diversity and outage performance in space–time block coded ... · fading. space–time codes...

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 5, SEPTEMBER 2005 2519

Diversity and Outage Performance in Space–TimeBlock Coded Ricean MIMO Channels

Rohit U. Nabar, Member, IEEE, Helmut Bölcskei, Senior Member, IEEE,and Arogyaswami J. Paulraj, Fellow, IEEE

Abstract—The goal of this paper is to assess the impact ofreal-world propagation conditions on the maximum achievablediversity performance of communication over Ricean multiple-input multiple-output (MIMO) channels. To this end, we examinea MIMO channel employing orthogonal space–time block codes(OSTBCs) and study the diversity behavior of the resulting effec-tive single-input single-output (SISO) channel. The performancecriteria employed are symbol error rate, outage capacity, andwideband spectral efficiency. For general propagation conditions,we establish key quantities that determine performance irrespec-tive of the performance criterion used. Furthermore, we discussthe relation between the notion of diversity order related to theslope of the average error probability versus signal-to-noise ratio(SNR) curve and diversity order related to the slope of the outageprobability versus SNR curve. For Ricean fading MIMO channels,we demonstrate the existence of an SNR-dependent critical rateRcrit, below which signaling with zero outage is possible and,hence, the fading channel behaves like an additive white Gaussiannoise (AWGN) channel. For SISO channels, Rcrit is always zero.In the MIMO case, Rcrit is a simple function of the angle betweenthe vectorized Ricean component of the channel and the subspacespanned by the vectorized Rayleigh fading component.

Index Terms—Multiple-input multiple-output (MIMO), outageprobability, Ricean fading, spatial diversity, space–time coding.

I. INTRODUCTION

W IRELESS links are impaired by random fluctuations insignal level known as fading. Diversity provides the

receiver with multiple (ideally independent) replicas of thetransmitted signal and is therefore a powerful means to combatfading. Space–time codes [1]–[4] are capable of extractingspatial diversity gain in systems employing multiple antennasat the transmitter and receiver (multiple-input multiple-output(MIMO) systems) without requiring channel knowledge inthe transmitter. Orthogonal space–time block codes (OSTBCs)[3], [4] are particularly attractive since they yield maximumspatial diversity gain and, at the same time, decouple the vec-tor detection problem into scalar detection problems, therebysignificantly reducing decoding complexity (at the expense ofspatial transmission rate).

Manuscript received December 20, 2003; revised May 15, 2004 and June 23,2004; accepted June 28, 2004. The editor coordinating the review of this paperand approving it for publication is H. Li.

R. U. Nabar is with the DSP Technology Group, Marvell Semiconductor,Inc., Sunnyvale, CA 94089 USA (e-mail: [email protected]).

H. Bölcskei is with the Communication Technology Laboratory, SwissFederal Institute of Technology (ETH) Zürich, Zürich CH-8092, Switzerland(e-mail: [email protected]).

A. J. Paulraj is with the Information Systems Laboratory, Stanford Univer-sity, Stanford, CA 94305 USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/TWC.2005.853835

The performance of any space–time coding scheme dependsstrongly on the MIMO channel statistics, which, in turn, dependon antenna characteristics, height and spacing, and scatter-ing richness. The classical independent identically distributed(i.i.d.) Rayleigh frequency-flat fading MIMO channel model[2] assumes that the elements of the matrix channel are i.i.d.zero-mean circularly symmetric complex Gaussian distributed.Measurements reveal, however, that, in practice, the MIMOchannel deviates significantly from this idealistic behavior [5].In this paper, we consider a general MIMO channel model thatallows Rayleigh or Ricean fading and arbitrary scalar channelgains and correlation between matrix elements.

A. Contributions

The goal of this paper is to assess the impact of real-world propagation conditions on maximum achievable diversityperformance in MIMO channels. OSTBCs convert the MIMOchannel into a single-input single-output (SISO) channel withthe effective channel gain given by the square-root of thesum of the squared magnitudes of the complex-valued scalarsubchannel gains [4]. All the degrees of freedom in the channelare utilized to realize diversity gain, and the multiplexinggain [6] equals 1. The ultimate diversity performance of ageneral MIMO channel can therefore be assessed by studyingthe performance limits of the effective SISO channel resultingfrom the application of OSTBCs. Commonly used performancemeasures in fading channels are outage capacity [7], averagesymbol error rate [8], and wideband spectral efficiency [9]. Thedetailed contributions reported in this paper can be summarizedas follows.

1) We quantify analytically the impact of Ricean fading,spatial fading correlation, and channel gain imbalance onoutage capacity, average symbol error rate, and widebandspectral efficiency and establish key quantities (as a func-tion of the propagation parameters) determining perfor-mance, irrespective of the performance criterion used.

2) We exhibit and quantify the critical performance deter-mining role played by the angle between the vectorizedRicean component and the range space of the fadingcorrelation matrix pertaining to the Rayleigh componentof the channel.

3) For Ricean channels, we demonstrate the existence ofa critical rate, below which signaling with zero outageis possible, or, equivalently, the channel behaves like anadditive white Gaussian noise (AWGN) channel. For fad-ing SISO channels, this critical rate is always zero. In the

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2520 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 5, SEPTEMBER 2005

MIMO case, the critical rate is a simple function of theangle between the vectorized Ricean component and therange space of the correlation matrix corresponding tothe Rayleigh component. Moreover, we show that for afixed transmission rate, channels with nonzero criticalrate have a diversity order of !, whereas in the case ofthe critical rate being equal to zero, the diversity order isgiven by the rank of the fading correlation matrix.

4) We demonstrate that the notions of diversity order relatedto the slope of the error probability versus signal-to-noise ratio (SNR) curve, and diversity order related to theslope of the outage probability versus SNR curve yieldequivalent results.

5) Finally, we provide analytical expressions for outageprobability (as a function of the propagation parameters),which lend themselves to efficient numerical evaluation.

B. Relation to Previous Work

We note that the effective channel for OSTBCs resemblesthe effective channel obtained for maximum ratio combining(MRC) [10], [11], the performance of which has been studiedextensively [11]–[14]. Noting that Ricean fading is a specialcase of Nakagami fading, the results in [11]–[13] can be usedto compute (numerically) the outage probability of OSTBCsunder Ricean fading. In this paper, we provide alternativepower and Laguerre series-based expressions for the outageprobability of OSTBCs under Ricean fading. In particular, thepower series expansion allows us to characterize the outageperformance analytically.

