finite-snr diversity-multiplexing tradeoff for correlated rayleigh and rician mimo channels

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 52, NO. 9, SEPTEMBER 2006 3965

Finite-SNR Diversity–Multiplexing Tradeoff forCorrelated Rayleigh and Rician MIMO Channels

Ravi Narasimhan, Senior Member, IEEE

Abstract—A nonasymptotic framework is presented to analyzethe diversity–multiplexing tradeoff of a multiple-input–mul-tiple-output (MIMO) wireless system at finite signal-to-noiseratios (SNRs). The target data rate at each SNR is proportional tothe capacity of an additive white Gaussian noise (AWGN) channelwith an array gain. The proportionality constant, which can beinterpreted as a finite-SNR spatial multiplexing gain, dictates thesensitivity of the rate adaptation policy to SNR. The diversity gainas a function of SNR for a fixed multiplexing gain is defined bythe negative slope of the outage probability versus SNR curve on alog-log scale. The finite-SNR diversity gain provides an estimate ofthe additional power required to decrease the outage probabilityby a target amount. For general MIMO systems, lower bounds onthe outage probabilities in correlated Rayleigh fading and Ricianfading are used to estimate the diversity gain as a function of multi-plexing gain and SNR. In addition, exact diversity gain expressionsare determined for orthogonal space–time block codes (OSTBC).Spatial correlation significantly lowers the achievable diversitygain at finite SNR when compared to high-SNR asymptotic values.The presence of line-of-sight (LOS) components in Rician fadingyields diversity gains higher than high-SNR asymptotic values atsome SNRs and multiplexing gains while resulting in diversitygains near zero for multiplexing gains larger than unity. Further-more, as the multiplexing gain approaches zero, the normalizedlimiting diversity gain, which can be interpreted in terms of thewideband slope and the high-SNR slope of spectral efficiency,exhibits slow convergence with SNR to the high-SNR asymptoticvalue. This finite-SNR framework for the diversity–multiplexingtradeoff is useful in MIMO system design for realistic SNRs andpropagation environments.

Index Terms—Finite signal-to-noise ratio (SNR), multiple an-tennas, outage probability, random matrices.

I. INTRODUCTION

THE benefits of multiple antennas at both the transmitterand the receiver in a wireless link consist of increased

reliability as well as high data rates. Techniques such asspace–time coding [1] are primarily concerned with increasingreliability through spatial diversity, while spatial multiplexing[2], [3] achieves high data rates by using the spatial degrees offreedom to transmit independent data streams. For a particulartransmission scheme, the conventional definition of diversitygain refers to the slope as the signal-to-noise ratio (SNR) tendsto infinity of the error rate versus SNR curve for the ideal

Manuscript received September 19, 2005; revised March 1, 2006. The mate-rial in this paper was presented in part at the IEEE Globecom, St. Louis, MO,November 2005.

The author is with the Department of Electrical Engineering, University ofCalifornia, Santa Cruz, CA 95064 USA (e-mail: [email protected]).

Communicated by M. Médard, Associate Editor for Communications.Digital Object Identifier 10.1109/TIT.2006.880057

channel conditions of independent and identically distributed(i.i.d.) Rayleigh fading. With this asymptotic definition ofdiversity, full-rank space–time codes ensure maximal spatialdiversity [1]. In practice, however, the SNR is often sufficientlylow such that the effective diversity, measured by the slope ofthe error rate versus SNR at a particular point, is significantlyless than the slope at high SNR. This behavior is further exac-erbated for multiple-input–multiple-output (MIMO) channelswith a high-spatial correlation or line-of-sight (LOS) compo-nent, propagation conditions that are frequently encountered inpractice.

In addition to exploiting the available spatial degrees offreedom, MIMO wireless systems typically employ rate adap-tation algorithms based on the average received SNR. Thelevel of rate adaptation, related to the multiplexing gain, mustbe selected for optimal performance at a given SNR. In thiscontext, the following question is of interest: how does the errorrate change for a change in the received SNR, given a certainrate adaptation strategy?

In order to address this question, it is necessary to charac-terize the behavior of the error probability as a function of therate adaptation parameters. Fundamentally, this characteriza-tion involves the tradeoff between diversity and multiplexingin MIMO systems. Such a tradeoff was obtained in [4], whichprovides a framework to bridge the two extremes of purediversity gain, related to link reliability, and pure multiplexinggain, related to spectral efficiency. The diversity–multiplexingtradeoff curve of a given space–time code enables one todetermine the set of diversity and multiplexing gains that canbe obtained simultaneously. This tradeoff was obtained as theSNR approaches infinity for i.i.d. Rayleigh-fading channels.The asymptotic characterization is typically valid for high datarates and correspondingly low error rates.

