt+1[k, n] ={(

1 − 1τ

)Rt[k, n] + 1

τ Rt[k, n], k = m(1 − 1

τ

)Rt[k, n], k �= m.

(40)

By adjusting τ , the desired tradeoff between fairness andthroughput can be achieved. In general, larger values of τprovide larger total throughput and more unfairness among theusers.

For the analysis of ASE with PFS given by (40), we consider

Rt[k, n] = Rc log2

⎛⎝1 +1.6γ ‖Ht[k, n]‖2

F

nT Rc ln(

0.2BER t

)⎞⎠ (41)

and for the channel capacity analysis with PFS given by (40),we consider

Rt[k, n] = Rc log2

(1 +

γ

nT Rc‖Ht[k, n]‖2

F

). (42)

V. IMPACT OF SPATIAL CORRELATION

ON USER SCHEDULING

In the previous sections, the antennas at the base stationand the antennas at the users ends have been assumed to beuncorrelated. In the following, we study the impact of spatial(antenna) correlation on the performance of user scheduling forthe system under study. To facilitate this, we first derive thepdf and cdf of the SNR of the scheduled user (i.e., the bestuser). Then, we provide analytical results evaluating the effectsof spatial correlation.

Similar to the model used in [34], the spatially correlatedMIMO channel between the kth user and the base stationon the nth frequency slot can be modeled as H[k, n] =R1/2

R [k, n]W[k, n]R1/2T [k, n], where R1/2

R is an nR × nR ma-trix representing the correlation between receive antennas atthe kth user station, and R1/2

T is an nT × nT matrix repre-senting the correlation between transmit antennas at the basestation. W[k, n] is an nR × nT matrix, where its elements arei.i.d. complex Gaussian random variables N (0, 0.5) per dimen-sion. In this model, the channel covariance matrix is expressedas R = Rt

T ⊗ RR, where (.)t is the transpose operator, and⊗ denotes the Kronecker product. We assume that there existz distinct and nonzero eigenvalues (λi, i = 1, . . . , z) for thecovariance matrix R repeated wi times such that

∑zi=1 wi =

Rank(R). Then, the pdf expression of the SNR given in (2) canbe written as [34]

fγ(γ) =z∑

i=1

wi∑j=1

μi,jγj−1

(j − 1)!

(η

γλi

)j

exp(− ηγ

γλi

)(43)

where η = nT Rc, and

μi,j =1

(wj − j)!(γλi

η )(wj−j)

×

⎧⎪⎨⎪⎩ ∂wj−j

∂swj−j

⎡⎢⎣ z∏k=1k �=i

(1 + s

γλk

η

)−wk

⎤⎥⎦⎫⎪⎬⎪⎭∣∣∣∣∣∣∣s=− η

γλi

(44)

where ∂wj−j/∂swj−j denotes the derivative of order wj − jwith respect to s for j = 1, . . . , wi. Then, the cdf of γ isobtained by taking the integral of the pdf in (43) with respectto γ as

Fγ(γ) = 1 −z∑

i=1

wi∑j=1

μi,j exp(− ηγ

γλi

) j−1∑k=0

1k!

(ηγ

γλi

)k

.

(45)

Similar to the uncorrelated antennas case, substituting (43) and(45) into (6) and (7) in the full-feedback load scenario (andinto (10) and (11) in the limited-feedback load scenario), wecan obtain the cdf Fγm

(γ) and pdf fγm(γ) expressions for γm.

Then, we can obtain mathematical expressions of the ASE, theaverage BER, and the average channel capacity. However, inthis paper, we only present analytical expressions of the ASEfor the full-feedback load scenario to evaluate the impact ofspatial correlation on the performance of the system understudy.

In the full-feedback load scenario, substituting (43) and (45)into (6) and (7), we obtain the cdf and pdf of the SNR of thebest user γm as

Fγm(γ) =

⎡⎣1 −z∑

i=1

wi∑j=1

μi,j exp(−ηγ

γλi

)

×j−1∑k=0

1k!

(ηγ

γλi

)k⎤⎦K

(46)

fγm(γ) =K

z∑i=1

wi∑j=1

μi,jγj−1

(j − 1)!

(η

γλi

)j

× exp(−ηγ

γλi

)⎡⎣1 −z∑

i=1

wi∑j=1

μi,j exp(− ηγ

γλi

)

×j−1∑k=0

1k!

(ηγ

γλi

)k⎤⎦K−1

. (47)

Substituting the cdf expression in (46) into (15), the ASE forthe discrete-rate adaptive modulation under correlated antennascan be written as (48), shown at the bottom of the next page.The ASE for the continuous-rate adaptive modulation undercorrelated antennas can be obtained by substituting the pdfexpression in (47) into (31) and (33), yielding (49), shown atthe bottom of the next page, where the integral can be computednumerically.


Finally, note that MIMO OSFBC-OFDM can achieve a space(antenna) diversity gain of nT × nR. To obtain the maxi-mum diversity order achievable in frequency-selective fadingMIMO channels that is shown to be equal to nT × nR × P[4], [35], [36], where P is the number of channel paths, onemay use other system designs for MIMO-OFDM that havebeen proposed in [35] and [36]. Then, user-scheduling schemesand adaptive modulation can be applied for multiuser systemsemploying those MIMO-OFDM systems.