Previously, the equivalence of diversity characterizationthrough average and outage error rate performance for a broaderclass of fading channels (Nakagami-q, n, m distributions) withfinite code books has been demonstrated in [14]. We showthat the same equivalence result is obtained in Ricean channelsthrough an average symbol error rate and outage probability[packet error rate (PER)] analysis assuming Gaussian codebooks in the latter case.

The performance of space–time codes under correlatedRayleigh fading and Ricean fading has been studied in [2] and[15]–[21]. The impact of propagation conditions on the wide-band spectral efficiency of MIMO channels has been studiedpreviously in [22]. Compared to [2] and [15]–[22], our analysisreveals the critical performance determining role of the anglebetween the vectorized Ricean component and the range spaceof the correlation matrix associated with the Rayleigh fadingcomponent. Moreover, our results establish the presence of apreviously unknown critical transmission rate, below whicha Ricean MIMO channel (in conjunction with an OSTBC)behaves like an AWGN channel, and above which it appearsas Rayleigh fading.

C. Notation

The superscripts T, H, " stand for transposition, conjugatetransposition, and elementwise conjugation, respectively. Edenotes the expectation operator. 0m,n stands for the m # nall-zeros matrix. Im is the m # m identity matrix. Tr(A),

$A$F =!

Tr(AAH), r(A), R(A), N (A), and A† denotethe trace, Frobenius norm, rank, range space, null space, andpseudo-inverse, respectively, of the matrix A. %(A) and &(A)stand for the real and imaginary parts of A, respectively. $a$denotes the Euclidean norm of the vector a. For an m # nmatrix A = [a1 a2 . . . an], we define the mn # 1 vectorvec(A) = [aT

1 aT2 . . . aT

n ]T. u(y) denotes the unit-step func-tion defined as u(y) = 1 for y ' 0 and 0 otherwise. A circularlysymmetric complex Gaussian random variable is a randomvariable Z = X + jY ( CN (0,!2), where X and Y are i.i.d.N (0,!2/2). Unless specified otherwise, all logarithms are tothe base 2.

D. Organization of the Paper

The remainder of this paper is organized as follows. InSection II, we present the general MIMO channel model, theassociated signal model, and a simplified channel model used inour simulation examples. In Section III, we discuss the differentperformance criteria analyzed in the paper. In Section IV, weprovide the power series and Laguerre series expansions for thecumulative distribution function (cdf) of the squared Frobeniusnorm of the MIMO channel matrix. In Sections V, VI, and VII,we study the impact of propagation conditions on OSTBC av-erage symbol error rate, outage capacity, and wideband spectralefficiency, respectively. We conclude in Section VIII.

II. MIMO CHANNEL AND SIGNAL MODEL

In this section, we introduce the MIMO channel model usedthroughout this paper, followed by a brief description of thesignal model for OSTBCs.

A. General MIMO Channel Model

We consider a frequency-flat fading MIMO channel withMT transmit and MR receive antennas. The correspondingMR # MT random channel matrix H = H + "H is decom-posed into the sum of a fixed [possibly line-of-sight (LOS)]component H = E{H} and a variable (or scattered) component"H. In the case of pure Rayleigh fading, H = 0MR,MT , whereasin the case of Ricean fading, H )= 0MR,MT . Throughout the re-mainder of this paper, we let N = MTMR and apply the chan-nel power normalization E{$H$2

F } = $H$2F + E{$ "H$2

F } *N . Furthermore, we assume that the channel is block fading[23] so that H remains constant over T ' MT symbol periodsand changes in an independent fashion from block to block.The elements of "H are circularly symmetric complex Gaussianrandom variables. With large antenna spacing, rich scatteringin the propagation environment, and all antenna elements em-ploying identical polarization, the scalar subchannels of "H can,furthermore, be assumed i.i.d. In practice, however, these condi-tions are hardly met so that the elements of "H will be correlatedwith possibly different variances resulting from power/gainimbalance and/or the use of different antenna polarizations.Defining "h = vec( "H), the statistics of "H are characterizedby the N # N correlation matrix R = E{"h"hH} with eigende-composition R = U!UH, where ! = diag{!j}N

j=1 with theordering !j ' !j+1(j = 1, 2, . . . , N + 1). Furthermore, we set


NABAR et al.: DIVERSITY AND OUTAGE PERFORMANCE IN SPACE–TIME BLOCK CODED RICEAN MIMO CHANNELS 2521

h = vec(H) and h = vec(H). Note that h and R completelycharacterize the statistics of the Ricean MIMO channel. Wefinally note that the classical i.i.d. Rayleigh fading MIMOchannel is recovered for h = 0N,1 and R = IN .

B. Signal Model

The input–output relation is given by

y =#

Es

MTHs + n

where y denotes the MR # 1 received signal vector, Es is thetotal average energy available at the transmitter over a symbolperiod, s is the MT # 1 transmit signal vector, and n is MR #1 spatiotemporally white noise with E{nnH} = N0IMR . Weimpose the transmit power constraint Tr(E{ssH}) = MT.Throughout the paper, we assume that the channel matrixH is unknown at the transmitter and perfectly known at thereceiver. Finally, we note that the use of OSTBCs combinedwith appropriate processing at the receiver [4] turns the matrixchannel H into an effective SISO channel with input–outputrelation

y =#

Es

MT$H$F s + n (1)

where y denotes the scalar processed received signal, s is thescalar transmitted signal, and n is CN (0, N0) white noise.We note that the results in this paper are not restricted toOSTBCs but apply more generally to any scheme resulting inthe signal model (1). Examples of such schemes include MRCin a single-input multiple-output (SIMO) channel or RAKEreception in a code division multiple access (CDMA) channelwith intersymbol interference (ISI).

C. Simplified Channel Model for Simulation Examples

For the sake of simplicity, in the simulation examples pre-sented in the remainder of the paper, we limit the number ofvariable channel parameters and consider a system with twotransmit and one or two receive antennas. The fixed and variablechannel components are given by

H =#

K

1 + K

$1 11 1

%"H =

#1

1 + K

$"g1,1 "g1,2

"g2,1 "g2,2

%(2)

where K ' 0 denotes the Ricean K-factor [24]. Furthermore,we define the following correlation coefficients

t =E&"g1,1"g"1,2

'(

E {|"g1,1|2} E {|"g1,2|2}=

E&"g2,1"g"2,2

'(

E {|"g2,1|2} E {|"g2,2|2}

r =E&"g1,1"g"2,1

'(

E {|"g1,1|2} E {|"g2,1|2}=

E&"g1,2"g"2,2

'(

E {|"g1,2|2} E {|"g2,2|2}

where t and r will be referred to as the transmit and re-ceive correlation coefficients, respectively. Measurements have

shown [5] that the correlation between the diagonal elements"g1,1 and "g2,2, and the antidiagonal elements "g1,2 and "g2,1, istypically very small compared to t and r. In the simplifiedchannel model, we shall therefore assume that E{"g1,1"g"2,2} =E{"g1,2"g"2,1} = 0. Note that the simplified correlation model isequivalent to the Kronecker correlation model in [16], [25], [26]if rt is small. Furthermore, unless specified otherwise, we setE{|"gi,j |2} = 1(i, j = 1, 2). When simulating a channel with asingle receive antenna, we consider the two-input one-outputchannel corresponding to only the first row of the matrixH = H + "H, with H and "H as defined in (2).