Systems such as wireless local area networks (WLANs) havemoderate target packet error rates (PERs) around –[5]. Furthermore, the SNRs encountered are usually in the range3–20 dB. In addition, the wireless channel often exhibits highspatial correlation and may contain LOS components. Severalrecent papers [6]–[9] have noted that codes designed for max-imum performance at high SNR in i.i.d. Rayleigh-fading chan-nels are not optimal at low to medium SNR, especially in thepresence of correlated fading. In such situations, asymptotictheory for high SNR and i.i.d. Rayleigh channels yields op-timistic results. Thus, it is essential to characterize the diver-sity–multiplexing tradeoff for realistic SNRs, data rates, PERs,and nonideal fading channels.

This paper presents a nonasymptotic, finite-SNR analysisof the diversity–multiplexing tradeoff in MIMO systems for

0018-9448/$20.00 © 2006 IEEE

3966 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 52, NO. 9, SEPTEMBER 2006

realistic propagation conditions. The two cases of spatially cor-related Rayleigh fading at either the transmitter or receiver andRician fading are considered. Furthermore, no channel-stateinformation (CSI) is assumed at the transmitter. The finite-SNRframework, necessary to evaluate the impact of these nonidealpropagation conditions, is derived for quasistatic channels, suchas those encountered in WLANs. In quasi-static fading, thechannel is constant over one coding block, or packet, and variesindependently across packets. Codes that approach capacity,such as turbo codes [10] or low-density parity-check (LDPC)codes [11], [12], are typically used on each packet such that thePER is well approximated by the channel outage probability.The outage probability is the probability that the mutual in-formation, conditioned on the fading realization, between thetransmitted and received signals is less than the target spectralefficiency. Based on the outage probability of MIMO systems,new definitions of diversity and multiplexing gains as functionsof SNR are presented. The multiplexing gain is the ratio of thespectral efficiency at each SNR to the capacity of an additivewhite Gaussian noise (AWGN) channel with an array gain. Fora constant multiplexing gain, the diversity gain is the negativeslope of the outage probability versus SNR curve on a log-logscale.

Although there exist closed-form expressions for the momentgenerating and characteristic functions of the MIMO mutual in-formation [13], [14], closed-form solutions for the cumulativedistribution function (cdf), needed for the outage probability,are not tractable. Exact numerical computation of the cdf, whichinvolves determinants of matrices of hypergeometric functions,provides little insight into the finite-SNR diversity–multiplexingtradeoff. Thus, a lower bound on the MIMO outage probabilityis derived to obtain an estimate of the diversity gain as a func-tion of the multiplexing gain and SNR. This estimate is conser-vative in that the exact diversity gain is typically lower. The spe-cial case of orthogonal space–time block codes (OSTBC) is alsoconsidered since closed-form expressions for the outage prob-ability are available. For spatially correlated Rayleigh fading,the achievable diversity gains at realistic SNRs are significantlylower than the high-SNR asymptotic values. For Rician fading,the maximum diversity gain is not always achieved at zero mul-tiplexing gain. In fact, the diversity gain at some SNRs and mul-tiplexing gains in Rician fading is larger than the asymptoticvalue for i.i.d. Rayleigh fading. These qualitative characteristicsdemonstrate the usefulness of the finite-SNR analysis describedin this paper.

The remainder of the paper is organized as follows. Section IIintroduces the system model and the definitions of diversity andmultiplexing gains as functions of SNR. Lower bounds on theoutage probability are derived in Section III for the cases ofspatially correlated Rayleigh fading and Rician fading. Fromthese bounds, the diversity gain is estimated as a function ofthe multiplexing gain and SNR. Exact diversity gain expressionsare computed for the case of OSTBC in Section IV. Section Vpresents outage probability and diversity gain curves at finiteSNRs for various system and propagation conditions. Compar-isons are also provided with high-SNR asymptotic results in theliterature. A discussion of the finite-SNR results and interpre-tations of the limiting diversity gain as the multiplexing gain

approaches zero are given in Section VI. Conclusions are givenin Section VII.