VI. NUMERICAL RESULTS AND DISCUSSIONS

In this section, we provide the numerical results obtainedfrom Monte Carlo simulations and from the mathematicalexpressions for the multiuser MIMO-OFDM system with userscheduling. The considered channels are multipath fading chan-nels (P = 4) with coherence bandwidth smaller than the totalbandwidth of the OFDM system and thus seen as frequency-selective channels. The OFDM system includes N = 128 sub-carriers and a cyclic prefix that is longer than the channeldelay spread. For discrete-rate adaptive modulation, we adoptan adaptive modulation with 8-ary QAM modulation: Ml ∈{1, 21, . . . , 28}, which corresponds to the transmission ofbl[m,n] ∈ {0, 1, 2, . . . , 8} bits/symbol. The modulationschemes include BPSK, 4-QAM, 8-QAM, 16-QAM, 32-QAM,64-QAM, 128-QAM, and 256-QAM, respectively. Thus,using (26) for a target BER of 10−4, we obtain eight SNRswitching levels as αl ∈ {8.39, 11.82, 15.49, 18.81, 21.96,25.04, 28.08, 31.11} dB. In most of the figures (see Figs. 3–17),we assume a full-feedback load scenario with a non-PFSscheme, unless specified in the figures.

The results shown in Figs. 3–17 are obtained from theanalytical formulas expressed in this paper. However, in Figs. 3,

Fig. 3. ASE versus average SNR. γ for continuous-rate and discrete-rate adaptive OSFBC-OFDM (2Tx, 1Rx) with multiuser scheduling(BER t = 10−4).

4, 7, 8, and 12, simulation results are also provided to verifythe analysis. The simulation results for Figs. 3 and 12 areobtained by evaluating the mathematical expressions [e.g., (29)]in which the channel coefficients we used had been obtainedfrom Monte Carlo simulations. For Figs. 3, 4, 7, and 8, we firstgenerate several realizations of the random channel coefficientsfor corresponding users as obtained from Monte Carlo simula-tions. We then calculate instantaneous SNR values for a givennumber of users using (2) for each frequency slot, and then, weselect the best user, i.e., the user with the largest channel SNR.Then, after calculating the required switching thresholds for agiven BER t using (26), the discrete-rate adaptive-modulation

ASE =Rc

N

N−1∑n=0

L∑l=1

bl[m,n] ·

⎧⎪⎨⎪⎩⎡⎣1 −

z∑i=1

wi∑j=1

μi,j exp(−ηαl+1

γλi

) j−1∑k=0

1k!

(ηαl+1

γλi

)k⎤⎦K

−

⎡⎣1 −z∑

i=1

wi∑j=1

μi,j exp(−ηαl

γλi

) j−1∑k=0

1k!

(ηαl

γλi

)k⎤⎦K⎫⎪⎬⎪⎭ (48)

ASE =Rc

N

N−1∑n=0

∞∫0

log2

⎛⎝1 +1.6γm

ln(

0.2BER t

)⎞⎠ fγm

(γm)dγm

=KRc

N

N−1∑n=0

z∑i=1

wi∑j=1

μi,j

(j − 1)!

(η

γλi

)j∞∫

0

γj−1m exp

(−ηγm

γλi

)log2

⎛⎝1 +1.6γm

ln(

0.2BER t

)⎞⎠

×

⎡⎣1 −z∑

i=1

wi∑j=1

μi,j exp(−ηγm

γλi

) j−1∑k=0

1k!

(ηγm

γλi

)k⎤⎦K−1

dγm (49)


Fig. 4. ASE versus average SNR. γ for discrete-rate adaptive OSFBC-OFDM(2Tx, nRRx) with multiuser scheduling for K = 8 users and BER t = 10−4.

Fig. 5. Probability of mode selection versus γ for discrete-rate adaptiveOSFBC-OFDM (2Tx, 1Rx) with multiuser scheduling (BER t = 10−4).

scheme is used to assign the highest modulation mode (bit allo-cation) for that best user according to its instantaneous channelSNR in that frequency slot, as explained in the first para-graphs of Sections IV and IV-A. In this section, the term ASEmeans ASE.

Fig. 3 shows the ASE of the OSFBC-OFDM system versusthe average SNR; γ using code G2 with two transmit and Ksingle-antenna users for both continuous-rate and discrete-rateadaptive modulation schemes for BER t = 10−4. For the sakeof comparison, the Shannon capacity limit is also shown in thefigure, along with the ASE of nonadaptive BPSK modulation.It is observed that nonadaptive BPSK has the smallest spectralefficiency compared with the adaptive cases. The ASE ofnonadaptive BPSK modulation for K = 1 user is obtained bydetermining the SNR value where the average BER is equal tothe target BER of 10−4. As shown in Fig. 6, this SNR value isequal to 19.28 dB. Obviously, for a larger K, i.e., K ≥ 2, this

Fig. 6. Average BER versus γ for discrete-rate adaptive OSFBC-OFDM (2Tx,1Rx) with multiuser scheduling (BER t = 10−4).