We finally emphasize that the analytical results presented inthis paper hold for the general MIMO channel model describedin Section II-A. The simplified channel model introduced in thissubsection will be employed in our simulation examples only.

III. PERFORMANCE CRITERIA

The performance limits of communication over fadingchannels are often quantified through average symbol errorrate, outage capacity, and wideband spectral efficiency. In thefollowing, we briefly describe these criteria and introduce thedefinitions used subsequently.

A. Uncoded Average Symbol Error Rate

Assuming quadrature amplitude modulation (QAM), theprobability of (uncoded) symbol error for a given channelrealization H can be approximated by [27], [28]

Pe = NeQ

)

*+

"d2min

2MT$H$2

F

,

- (3)

where " = Es/N0 denotes the SNR and Ne and dmin standfor the average number of nearest neighbors and the minimumdistance of separation of the underlying constellation, respec-tively. Applying the Chernoff bound Q(x) * exp[+(x2/2)] to(3), we get

Pe * Nee+

!d2min!H!2

F4MT .

Consequently, the average (over the channel H) uncodedsymbol error rate P e = E{Pe} can be upper bounded as

P e * Ne#($") (4)

where $ = d2min/(4MT) is a constellation-specific constant and

#(s) = E{exp(+s$H$2F )} is the moment-generating function

(MGF) of $H$2F . Traditionally, the diversity order offered by

a fading channel is defined as the magnitude of the high-SNRslope of average error probability as a function of SNR (on alog–log scale) [8], [29]. Based on this concept, the maximumachievable diversity order over a general MIMO channel (i.e.,the diversity order achieved by OSTBCs) is given by

dE = + lim!,!

log P e

log "' + lim

!,!

log#($")log "



where the subscript E in dE relates to the average errorprobability. Since the Chernoff bound is tight for high SNR[29], we can take [2]

dE = + lim!,!

log#($")log "

. (5)

B. Outage Capacity

By virtue of the scalar nature of the effective channel inducedby OSTBCs [cf. (1)], assuming a scalar Gaussian code book,the corresponding mutual information is given by [30]

I = rc log.

1 +"

MT$H$2

F

/bps/Hz

where rc * 1 denotes the spatial rate of the space–time blockcode as defined in [4].

Since H is random, the mutual information I will be a ran-dom variable. The outage probability corresponding to trans-mission rate R is defined as Pout(R) = P (I * R) [7], [23].Equivalently, one can characterize the outage behavior by spec-ifying the q% outage capacity Cout(q%) as the capacity thatis guaranteed for (100 + q)% of the channel realizations, i.e.,P (I * Cout(q%)) = q%. Outage capacity is relevant when thelength of the transmitted (temporal) codeword is restricted tobe less than or equal to the fading block length. We can see thatcalculating the outage probability or outage capacity requiresknowledge of the cdf of the random variable I . Denoting thecdf of $H$2

F as F (y), we have

Pout(R) = F

)

*

02

Rrc + 1

1MT

"

,

-

and1

Cout(q%) = rc log.

"

MTF+1(q%) + 1

/.

For the case where codewords span only one fading block,the outage probability can be related to PER as follows. Sup-pose we fix a transmission rate R. Since the channel is drawnrandomly according to a given fading distribution, there willalways be a nonzero probability that the mutual informationcorresponding to a given channel realization falls below R(no matter how small R is). Assuming that the transmittedcodeword (packet) is decoded successfully if R is less thanthe mutual information of the given channel realization anddeclaring a decoding error otherwise, the outage probabilityequals the PER. This leads to the notion of a rate-dependentdiversity order as defined in [31]

dO(R) = + lim!,!

log Pout(R)log "

(6)

where the subscript O in dO(R) indicates that this definitionrelates to outage probability. Clearly, the statistics of $H$2

F(which, in turn, depend on the channel conditions) will have

1F!1(y) denotes the inverse function of F (y).

a significant influence on the PER and, hence, the (rate-dependent) diversity order.

We finally note that in contrast to dE, the quantity dO(R)explicitly reflects the dependence of achievable diversity or-der on transmission rate R. Moreover, dO(R) is relevant forGaussian code books, whereas dE pertains to finite code books.Since both definitions are frequently used in the literature, wewill study the impact of propagation conditions on diversityorder through an analysis of both dE and dO(R) and discussthe relation between the two quantities.

C. Spectral Efficiency in the Wideband Regime

A third frequently used measure to describe the severity ofrandom fluctuations in the fading channel relates to the spectralefficiency in the wideband regime [9]. Assuming that the fadingprocess is ergodic and coding can be performed over an infinitenumber of independently fading blocks, the spectral efficiencyachieved by OSTBCs is given by

C(") = rc E2

log.

1 +"

MT$H$2

F

/3bps/Hz (7)

where the expectation is taken with respect to the randomchannel H. It is important to note that the use of OSTBCs canresult in a considerable loss in spectral efficiency (comparedto the true MIMO capacity). However, OSTBCs will performwell 1) in the very low-SNR regime (i.e., " - 1) or 2) if 2 therandom MIMO channel has rank 1 with probability 1 (such asthe pin-hole channel [26]). Following [9], we define Eb/N0 ="/C(") as the SNR per information bit and (Eb/N0)min asthe minimum Eb/N0 required to sustain error-free communi-cation. It is shown in [9, Th. 1] that (Eb/N0)min = ln 2/C(0)where C(·) denotes the derivative of C(") in nats/s/Hz.It now follows that C(0) = (rcµ)/MT with µ = E{$H$2

F }.Invoking the power constraint µ * N = MTMR, we ob-tain (Eb/N0)min ' ln 2/(rcMR), which reflects the fact thatadding receive antennas reduces (Eb/N0)min due to receivearray gain [6]. Since the channel is unknown to the transmitter,no transmit array gain can be realized.

The quantity we are interested in is the slope of the spectralefficiency as a function of Eb/N0 (in bits per second per hertzper 3 dB) at (Eb/N0)min (i.e., the wideband slope). Applying[9, Th. 9] to (7), this slope is obtained as

S =2rc

%(8)

where % = E{$H$4F }/(E{$H$2

F })2 is the kurtosis of $H$F .Denoting the standard deviation of $H$2

F by ! =(E{$H$4

F }+ (E{$H$2F })2)1/2 it is easily seen that

% = AF + 1 (9)

where AF = (!/µ)2 denotes the amount of fading as definedin [11] and [32]. The amount of fading is a widely used measurefor quantifying the severity of random fluctuations in fadingchannels.

2OSTBCs are, in fact, capacity optimal for rank-1 MIMO channels if rc = 1.



We conclude this section by noting that kurtosis is a mea-sure of the peakiness of a distribution (or, equivalently, therandomness of the underlying random variable). The minimumvalue of kurtosis is 1, which is achieved if and only if therandom variable is deterministic. As a consequence of theseobservations, we have

S * 2rc

with the upper bound being achieved if and only if the channelis AWGN. The wideband slope (8) can therefore be interpretedas a measure of the deviation of the fading channel from anAWGN channel. We finally note that the wideband slope for theMIMO channel SMIMO is lower bounded by the correspondingwideband slope for OSTBCs with rc = 1, i.e., SMIMO ' 2/%,where % is defined in (8).

IV. DERIVATION OF THE CDF OF $H$2F

In the previous section, we have seen that specification of thediversity performance through average uncoded symbol errorrate or PER requires computation of the MGF #(s) and the cdfF (y) of $H$2

F , respectively. Modifying the approach describedin [33] to apply to the complex-valued case, we derive a closed-form expression for #(s) in the Appendix [cf. (22)], based onwhich, again suitably modifying the approach in [33], we thenfind a power series expansion [cf. (24)] and a Laguerre seriesexpansion [cf. (25)] for F (y). We hasten to add that the resultspresented in the Appendix follow from a simple generalizationof the approach reported in [33]. We have therefore restrictedour exposition to providing only the key steps in the derivations.We note that an alternative series expansion for F (y) has beenpresented in [34]. The resulting expressions for F (y) in [34]are, in general, complicated and amenable to analytical studiesonly in certain special cases [35].

The convergence properties of the power series expansionin the low-outage regime allow us to analytically characterizePER-based diversity performance with relative ease. As notedin [33], the Laguerre series expansion on the other hand showsbetter convergence over a wider range of y. Generally, weobserved that for y * 2, the power series converges faster thanthe Laguerre series. In the remainder of this paper, we shalluse the power series expansion of F (y) for analytical studiesof the impact of the propagation conditions on the outage prob-ability (or, equivalently, PER) in the low-outage regime. TheLaguerre series expansion has been provided to allow numericalcomputation of the PER over a wide range of outage rates. Wefinally note that the truncation analysis in [36] can readily beextended to the complex-valued case to derive bounds on theresidual error associated with summing a finite number of termsin the power and Laguerre series expansions. Moreover, denot-ing the truncation error that results from using only the first term(corresponding to k = 0) in the power series expansion (24) asE(y), it follows from [36, eq. (121)] that E(y) . y(r(R)+1)ey/c

(with c arbitrarily chosen to satisfy 0 < c < !r(R)) and, hence,

limy,0

E(y) = 0. (10)

A. Numerical Example 1

This example is designed to demonstrate the differencein the convergence behavior of the power series and La-guerre series expansions of F (y). In the following, FE(y)and FA(y) denote the empirically (through Monte Carlomethods) obtained cdf and the analytical cdf [obtained from(24) or (25)] of $H$2

F , respectively. We define the meansquare error (MSE) between FE(y) and FA(y) as MSE =(1/M)

4Mi=1 |FE(yi) + FA(yi)|2 with M uniformly spaced

data points yi(i = 1, 2, . . . ,M) in the interval under consid-eration. For M = 1000, Fig. 1 shows the MSE as a functionof the number of terms summed in the power and Laguerre(& = 1) series expansions, respectively, of F (y)(0 * y * 8)for a channel with K = 2, r = 0.1, and t = 0.2 (cf. simplifiedchannel model described in Section II-C). It is clearly seenthat for a given MSE, the Laguerre series expansion requiressignificantly fewer terms than the power series expansion. Theerror floors in Fig. 1 can be attributed to the fact that beyonda certain number of terms in the series expansion, the errorin empirical estimation of F (y) dominates.

We shall next demonstrate the superiority of the power seriesexpansion over the Laguerre series expansion for small valuesof y. For the same channel as in Fig. 1, Fig. 2 shows theempirically obtained cdf (through Monte Carlo methods) of$H$2

F along with the power series expansion (24) of F (y),calculated using the first term in the series expansion, and theLaguerre series expansion (25) of F (y), calculated using thefirst and the first ten terms, respectively. The figure clearlyshows that for small y, using the first term in the power seriesexpansion (24) yields an excellent approximation of F (y).

V. UNCODED AVERAGE SYMBOL

ERROR RATE PERFORMANCE

In this section, combining (4) and the expression for theMGF of $H$2

F derived in the Appendix, we shall quantifythe impact of the propagation conditions on P e as well as therelated definition of diversity order dE given in (5). We start byinserting (22) into (4) and taking the logarithm of both sides toobtain

log P e * log Ne + $"$H$2F

+r(R)5

j=1

($"|bj |)2

1 + $"!j+

r(R)5

j=1

log(1 + $"!j). (11)

Using (5) and setting ' = $H$2F +

4r(R)j=1 (|bj |2/!j), it now

follows that

dE = lim!,!

.$"'

log "

/+ r(R) (12)

which, upon application of L’Hopital’s rule, yields

dE =2

!, ' > 0r(R), ' = 0. (13)



Fig. 1. Comparison of the convergence properties of the Laguerre series and the power series expansions of F (y). The superior convergence property of theLaguerre series expansion is evident.

Fig. 2. Comparison of the approximation properties of the Laguerre series and the power series expansions of F (y) for small y. The power series expansionexhibits superior numerical properties.