Fig. 7. ASE for discrete-rate adaptive OSFBC-OFDM (2Tx, nRRx) withmultiuser scheduling (BER t = 10−4).

SNR value becomes smaller than that for K = 1. As a conse-quence, the ASE curve for nonadaptive modulation will shift tothe left, depending on the value of that SNR, and also ASE inthe nonadaptive case remains smaller than that for the adaptivecases, as expected. Finally, the ASE is calculated by averagingthe allocated bits over 10 000 random channel realizations. InFig. 3, it is observed that, in each case, increasing the numberof users increases the ASE. This is due to the fact that theuser diversity increases as the number of users increases. Wecan also see that the achievable ASE in the continuous-rateadaptive modulation scheme shows about 2-dB improvementcompared with the discrete-rate scheme. This is due to theuse of continuous-rate constellation size and shows a higherbound for the ASE. However, for a target BER of 10−4, theachievable spectral efficiency in the continuous-rate adaptivemodulation comes within about 7 dB of the Shannon capacitylimit. Naturally, if an outer error correcting code is used in


Fig. 8. ASE versus the average SNR γ for OSFBC-OFDM (2Tx, 1Rx) withmultiuser scheduling for K = 8 users.

Fig. 9. ASE versus number of users for discrete-rate adaptive OSFBC-OFDM(2Tx, nRRx) with multiuser scheduling and with limited-feedback load forγ = 20 dB.

addition to the presented MIMO OSFBC-OFDM system, thedistance from the Shannon limit can be reduced. As can alsobe observed, simulation results are matched with the resultsobtained from the analytical formula, verifying the accuracy ofthe analysis.

We can observe from Fig. 4 that increasing nR, i.e., thenumber of receive antennas, improves the ASE even in a single-user scenario. This is due to the increased receiver antennadiversity. In Figs. 3 and 4, it is observed that for very large Kand/or large nR, the corresponding curves undulate and exhibita staircase-like shape. This is because increasing K (and/or nR)increases the multiuser diversity gain (and antenna diversity)and makes the effective subchannel a somewhat “less severe”fading channel. Therefore, the mode selection becomes more“selective” in comparison to the single-user case (and single-receive antenna).

Fig. 10. ASE versus the average SNR γ for continuous-rate adaptive OSFBC-OFDM (nT Tx, 1Rx) with the codes G2 (nT = 2, Rc = 1), H3 (nT =3, Rc = 3/4), and G3 (nT = 3, Rc = 1/2) with multiuser scheduling(BER t = 10−4).

Fig. 11. ASE for continuous-rate adaptive OSFBC-OFDM (nT Tx, 1Rx)with the codes G2 (nT = 2, Rc = 1), H4 (nT = 4, Rc = 3/4), H3 (nT =3, Rc = 3/4), G4 (nT = 4, Rc = 1/2), and G3 (nT = 3, Rc = 1/2) withmultiuser scheduling (γ = 10 dB and BER t = 10−5) with non-PFS.

For further explanation of this phenomenon, we show thecorresponding mode-selection probabilities in Fig. 5 with thesame parameter settings, as explained in Fig. 3. By comparingFig. 5(a) with 5(b), we can see that user selection increasesthe probability of the most appropriate modulation mode ata certain channel SNR and causes the corresponding curveto exhibit a narrower shape. This is clearly demonstrated inFig. 5(b) by the increased peaks of each modulation modearound certain SNR regions for BPSK to 256-QAM modes,where the staircase-like behavior happens in Fig. 5(b). Thisexplains why the throughput curve for the user-selection caseincreases in a staircase-like manner, particularly when thenumber of users and/or the number of receive antennas are high.


Fig. 12. ASE for continuous-rate adaptive OSFBC-OFDM (2Tx, nRRx) withmultiuser scheduling (γ = 10 dB and BER t = 10−5) with PFS and non-PFS.

Fig. 13. ASE for continuous-rate adaptive OSFBC-OFDM (2Tx, nRRx)with multiuser scheduling for limited-feedback scenario (γ = 10 dB andBER t = 10−5).

Fig. 6 shows the average BER versus γ for different numbersof users with nT = 2, nR = 1. As can be observed, the averageBER is always below BER t = 10−4. This is due to the fact thatthe switching levels are chosen so that the instantaneous BERis guaranteed to be always lower than the target BER. For highSNRs, increasing the number of users decreases the averageBER and reaches its achievable value in the high SNR region,which is equal to the BER of the highest available modulationmode (256-QAM) in high SNRs. The fluctuation in the BERcurves is due to the use of discrete-rate adaptive modulationwith user scheduling, as explained in Fig. 5, whereas eachfluctuation corresponds to the BER of a certain modulationmode in that SNR region. This fluctuating behavior occurswhen the number of users (K) and/or number of receive

Fig. 14. Average capacity for OSFBC-OFDM (2Tx, nRRx) with multiuserscheduling for limited-feedback scenario (γ = 10 dB).