The case ' < 0 is excluded since

' = hHU$

0r(R),r(R) 0r(R),N+r(R)

0N+r(R),r(R) IN+r(R)

%

6 78 9A

UHh ' 0. (14)

Note that UAUH = IN + R(RHR)†RH, and, hence, ' maybe interpreted as the squared Euclidean norm of the projectionof h onto N (R). Recalling that U contains the eigenvectors ofR, it follows that UA spans N (R), while U(IN + A) spans

R(R). We can now conclude that equality in (14) is achievedin either of the following cases: 1) h )= 0N,1 and r(R) = N ;2) h = 0N,1 and R arbitrary; or 3) h )= 0N,1, r(R) < N , andh lies completely in R(R). This immediately implies that inthe case of pure Rayleigh fading (i.e., h = 0N,1), the diversityorder is given by dE = r(R). In the Ricean case, the situationis more complicated as both dE = ! and dE = r(R) arepossible. If the vector h lies entirely in R(R), equality in (14)is achieved and dE = r(R). If h has nonzero components inN (R), we have ' > 0 and, consequently, dE = !. This resulthas a nice physical interpretation: Let us start by investigating



Fig. 3. Schematic illustrating the impact of the Hermitian angle between h and R(R) on diversity order. If the Hermitian angle is nonzero, at least one of theMRMT vectorized channel dimensions does not experience fading and the channel is effectively AWGN.

the SISO case where h = h + "h and (14) reduces to ' = (|h|2with

( =2

0, K < !1, K = !

where K = |h|2/E{|"h|2} denotes the K-factor of the SISOchannel. Employing (13), we can immediately conclude thatdE = 1 for K < ! and dE = ! if K = !. Equivalently, thisresult says that the diversity order equals 1 as long as there isa Rayleigh fading component in the channel and the diversityorder becomes ! for an AWGN channel.

We are now in a position to generalize this discussion toarbitrary MIMO channels with N = MTMR. The realizationsof the N -dimensional vector h = h + "h span a subspace thatdepends on h and the subspace spanned by the realizationsof "h and, hence, specified by R. Now, in the case where hlies entirely in the range space of R (and, hence, the subspacespanned by the realizations of "h), all the dimensions “excited”by h will have a Rayleigh fading component so that, usingour result above for the SISO case, the diversity order issimply the number of dimensions excited by h, or, equivalently,dE = r(R). In the case where the projection of h onto N (R) isnonzero, we have at least one dimension that is purely AWGN,and, hence, by (13), we get dE = !. With the projection of honto R(R) given by

p/(h,R) = R(RHR)†RHh

we can rewrite

' = $h$2 +::p/(h,R)

::2

and, hence, recover more formally that dE = ! whenever theprojection of h onto N (R) is nonzero. A more geometricinterpretation is obtained by considering the Hermitian anglebetween h and R(R) (see the equation at the bottom of thepage) [37].

Note that 0 * !(h,R(R)) * )/2. We can now concludethat 0 < !(h,R(R)) * )/2 implies a diversity order of !,whereas !(h,R(R)) = 0 results in dE = r(R). The situationis summarized in Fig. 3.

Besides the high-SNR slope of P e, the offset of the P e

curve will indicate the impact of the propagation conditionson uncoded average error rate performance. We shall there-fore briefly discuss this aspect. Let us start with the high-SNRcase (i.e., " 0 1) where (11) simplifies to

log P e * log Ne + $"' + r(R) log($") +r(R)5

j=1

log !j .

For fixed R with r(R) < N , the upper bound on P e is min-imum if ' is maximum (i.e., the projection of h onto N (R)is maximum) and all nonzero eigenvalues of R are equal.Hence, the Hermitian angle !(h,R(R)) has an importantimpact on the offset as well. For r(R) = N and, hence,!(h,R(R)) = 0, the error rate performance is optimum ifall the !j are equal, or, equivalently, R = *IN with * =E{$ "H$2

F }/N . In the low-SNR regime (i.e., " - 1), we have

log P e * log Ne + $";$H$2

F + Tr(R)<

(15)

which shows that the error rate performance is simply a func-tion of the total power in the channel. Moreover, we can seethat in the low-SNR regime, the geometry of h relative to R as

!;h,R(R)

<=

=>>?

>>@

0, h = 0N,1

cos+1

.AAAAh

Hp"(h,R)

$h$$p"(h,R)$

AAAA

/, h )= 0N,1 and p/(h,R) )= 0N,1

"2 , h )= 0N,1 and p/(h,R) = 0N,1



Fig. 4. Impact of the Hermitian angle between h and R(R) on average uncoded symbol error rate performance. Both channels perform equally well whent = 0, but show drastically different behavior for t = 1.

Fig. 5. Simplified graphical interpretation of Numerical Example 1 (for ease of visualization, we take R(R) to be two-dimensional).

well as the eigenvalue spread of R no longer have an impact onaverage error rate performance.


This example demonstrates the critical impact of the Her-mitian angle !(h,R(R)) on uncoded average symbol errorrate performance. We consider a system with MT = 2 andMR = 1 employing the Alamouti scheme (rc = 1) and two dif-ferent channels H1 and H2 having different fixed componentsH1 = [1 1] and H2 = [1 + 1], respectively, but the same fad-

ing components. Fig. 4 shows the empirically (through MonteCarlo simulation) calculated average symbol error rate forboth channels for K = 2 and for transmit correlation coef-ficients of t = 0 and t = 1. The symbol error rate for anAWGN channel with $H$2

F = 2 is plotted for comparison.For t = 0, we have R = (2/3)I2 and, hence, !(h1,R(R)) =!(h2,R(R)) = 0 so that for both channels dE = 2. However,for t = 1, the performance of the two channels is completelydifferent. For Channel 1, !(h1,R(R)) = 0 (see Fig. 5) anddE = 1, reflected by the slope of the average error rate curveat high SNR. For channel 2, we have !(h2,R(R)) = )/2 (see



Fig. 5) so that dE = !, reflected by the close proximity of theerror rate curve to that for the AWGN channel. In the low-SNRregime, all channels perform identically as expected from (15).

VI. PACKET ERROR RATE PERFORMANCE

In this section, we study the impact of propagation conditionson PER as defined in Section III. More specifically, employingthe power series expansion in (24), we compute the outagerelated diversity order dO(R), defined in (6) as a function of thepropagation parameters, and show that the conclusions obtainedthrough an analysis of dE are equivalent to those obtainedfrom dO(R). Consequently, we find an interesting relationbetween the notion of diversity order obtained from the averageerror probability for finite code books and diversity order ob-tained from PER assuming Gaussian code books. Moreover, theanalysis presented in this section establishes the existence of acritical rate Rcrit, below which Ricean fading channels behavelike AWGN channels, in the sense that signaling with zerooutage is possible. For rates above Rcrit, the Ricean channelbehaves like a Rayleigh fading channel. Interestingly, it can beshown that Rcrit is a simple function of !(h,R(R)).