Fig. 15. ASE for continuous-rate adaptive OSFBC-OFDM (2Tx, 1Rx) withmultiuser scheduling for correlated transmit antennas (BER t = 10−4) andassuming the correlation coefficients ρ = 0.0, ρ = 0.75, and ρ = 1.0, which,respectively, correspond to uncorrelated, strongly correlated, and fully corre-lated transmit antennas.

antennas is large. For the sake of comparison, the averageBER of nonadaptive BPSK for K = 1 user is also shown,which is obtained by using Monte Carlo simulations. As can beobserved, above the SNR value of 19.28 dB, the average BERis always below the target BER value of 10−4.

Fig. 7 shows the ASE versus the number of users for differentaverage SNR values. As can be observed, the ASE may reacha maximum bits per second per hertz value, depending on theaverage SNR. It is shown that as the average SNR increases,the ASE is increased. This is because by increasing the averageSNR, a higher-level modulation mode can be selected. It is alsoobserved that the ASE increases as the number of users or thenumber of receive antennas increases.

We show in Fig. 8 the impact of BER t on ASE, assumingnT = 2, nR = 1, and K = 8 users. As expected, it can be


Fig. 16. ASE for continuous-rate adaptive OSFBC-OFDM (nT Tx, 1Rx)(BER t = 10−4) for γ = 10 dB with multiuser scheduling for correlatedtransmit antennas with the codes G2 (nT = 2, Rc = 1), H3 (nT = 3, Rc =3/4), H4 (nT = 4, Rc = 3/4), G3 (nT = 3, Rc = 1/2), and G4 (nT =4, Rc = 1/2) and assuming the correlation coefficients ρ = 0.0, ρ = 0.75,and ρ = 1.0, which, respectively, correspond to uncorrelated, strongly corre-lated, and fully correlated transmit antennas.

Fig. 17. ASE for continuous-rate adaptive OSFBC-OFDM (nT Tx, 1Rx)(BER t = 10−4) for γ = 20 dB with multiuser scheduling for correlatedtransmit antennas with the codes G2 (nT = 2, Rc = 1), H3 (nT = 3, Rc =3/4), H4 (nT = 4, Rc = 3/4), G3 (nT = 3, Rc = 1/2), and G4 (nT =4, Rc = 1/2) and assuming the correlation coefficients ρ = 0.0, ρ = 0.75,and ρ = 1.0, which, respectively, correspond to uncorrelated, strongly corre-lated, and fully correlated transmit antennas.

seen that for a given average SNR, the achievable ASE will beincreased when BER t increases.

Fig. 9 shows the ASE for the same system versus the numberof users in the limited-feedback load scenario for differentsystem configurations and with different feedback loads (F =1, 0.5, 0.1, and 0.05). As can be observed, by reducing the feed-back load by 50% (F = 0.5), the system performance remainsalmost the same as that of the full feedback load (F = 1). For

a smaller feedback load such as 10% (F = 0.1), a performanceloss is negligible when the number of users is above 30.

We show, in Fig. 10, the impact of the number of transmitantennas on ASE. This figure shows the ASE of the OSFBC-OFDM system versus the average SNR for continuous-rateadaptive modulation and using the codes G2 (nT = 2, Rc = 1),H3 (nT = 3, Rc = 3/4), and G3 (nT = 3, Rc = 1/2) with Kusers with a single receive antenna nR = 1. It can be observedthat for all the codes used, increasing the number of availableusers increases the ASE. It is also shown that the code G2 em-ploying two transmit antennas nT = 2 provides a higher ASEthan the codes employing three transmit antennas nT = 3. Thisis due to the fact that among all the possible orthogonal STBCcodes, the code G2 (Alamouti’s code) is the only full-rate codeRc = 1, whereas G3 and H3, employing three transmit antennas(nT = 3), are half rate and 3/4 rate codes, respectively.

Fig. 11 shows the ASE versus the number of users for thesame system as in Fig. 10 but with nR = 2 per user. We assumea target BER of 10−5 and an average SNR of γ = 10 dB. Ascan be observed in all cases, by increasing the number of users,the ASE is increased. Again, it can be observed that the code G2

provides the highest ASE, although it employs less number oftransmit antennas compared with the other codes. Comparingthe ASE of the system employing H3 and H4 having thesame code rate (Rc = 3/4), we observe that in the single-usercase (K = 1), the ASE is higher for H4 employing one moretransmit antenna compared with the code H3. On the contrary,for K ≥ 2, the ASE for H4 is lower than that of H3, which isdue to the channel hardening effects [37]. We can conclude thatincreasing the number of transmit antennas at the base stationlimits the performance gain that could be achieved by multiuserdiversity.

Fig. 12 shows the ASE for continuous-rate adaptivemodulation versus the number of users for the same systemusing code G2, presenting the impact of increasing the numberof receive antennas on ASE. We assume a target BER of 10−5

and an average SNR of γ = 10 dB. As can be observed byincreasing either the number of receive antennas or the numberof users, the ASE is increased. To provide more fairness in userscheduling, we have also used the PFS technique with the win-dow of length τ = 100. As can be observed, providing fairnesswith PFS slightly decreases the ASE. For the various MIMOcases, the first set of curves (Simulation) for the non-PFScase is obtained from calculating the ASE(n) from (29) withselection of the best user with the highest SNR from K activeusers and then averaging over several realizations of MIMOchannels. Then, using (33), the average over all N subchannelsis obtained. The second set of curves (Analysis) is obtainedfrom (32) and (33). As can be seen, the simulation results arematched with the results obtained from the analytical formula.The third set of curves corresponds to using the PFS technique(with τ = 100), and the results are obtained using channelsimulations. We first selected the best user with the givencriteria in (39), and then, we calculated the spectral efficiencyin (41) by considering the values of throughput updated as in(40). Then, we calculated the average of the spectral efficiencyas given by (41) over several realizations of MIMO channeland then averaging over all N subchannels. It can be seen that


at the cost of a slight loss in transmission spectral efficiency,scheduling fairness among the users can be guaranteed.

Figs. 13 and 14 show the ASE and the average channelcapacity versus the number of users for OSFBC-OFDM (2Tx,nRRx) in the limited-feedback load scenario for the averageSNR of 10 dB. In addition, different system configurations withdifferent feedback loads (F = 1, 0.5, 0.2, and 0.1) have beenconsidered. As can be observed, reducing the feedback loadby 50% (F = 0.5), the system performance remains almostthe same as in the full-feedback load (F = 1). For a smallerfeedback load such as F = 0.1, a performance loss is negligiblewhen the number of users is above 30.

Finally, Figs. 15–17 show the ASE for the system under studywhen the transmit antennas at the base station are correlated.We assume that the underlying covariance matrices for the usedcodes are as follows. Assuming the single-antenna receiverat the user stations nR = 1, the covariance matrix R is annT × nT matrix with elements rj,i = 1 if j = i and rj,i =ρ if j �= i, where ρ is the correlation coefficient such that0 ≤ ρ ≤ 1.

Fig. 15 shows the ASE versus γ for the different numberof users and for a target BER denoted BER t = 10−4, wherewe assume nT = 2, nR = 1, and using code G2. In this case,the underlying covariance matrix has two distinct eigenvaluesλ1 = 1 + ρ and λ2 = 1 − ρ, each of which is repeated once(w1 = w2 = 1). We assume the correlation coefficients ρ =0.0, ρ = 0.75, and ρ = 1.0, which respectively correspond touncorrelated, strongly correlated, and fully correlated transmitantennas. As can be observed, a spatial correlation reducesthe ASE in the single-user scenario (K = 1), whereas in themultiuser case (K ≥ 2), spatial correlation is beneficial to thesystem and can increase the ASE of the system.

Figs. 16 and 17 show the ASE versus the number of usersfor a target BER value of 10−4 and the average SNR of γ =10 and 20 dB, respectively. The results are provided for severalpossible cases using different codes, such as G2, H3, H4, G3,and G4. The underlying covariance matrices R for the usedcodes in each case is an nT × nT matrix with elements definedearlier, where depending on the employed code, the number oftransmit antennas nT may be different [20]. For the matrix Rcorresponding to the codes G3 and H3, there exist two distincteigenvalues λ1 = 1 + 2ρ and λ2 = 1 − ρ, each repeated onceand twice, respectively (w1 = 1 and w2 = 2). For the matrix Rcorresponding to the codes G4 and H4, there exist two distincteigenvalues λ1 = 1 + 3ρ and λ2 = 1 − ρ, each repeated oneand three times, respectively (w1 = 1 and w2 = 3). However,when all transmit antennas are fully correlated (ρ = 1.0), thesystem acts like a single transmit antenna.

From Figs. 16 and 17, it can be observed that when ρ = 1.0,the ASE for code G2 is equal to that of the SISO case, whereasthe average spectral efficiencies for other codes such as G3

and H3 (that are equivalent for both codes) are smaller thanthat for the SISO case, since the code rate (Rc) of codes G3

and H3, which are 1/2 and 3/4, respectively, are smaller thanRc = 1.

As also observed in Figs. 11, 16, and 17, for the uncorrelatedand correlated cases with 0 ≤ ρ < 1, when K ≥ 2, the ASE forH4 (and for G4) is a little lower than that of H3 (and that of G3),

which is due to the channel-hardening effects [37], where wealso concluded in Fig. 11 that increasing the number of transmitantennas at the base station limits the performance gain thatcould be achieved by multiuser diversity.

Comparing uncorrelated and correlated cases for each codein Figs. 15–17, we observe that in the single-user case (K = 1),ASE is higher for uncorrelated case, whereas in the multiusercase (K ≥ 2), ASE for correlated case is higher than that ofuncorrelated case. This clearly states that unlike the single-user MIMO case, spatial correlation is beneficial for multiuserMIMO systems employing user scheduling. However, a sim-ilar conclusion is given in [38] for the capacity of broadcastchannels with MIMO maximal ratio combining systems underspatial correlation.