Our analysis is for the high-SNR case (i.e., " 0 1), whichallows us to approximate the cdf of $H$2

F by retaining the firstterm in the power series expansion (24)

Pout(R) 1

)

*r(R)B

j=1

1!j

,

-

)

*e+4r(R)

j=1

|bj |2

"2j

,

-

#

)

*

02

Rrc +1

1MT

! + '

,

-r(R)

r(R)!u

)

*

02

Rrc + 1

1MT

"+ '

,

- (16)

where ' was defined in (12). We shall next distinguish thecases ' = 0 and ' > 0.

For ' = 0, the unit-step function in (16) will be equal to 1 forall rates R so that

log Pout(R) = +r(R)5

j=1

log !j +r(R)5

j=1

|bj |2

!2j

+ r(R) log00

2Rrc + 1

1MT

1+ r(R) log "+ log(r(R)!).

Invoking (6), we find that for a fixed (SNR-independent)transmission rate R

dO(R) = r(R)

which is consistent with the finite code book result in (13).Note, however, that if the transmission rate grows as a functionof SNR so that R(") = rc log(1 + (Z"/MT)), where Z is anarbitrary positive constant, then Pout(R) = F (Z) for any SNRand, hence, dO(R) = 0, conforming with [31].

Let us proceed with the case ' > 0. Investigating the ar-gument of the unit-step function in (16), we obtain Pout(R)(see the equation at the bottom of the page), where Rcrit =rc log(1 + ('"/MT)) and may be interpreted as the rate sup-ported by the projection of h onto N (R). This result pointsto an interesting behavior. At transmission rates R < Rcrit,signaling with zero outage is possible, and the Ricean chan-nel behaves like an AWGN channel. Consequently, we obtaindO(R) = ! in this case. Interestingly, up to the pre-log rc, thecritical rate Rcrit is the capacity of an SISO AWGN channelwith SNR = '"/MT. Equivalently, we can say that the gainof the effective SISO AWGN channel is given by the lengthof the projection of h onto the null space of R (see Fig. 3).Note that while (17), shown at the bottom of the page, presentsthe outage probability for high SNR, it follows from (24)or (25) (cf. the argument of the unit-step function) that thepresence of a critical rate can be established at any SNR andwithout invoking any approximation for Pout(R). Finally, ifthe transmission rate is allowed to grow with SNR so thatR(") = rc log(1 + (Z"/MT)) with Z ' ', then dO(R) = 0. Ifon the other hand, the transmission rate grows with SNR so thatR(") = rc log(1 + (Z"/MT)) with Z < ', then we are assuredzero outage and once again we have dO(R) = !.

We can now summarize our findings by noting that forfixed (SNR-independent) transmission rate R

dO(R) =2

!, ' > 0r(R), ' = 0 (18)

which is fully consistent with the behavior of dE [cf. (13)] andconfirms a similar equivalence made for average and outageerror rates in the finite code book case [14]. Furthermore,we note that Rcrit = 0 if !(h,R(R)) = 0 and Rcrit > 0 if0 < !(h,R(R)) * )/2. Consequently, Rcrit = 0 for an SISOfading channel (unless K = !). Finally, we note that for ' = 0and arbitrary R, or for ' > 0 with R > Rcrit, fixed r(R), andsmall |bj | (j = 1, 2, . . . , N) (i.e., small K-factor), the spreadof the !j determines the offset of the PER with minimumPER obtained if R = *IN . This result is consistent with ourfindings for P e in Section V.


This example serves to demonstrate the impact of!(h,R(R)) on Rcrit and outage performance. For K = 2,

Pout(R)

=>>>>>?

>>>>>@

= 0, R < Rcrit

10Cr(R)

j=11#j

1)

*e+4r(R)

j=1

|bj |2

"2j

,

-

)

*

02

Rrc #1

1MT

! +$

,

-r(R)

r(R)! , R ' Rcrit

(17)



Fig. 6. Impact of the Hermitian angle between h and R(R) on PER performance. Both channels perform equally well under independent fading (i.e., t = 0).For t = 1, channel 2 behaves like an AWGN channel for SNRs above 6.53 dB.

R = 2 bps/Hz and H1 and H2 defined in NumericalExample 2, Fig. 6 shows the empirically (through Monte Carlosimulation) calculated PER for the two channels for t = 0 andt = 1. In the uncorrelated case t = 0, ' = 0, and, consequently,Rcrit = 0 for both channels. Fig. 6 indeed shows that thetwo channels yield equal performance and exhibit Rayleighfading behavior with dO = 2 as suggested by (18). In the fullycorrelated case t = 1, Rcrit = 0 for channel 1, which, combinedwith r(R) = 1, yields a Rayleigh fading behavior as seen inFig. 6. Channel 2, however, has a nonzero critical rate for t = 1given by Rcrit = log(1 + (2"/3)) and, hence, behaves like anAWGN channel for SNRs above 6.53 dB. (The performance ofan AWGN channel with $H$2

F = 2 is shown for comparison.)

VII. WIDEBAND SPECTRAL EFFICIENCY OF OSTBCS

From our discussion in Section III, it is clear that under atotal channel power constraint (i.e., $H$2

F + E{$ "H$2F } * N ),

the exact statistics of H do not have an impact on the min-imum SNR per information bit required to sustain reliablecommunication, confirming the observation made previously in[9]. However, as will be seen in the following, the statisticsof H do have a profound impact on the slope of widebandspectral efficiency as defined in [9]. Using (8), it is shown inthe Appendix that

S = 2rc

)

D*

)

D*Tr(!2) + 2hHRh0Tr(!) + hHh

12

,

E- + 1

,

E-

+1

. (19)

It is easily verified that (19) is equivalent to [22, eq. (18)]for the special case of MISO channels with rc = 1, a conse-

quence of the fact that OSTBCs are capacity optimal in sucha scenario.

A. Pure Rayleigh Fading

Constraining the total average power in the channel toTr(!) = N , it is straightforward to show that S is maximized(or, equivalently, Tr(!2) is minimized) when !j = 1(j =1, 2, . . . , N). This condition corresponds to i.i.d. fading, inwhich case S = 2rcN/(N + 1). We can see that in the infinitediversity limit N , !, the wideband slope of the Rayleighfading channel approaches the wideband slope of the AWGNchannel (up to the factor rc). Correlation matrix eigenvaluespread resulting from fading correlation and/or gain imbalancebetween channel elements will therefore reduce the widebandslope S through the term Tr(!2), which is directly related tothe “correlation number” defined in [22].