VII. CONCLUSION

In this paper, we have presented and analyzed the perfor-mance of multiuser scheduling schemes for a MIMO-OFDMsystem over multipath frequency-selective MIMO fading chan-nels. Two schemes, including discrete-rate and continuous-rateadaptive modulations, have been employed to increase the spec-tral efficiency of the system. Two channel feedback scenarioshave been considered: 1) full feedback and 2) limited feedback.For both scenarios, a performance evaluation using mathemat-ical analysis and numerical simulation has been performedshowing the significant advantages of the proposed schemes.It is shown that rate-adaptive modulation and user schedulingcan increase the ASE. It is observed that in using the proposeduser scheduling in conjunction with the reduced-feedback loadscenario (reduced up to 90%), the channel capacity and theASE of the system under study remain almost the same asin the full-feedback load scenario when the number of usersis greater than 30. It is observed that due to the channel-hardening effects, increasing the number of transmit antennasat the base station limits the performance gain. It is alsoshown that unlike the single-user MIMO case, spatial correla-tion is beneficial for multiuser MIMO systems employing userscheduling.

APPENDIX

Substituting into (31) the expressions for fγm(γ), which is

given by (9), we obtain the corresponding ASE expression as

ASE(n) =KRc

(nT nR − 1)!

K−1∑i=0

(K − 1

i

)(−1)i

×i(nT nR−1)∑

t=0

at,i

(nT Rc

γ

)

×∞∫

0

log2

⎛⎝1 +1.6γm

ln(

0.2BER t

)⎞⎠

×(

nT Rcγm

γ

)t+nT nR−1

× exp(−(i + 1)nT Rcγm

γ

)dγm. (50)


For solving the preceding integral, we use the followingresult [39]:

μ

(M − 1)!

∞∫0

ln(1 + x)(μx)M−1e−μxdx

= PM (−μ)E1(μ) +M−1∑j=1

1jPj(μ)PM−j(−μ) (51)

where PM (.) is the Poisson distribution defined by PM (x) =∑M−1v=0 (xv/v!)e−x, and E1(.) is the exponential integral of first

order, which is defined by E1(x) =∫∞

x t−1e−tdt for x > 0.Therefore, using (51) and after performing changes of vari-

ables together with some simplifications, we obtain the solutionof integral in (50) as

∞∫0

log2

⎛⎝1 +1.6γm

ln(

0.2BER t

)⎞⎠

×(

nT Rcγm

γ

)t+nT nR−1

exp(−(i + 1)nT Rcγm

γ

)dγm

=γ (t + nT nR − 1)! log2(e)

nT Rc(i + 1)t+nT nR

×{Pt+nT nR

(− (i + 1)nT Rc

ξγ

)E1

((i + 1)nT Rc

ξγ

)

+t+nT nR−1∑

j=1

1jPj

((i + 1)nT Rc

ξγ

)

× Pt+nT nR−j

(− (i + 1)nT Rc

ξγ

)}(52)

where ξ = 1.6/ ln(0.2/BER t). Finally, replacing (52) in (50)yields (32), which concludes the proof. �

REFERENCES

[1] H. Yang, “A road to future broadband wireless access: MIMO-OFDM-based air interface,” IEEE Commun. Mag., vol. 43, no. 1, pp. 53–60,Jan. 2005.

[2] T. Keller and L. Hanzo, “Adaptive multicarrier modulation: A convenientframework for time–frequency processing in wireless communications,”Proc. IEEE, vol. 88, no. 5, pp. 611–640, May 2000.

[3] T. Liew and L. Hanzo, “Space–time trellis and space–time block codingversus adaptive modulation and coding aided OFDM for wideband chan-nels,” IEEE Trans. Veh. Technol., vol. 55, no. 1, pp. 173–187, Jan. 2006.

[4] M. Jiang and L. Hanzo, “Multiuser MIMO-OFDM for next-generationwireless systems,” Proc. IEEE, vol. 95, no. 7, pp. 1430–1469, Jul. 2007.

[5] P. Viswanath, D. Tse, and R. Laroia, “Opportunistic beamforming usingdumb antennas,” IEEE Trans. Inf. Theory, vol. 48, no. 6, pp. 1277–1294,Jun. 2002.

[6] N. Sharma and L. H. Ozarow, “A study of opportunism for multiple-antenna systems,” IEEE Trans. Inf. Theory, vol. 51, no. 5, pp. 1804–1814,May 2005.

[7] J. Jiang, R. M. Buehrer, and W. H. Tranter, “Antenna diversity in multiuserdata networks,” IEEE Trans. Commun., vol. 52, no. 3, pp. 490–497,Mar. 2004.

[8] W. Ajib and D. Haccoun, “An overview of scheduling algorithms inMIMO-based fourth-generation wireless systems,” IEEE Netw., vol. 19,no. 5, pp. 43–48, Sep./Oct. 2005.

[9] D. Gesbert and M. Alouini, “How much feedback is multi-user diversityreally worth?” in Proc. IEEE ICC, 2004, pp. 234–238.

[10] B. Holter, M. Alouini, and G. E. Øien, “Multiuser switched diversitytransmission,” in Proc. IEEE VTC—Fall, 2004, pp. 2038–2043.

[11] D. Niyato, E. Hossain, and V. Bhargava, “Scheduling and admissioncontrol in power-constrained OFDM wireless mesh routers: Analysis andoptimization,” IEEE Trans. Wireless Commun., vol. 6, no. 10, pp. 3738–3748, Oct. 2007.

[12] S. Chieochan and E. Hossain, “Adaptive radio resource allocation inOFDMA systems: A survey of the state-of-the-art approaches,” WirelessCommun. Mobile Comput., vol. 9, no. 4, pp. 513–527, Apr. 2009.

[13] D. Niyato and E. Hossain, “Adaptive fair subcarrier/rate allocation inmultirate OFDMA networks: Radio link level queuing performanceanalysis,” IEEE Trans. Veh. Technol., vol. 55, no. 6, pp. 1897–1907,Nov. 2006.

[14] Y. J. Zhang and K. B. Letaief, “Multiuser adaptive subcarrier-and-bitallocation with adaptive cell selection for OFDM systems,” IEEE Trans.Wireless Commun., vol. 3, no. 5, pp. 1566–1575, Sep. 2004.

[15] Y. J. Zhang and K. B. Letaief, “An efficient resource-allocation schemefor spatial multiuser access in MIMO/OFDM systems,” IEEE Trans.Commun., vol. 53, no. 1, pp. 107–116, Jan. 2005.

[16] C. Y. Wong, R. S. Cheng, K. B. Letaief, and R. D. Murch, “MultiuserOFDM with adaptive subcarrier, bit, and power allocation,” IEEE J. Sel.Areas Commun., vol. 17, no. 10, pp. 1747–1758, Oct. 1999.

[17] J. Jang and K. B. Lee, “Transmit power adaptation for multiuser OFDMsystems,” IEEE J. Sel. Areas Commun., vol. 21, no. 2, pp. 171–178,Feb. 2003.

[18] A. Czylwik, “Adaptive OFDM for wideband radio channels,” in Proc.IEEE GLOBECOM, 1996, pp. 713–718.

[19] H. Shin and J. H. Lee, “Performance analysis of space–time block codesover keyhole Nakagami-m fading channels,” IEEE Trans. Veh. Technol.,vol. 53, no. 2, pp. 351–362, Mar. 2004.

[20] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space–time blockcodes from orthogonal designs,” IEEE Trans. Inf. Theory, vol. 45, no. 5,pp. 1456–1467, Jul. 1999.

[21] M. Torabi, S. Aissa, and M. Soleymani, “On the BER performance ofspace–frequency block coded OFDM systems in fading MIMO channels,”IEEE Trans. Wireless Commun., vol. 6, no. 4, pp. 1366–1373, Apr. 2007.

[22] J. G. Proakis, Digital Communications. New York: McGraw-Hill, 2000.[23] M. Torabi, W. Ajib, and D. Haccoun, “Multiuser scheduling for MIMO-

OFDM systems with continuous-rate adaptive modulation,” in Proc. IEEEWCNC, 2008, pp. 946–951.

[24] A. Papoulis, Probability, Random Variables, and Stochastic Process.New York: McGraw-Hill, 1991.

[25] C. J. Chen and L. C. Wang, “A unified capacity analysis for wireless sys-tems with joint multiuser scheduling and antenna diversity in Nakagamifading channels,” IEEE Trans. Commun., vol. 54, no. 3, pp. 469–478,Mar. 2006.

[26] L. Hanzo, C. H. Wong, and M. S. Yee, Adaptive Wireless Transceivers:Turbo-Coded, Turbo-Equalised and Space–Time Coded TDMA, CDMAand OFDM Systems. Hoboken, NJ: Wiley, 2002.

[27] M. Torabi, W. Ajib, and D. Haccoun, “Discrete-rate adaptive multiuserscheduling for MIMO-OFDM systems,” in Proc. IEEE VTC—Fall, 2008,pp. 1–5.

[28] A. J. Goldsmith and S. G. Chua, “Variable-rate variable-power MQAMfor fading channels,” IEEE Trans. Commun., vol. 45, no. 10, pp. 1218–1230, Oct. 1997.

[29] I. Gradshteyn, I. Ryzhik, and A. Jeffrey, Table of Integrals, Series, andProducts. New York: Academic, 2007.

[30] W. T. Webb and R. Steele, “Variable rate QAM for mobile radio,” IEEETrans. Commun., vol. 43, no. 7, pp. 2223–2230, Jul. 1995.

[31] S. T. Chung and A. J. Goldsmith, “Degrees of freedom in adaptive modu-lation: A unified view,” IEEE Trans. Commun., vol. 49, no. 9, pp. 1561–1571, Sep. 2001.

[32] M. S. Alouini and A. J. Goldsmith, “Adaptive modulation over Nakagamifading channels,” Wireless Pers. Commun., vol. 13, no. 1/2, pp. 119–143,May 2000.

[33] G. D. Forney, Jr., R. G. Gallager, G. R. Lang, F. M. Longstaff, andS. U. Qureshi, “Efficient modulation for band-limited channels,” IEEEJ. Sel. Areas Commun., vol. SAC-2, no. 5, pp. 632–647, Sep. 1984.

[34] I. Kim, “Exact BER analysis of OSTBCs in spatially correlated MIMOchannels,” IEEE Trans. Commun., vol. 54, no. 8, pp. 1365–1373,Aug. 2006.

[35] W. Su, Z. Safar, M. Olfat, and K. Liu, “Obtaining full-diversityspace–frequency codes from space–time codes via mapping,” IEEE Trans.Signal Process., vol. 51, no. 11, pp. 2905–2916, Nov. 2003.