B. Ricean Fading

In the presence of Ricean fading, the wideband slope de-pends on the eigenvalue spread of R through the term Tr(!2)and through the quadratic form hHRh. The Hermitian anglebetween h and R(R) will impact the quadratic form hHRh;however, the delineation in performance is not as sharp as forthe case of average or outage error rates. For example, hHRh isminimum (i.e., 0) and, hence, the wideband slope is maximum(but finite) when !(h,R(R)) = )/2. For 0 * !(h,R(R)) <)/2, the exact value of the wideband slope will depend onthe geometry of h relative to the geometry of the eigenvec-tors of R. Unlike the cases of symbol error rate and PER,the wideband slope does not exhibit fundamentally differentbehavior in the cases of Rayleigh and Ricean fading. In both



Fig. 7. Impact of the Hermitian angle between h and R(R) on the slope of wideband spectral efficiency.

cases, the AWGN wideband slope of S = 2rc can be achieved.As already mentioned in Section III, the wideband slopeessentially measures the peakiness of the pdf of $H$2

F . Thepresence of a Ricean component results in faster convergenceto the AWGN wideband slope.

C. Numerical Example 4

Following the setup in Numerical Examples 2 and 3, Fig. 7plots S for the Alamouti scheme (i.e., rc = 1) as a func-tion of the K-factor. As expected, both channels exhibit thesame performance for t = 0. In the correlated case t = 1,channel 2 outperforms channel 1, in the sense that convergenceto the AWGN wideband slope is much faster. Furthermore, forK = 0, we can see that spatial fading correlation results inreduced S when compared to the uncorrelated case t = 0.

VIII. CONCLUSION

We characterized the impact of real-world propagation con-ditions on the maximum achievable diversity performanceof communication over Ricean multiple-input multiple-output(MIMO) channels. Our analysis identified the Hermitian angle!(h,R(R)) and the eigenvalue spread of R as key quantitiesgoverning performance, irrespectively of the performance crite-rion used. For Ricean channels, we established the existence ofa critical rate Rcrit, below which signaling with zero outage ispossible and, hence, the fading channel behaves like an AWGNchannel. For rates above Rcrit, the Ricean channel behaves likea Rayleigh fading channel. For fading SISO channels, Rcrit

is always zero, whereas in the MIMO case, Rcrit is a simplefunction of !(h,R(R)). Finally, we showed that the notionsof diversity order related to the slope of the average errorprobability versus SNR curve, and diversity order related tothe slope of the outage probability versus SNR curve, yieldequivalent results.

APPENDIX

MGF of $H$2F

For a given N # 1 vector h = vec(H), we start by definingthe 2N # 1 vector p = [%(h)T &(h)T]T = p + "p with p =[%(h)T &(h)T]T and "p = [%("h)T &("h)T]

T. The joint distri-

bution of the elements of p is completely specified by the meanvector p = E{p} and the 2N # 2N covariance matrix Rp =E{"p"pT}. Next, we establish a relation between the eigende-compositions of R and Rp. For R = U!UH where ! =diag{!j}N

j=1 (recall that !j ' !j+1 for j = 1, 2, . . . , N + 1)and Rp = Up"UT

p , where " = diag{+j}2Nj=1, it is straight-

forward to show that

" =12

$! 00 !

%Up =

$%(U) +&(U)&(U) %(U)

%.

Using the reduced eigendecomposition Rp = FUpF"FUT

p whereF" = diag{F+j}2r(R)

j=1 with F+j(j = 1, 2, . . . , 2r(R)) denotingthe nonzero eigenvalues of Rp, the MGF of pTp + (pTp+42r(R)

j=1 (r2j /F+j)) and the associated region of convergence

(ROC) are obtained from [33] as

F#(s) = E

=?

@e+s

0pTp+

0pTp+

42r(R)

j=1

r2j

F#j

11GH

I

= exp

=?

@+2r(R)5

j=1

r2j

2F+2j

+ 22r(R)5

j=1

r2j

(2F+j)3

# s+1

J1 +

1s2F+j

K+1GH

I

2r(R)B

j=1

(1 + s2F+j)+12 ,

ROC : %(s) > maxF%j

J+ 1

2F+j

K(20)



where rj(j = 1, 2, . . . , 2r(R)) denotes the jth element of

r = F"1/2 FUT

p p. Defining

b = [ b1 b2 · · · bN ]T = !12 UHh

it is easily seen that r = (1/2

2)[%(Fb)T &(Fb)T]T

, whereFb = [b1 b2 · · · br(R)]T. Hence, pTp + (pTp +

42r(R)j=1 (r2

j /F+j)) = $H$2

F + ($H$2F +

4r(R)j=1 (|bj |2/!j)) so that (20) can

be written as

F#(s) =r(R)B

j=1

(1 + s!j)+1

# exp

=?

@+r(R)5

j=1

|bj |2

!2j

+r(R)5

j=1

|bj |2

!3j

s+1

.1 +

1s!j

/+1GH

I

ROC : %(s) > max#j

.+ 1!j

/. (21)

Summarizing our results, we obtain the MGF of $H$2F as

#(s) = EL

e+s$H$2FM

= exp

=?

@+s

)

*$H$2F +

r(R)5

j=1

|bj |2

!j

,

-

GH

IF#(s). (22)

Power Series Expansion for the cdf of $H$2F

Following the series expansion approach in [33] with appro-priate modifications to reflect the complex Gaussian nature ofthe elements of H, we shall next use (21) to derive a powerseries expansion for the cdf of $H$2

F . Defining

M(,) =

)

*r(R)B

j=1

1!j

,

-

)

*r(R)B

j=1

.1 +

,

!j

/+1,

-

# exp

=?

@

r(R)5

j=1

|bj |2,!3

j

.1 +

,

!j

/+1

+r(R)5

j=1

|bj |2

!2j

GH

I

we can write (21) as

F#(s) = s+r(R)M

.1s

/. (23)

In [33], it is shown that M(,) can be expressed as a powerseries of the form M(,) =

4!k=0 ck,k with

c0 =

)

*r(R)B

j=1

1!j

,

- exp

)

*+r(R)5

j=1

|bj |2

!2j

,

-

ck =1k

k+15

r=0

dk+rcr for k ' 1

where d0 = +4r(R)

j=1 ln(!j) +4r(R)

j=1 (|bj |2/!2j ) and dk =

(+1)k4r(R)

j=1 ((1/!kj ) + (k|bj |2/!k+2

j )) for k ' 1. Using (23),it follows that3

F#(s) =!5

k=0

cks+(r(R)+k)

which lends itself readily to Laplace transform inversion fol-lowed by integration to give the cdf of $H$2

F as

F (y) =!5

k=0

ck(y + ')r(R)+k

(r(R) + k)!u(y + ') (24)

where ' = $H$2F +

4r(R)j=1 (|bj |2/!j).

Laguerre Series Expansion for the cdf of $H$2F

As for the power series expansion, the Laguerre seriesexpansion approach in [33] can be adapted to the complex-valued case with appropriate modifications. For the sake ofconciseness, we state the result without derivation

F (y) =

)

D*1 +r(R)+15

i=0

0y+$2&

1r(R)+1+i

(r(R) + 1 + i)!e+

y#$2%

,

E-u(y + ')

+

J !5

k=1

"ck(k + 1)!

(r(R) + k + 1)!

.y + '

2&

/r(R)

# e+y#$2% L(r(R))

k+1

.y + '

2&

//u(y + ') (25)

where & is an arbitrary positive constant whose role will bemade clear shortly and L(')

k (x) (- > +1, k = 0, 1, . . .) de-notes the generalized Laguerre polynomial given by

L(')k (x) =

k5

r=0

J(+x)r

r!(k + r)!

kB

n=r+1

(- + n)

K

and the coefficients "ck(k ' 1) can be calculated recursivelythrough the relation

"ck =1k

k+15

r=0

"dk+r"cr

where "c0 = 1 and

"dk = + k

2&

r(R)5

j=1

|bj |2

!j

.1 + !j

2&

/k+1

+r(R)5

j=1

.1 + !j

2&

/k

, k ' 1.

3We note that dk as specified in [33, eq. 4.2b.10] for the real-valued case isincorrect as the factor (!1)k is missing.



The convergence properties of the series expansion in (25)depend strongly on the choice of &. We observed that choosing& to satisfy 1 * & * 4 typically results in fast convergence.

Standard Deviation of $H$2F

We start by noting that $H$2F = hHh + "hH"h + 2%(hH"h).

Using 2%(hH"h) ( N (0, 2hHRh), it is easy to show that therandom variables "hH"h and 2%(hH"h) are uncorrelated. Con-sequently, !2 = var($H$2

F ) = var("hH"h) + var(2%(hH"h)).Noting that "hH"h = "pT"p and var("pT"p) = 2Tr(R2

p) =Tr(!2),we finally obtain

! =!

Tr(!2) + 2hHRh.

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewersand the associate editor for their useful comments on the paper.

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Rohit U. Nabar (S’02–M’03) was born in Bombay,India, on December 18, 1976. He received the B.S.degree (summa cum laude) in electrical engineeringfrom Cornell University, Ithaca, NY, in 1998, andthe M.S. and Ph.D. degrees in electrical engineeringfrom Stanford University, Stanford, CA, in 2000and 2003 respectively. His doctoral research focusedon signaling for real-world multiple-input multiple-output (MIMO) channels.

From 2002 to 2004, he was a PostdoctoralResearcher in the Communication Theory Group,

Swiss Federal Institute of Technology (ETH), Zurich, Switzerland. Between2004–2005, he was a Faculty Member at the Communications and SignalProcessing Research Group, Department of Electrical and Electronic Engineer-ing, Imperial College, London, U.K. Since July 2005, he has been a member ofthe DSP Technology Group at Marvell Semiconductor, Inc., Sunnyvale, CA.His research interests include signal processing and information theory forwireless communications and networks. He has coauthored (with A. Paulrajand D. Gore) the textbook Introduction to Space–Time Wireless Communica-tions (Cambridge, U.K.: Cambridge Univ. Press, 2003).

Dr. Nabar was a Member of the Smart Antennas Research Group at Stanfordand a recipient of the Dr. T. J. Rodgers Stanford graduate fellowship. Hecurrently serves as an Associate Editor for the IEEE COMMUNICATIONSLETTERS.

Helmut Bölcskei (M’98–SM’02) was born inAustria on May 29, 1970. He received the Dr.techn.degree in electrical engineering from the ViennaUniversity of Technology, Vienna, Austria, in 1997.

In 1998, he was a Postdoctoral Researcher atVienna University of Technology. From 1999 to2001, he was a postdoctoral researcher in the Infor-mation Systems Laboratory, Department of Electri-cal Engineering, Stanford University, Stanford, CA.During that period, he was also a consultant forIospan Wireless, Inc., Palo Alto, CA. From 2001 to

2002, he was an Assistant Professor of Electrical Engineering at the Universityof Illinois at Urbana-Champaign. Since 2002, he has been an Assistant Pro-fessor of Communication Theory at the Swiss Federal Institute of Technology(ETH) Zürich, Zürich, Switzerland. He was a Visiting Researcher at PhilipsResearch Laboratories Eindhoven, The Netherlands, ENST Paris, France, andthe Heinrich Hertz Institute Berlin, Germany. His research interests includecommunication and information theory with special emphasis on wirelesscommunications and signal processing.

Dr. Bölcskei received a 2001 IEEE Signal Processing Society Young AuthorBest Paper Award and was an Erwin Schrödinger Fellow (1999–2001) of theAustrian National Science Foundation (FWF). He was an Associate Editorfor the IEEE TRANSACTIONS ON SIGNAL PROCESSING and the EURASIPJournal on Applied Signal Processing, and is currently an Associate Editor forthe IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS and serves onthe editorial board of Foundations and Trends in Networking.

Arogyaswami J. Paulraj (SM’85–F’91) receivedthe Ph.D. degree from the Naval Engineering Collegeand the Indian Institute of Technology, New Delhi,India.

He is a Professor at the Department of Electri-cal Engineering, Stanford University, Stanford, CA,where he supervises the Smart Antennas ResearchGroup, working on applications of space–time tech-niques for wireless communications. His nonacad-emic positions included Head, Sonar Division, NavalOceanographic Laboratory, Cochin, India; Director,

Center for Artificial Intelligence and Robotics, Bangalore, India; Director,Center for Development of Advanced Computing, Pune, India; Chief Scien-tist, Bharat Electronics, Bangalore, India; Chief Technical Officer and Founder,Iospan Wireless, Inc., Palo Alto, CA. His research has spanned several disci-plines, emphasizing estimation theory, sensor signal processing, parallel com-puter architectures/algorithms and space–time wireless communications. Hisengineering experience has included development of sonar systems, massivelyparallel computers, and broadband wireless systems. He is the author of over300 research publications and holds 8 patents.

Dr. Paulraj has won several awards for his engineering and research contribu-tions, including the IEEE Signal Processing Society’s Technical AchievementAward. He is a member of the Indian National Academy of Engineering.


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