[36] X. Ma and G. Giannakis, “Full-diversity full-rate complex-fieldspace–time coding,” IEEE Trans. Signal Process., vol. 51, no. 11,pp. 2917–2930, Nov. 2003.


[37] B. M. Hochwald, T. L. Marzetta, and V. Tarokh, “Multi-antenna channelhardening and its implication for rate feedback and scheduling,” IEEETrans. Inf. Theory, vol. 50, no. 9, pp. 1893–1909, Sep. 2004.

[38] R. Louie, M. McKay, and I. Collings, “Impact of correlation on thecapacity of multiple access and broadcast channels with MIMO-MRC,”IEEE Trans. Wireless Commun., vol. 7, no. 6, pp. 1–11, Jun. 2008.

[39] C. G. Günther, “Comment on ‘Estimate of channel capacity in Rayleighfading environment,”’ IEEE Trans. Veh. Technol., vol. 45, no. 2, pp. 401–403, May 1996.

Mohammad Torabi (S’02–M’05) received the Ph.D. degree in electricalengineering from Concordia University, Montréal, QC, Canada, in 2004.

He was a Postdoctoral Fellow with the Department of Electrical and Com-puter Engineering, Concordia University. He was a Postdoctoral Fellow and iscurrently a Research Associate with the Department of Electrical Engineering,École Polytechnique de Montréal, Montréal, QC. His research interests includedigital signal processing, communication theory and wireless communica-tions with emphasis on orthogonal frequency division multiplexing, adaptivemodulation, resource allocation, user scheduling, cooperative communications,multiple-input–mutliple-output systems, and space–time coding.

David Haccoun (S’62–M’67–SM’84–F’93–LF’03)received the Engineer and B.A.Sc. (magna cumlaude) degrees in engineering physics from ÉcolePolytechnique de Montréal, Montréal, QC, Canada,the S.M. degree in electrical engineering from theMassachusetts Institute of Technology, Cambridge,and the Ph.D. degree in electrical engineering fromMcGill University, Montréal.

Since 1966, he has been with the Departmentof Electrical Engineering, École Polytechnique deMontréal, where he has been a Professor of electrical

engineering since 1980 and was the founding Head of the Communicationand Computer Section in 1980. He was a Visiting Research Professor withConcordia University, Montréal; INRIA Laboratories, Paris, France; the Uni-versity of Lund, Lund, Sweden, École de Technologie Supérieure, Montréal;the University of Victoria, Victoria, BC, Canada; and the Advanced StudyInstitute, University of British Columbia, Vancouver, BC. From 1990 to2002, he was a Researcher with the Canadian Institute for Telecommuni-cations Research under the National Centers of Excellence of the Govern-ment of Canada. He is the holder of a U.S. patent for an error-controltechnique and is a coauthor of the books The Communications Handbook(Boca Raton, FL: CRC, 1997 and Piscataway, NJ: IEEE, 2001), The Encyclo-pedia of Telecommunications (New York: Wiley, 2003), and Digital Commu-nications by Satellite: Modulation, Multiple-Access and Coding (New York:Wiley, 1981), a Japanese translation of which was published in 1984. Hisresearch interests include communication theory, theory and applications oferror-control coding, wireless and mobile communications, and digital commu-nication systems by satellite. He is the author or coauthor of over 300 journalpapers and conferences papers in these areas.

Dr. Haccoun is a Fellow of the Engineering Institute of Canada. He is aMember of the Order of Engineers of Quebec, Canada; Sigma Xi; The NewYork Academy of Sciences; and the American Association for the Advance-ment of Sciences. He is a Member of the board of Governors of the IEEEVehicular Technology Society, a Member of the Board of the Telecommuni-cations Engineering Management Institute of Canada, and was a Member ofthe Board of the Communications Research Centre, Ottawa, ON, Canada. Hewas the General Cochair of the IEEE Vehicular Technology Conference, Fall2006, in Montréal.

Wessam Ajib (M’05) received the EngineerDiploma degree in physical instruments from theInstitute National Polytechnique de Grenoble,Grenoble, France, in 1996 and the Diplome d’ÉtudesApprofondies degree in digital communication sys-tems and the Ph.D. degree in computer sciences andcomputer networks from École Nationale Supèrieuredes Télécommunication, Paris, France, in 1997 and2000, respectively.

From October 2000 to June 2004, he was anArchitect and a Radio Network Designer with Nortel

Networks, Ottawa, ON, Canada, where he had conducted many projects andintroduced different innovative solutions for the third generation of wirelesscellular networks. From June 2004 to June 2005, he was a PostdoctoralFellow with the Department of Electrical Engineering, École Polytechniquede Montréal, Montréal, QC, Canada. Since June 2005, he has been with theDepartment of Computer Science, Université du Québec à Montréal, where heis currently an Assistant Professor of computer networks. He is the author orcoauthor of many journal papers and conferences papers. His research inter-ests include wireless communications and wireless networks, multiple accessand medium-access control design, traffic scheduling, multiple-input–multiple-output systems, and cooperative communications.

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