II. SYSTEM MODEL AND DEFINITIONS

Consider a narrowband MIMO system with transmitand receive antennas. Let denote the channelmatrix. For the case of correlated Rayleigh fading, correlation atthe transmitter is assumed in this paper, although the results areeasily applied to the case of receive antenna correlation. Withtransmit antenna correlation, the channel matrix can be writtenas , where denotes the transmitcovariance matrix and is a matrix whose ele-ments are i.i.d. complex Gaussian random variables with zeromean and unit variance [15]. For the case of Rician fading, thechannel matrix can be written as , where

and denote the line-of-sight (LOS) and non-line-of-sight (NLOS) matrices, respectively. Frequently usedmodels for these matrices areand , where denotes the Ri-cian K-factor, denotes the matrix of allones, and no spatial correlation is assumed [15]. Note that therank-deficient LOS matrix arises in the usual case where theantenna separations at the transmitter and receiver arrays aremuch smaller than the range, or distance between transmitterand receiver. For simplicity, the phases of the LOS componentare chosen such that the transmitter and receiver are in thebroadside orientation. Diagonal phase matrices are needed toaccount for general array orientations. It can be shown thatan arbitrary diagonal receive phase matrix does not affect thediversity–multiplexing tradeoff. Furthermore, if the transmitarray orientation is known, a diagonal transmit phase matrixcan be compensated for by adjusting the transmit antennaphases. Let and denote the transmitted and received vec-tors, respectively. The MIMO system equation is given by

(1)

where is a complex AWGN vector with covariance ,and denotes the identity matrix.

Under the assumption of quasistatic fading with capacity-achieving codes per packet, the PER is equal to the outage prob-ability. With this model, the conventional asymptotic definitionsof multiplexing and diversity gains of a MIMO system are givenas follows [4]:

(2)

(3)

where and are the asymptotic mul-tiplexing and diversity gains, respectively, is the spectralefficiency (in bits per second/Hertz) of the MIMO system, isthe average SNR per receive antenna, and is the outageprobability. These definitions capture the high-SNR tradeoff ofdata rate (represented by the multiplexing gain) and reliability(represented by the diversity gain).

The need to characterize the diversity–multiplexing tradeoffat realistic SNRs, data rates, and PERs motivates the new fi-

NARASIMHAN: FINITE-SNR DIVERSITY–MULTIPLEXING TRADEOFF 3967

Fig. 1. Illustration of diversity gain at finite SNR.

nite-SNR definitions introduced in this paper. The multiplexinggain is defined as the ratio of to the capacity of an AWGNchannel at SNR with array gain :

(4)

The array gain is chosen by considering the MIMO mu-tual information at low SNR. In the limit

, where is the Frobeniusnorm of the channel matrix [15]. Thus, for a fair comparisonof diversity and outage performance across different and

at low to medium SNRs, the array gain is chosen suchthat .

Note that for a constant multiplexing gain , the spectral ef-ficiency increases as the SNR increases. The multiplexing gain

provides an indication of the sensitivity of a rate adaptationstrategy as the SNR changes. As the value of increases, a moresensitive rate adaptation strategy is used in which a moderatechange in SNR can result in a significant change in the data rate.

As illustrated in Fig. 1, the diversity gain of a rate-adaptive system with a fixed multiplexing gain at SNR isdefined by the negative slope of the log-log plot of outage prob-ability versus SNR

(5)

where , the outage probability of the MIMO link as afunction of the multiplexing gain and SNR, is given by

(6)

The significance of Definition (5) is that the diversity gain ata particular SNR can be used to estimate the additional SNRrequired to decrease the PER by a specified amount for a givenmultiplexing gain. The finite-SNR diversity gain (5) is wellsuited to a perturbation analysis of the MIMO system around aspecified SNR. For large deviations, the behavior of the outageprobability itself can be analyzed using (6).

III. OUTAGE PROBABILITY AND DIVERSITY GAIN

In this section, lower bounds on the outage probability arecomputed for rate-adaptive MIMO systems under spatially cor-related Rayleigh fading and Rician fading. The lower bounds

are then used to estimate the finite-SNR diversity gain using (5).The asymptotic diversity gain at high SNR is also computed.

The lower bounds on the outage probability are computedunder the assumption of no CSI at the transmitter and perfectCSI at the receiver. In this case, equal power is allocated to eachof the transmit antennas. Thus, the MIMO mutual informa-tion conditioned on the channel realization is given by [16]

(7)

where the superscript denotes conjugate transposition. Forsimplicity, it is assumed that in this paper, althoughthe results are easily generalized to arbitrary and .

A. Lower Bound on Outage Probability: Correlated RayleighFading

A lower bound on the outage probability is now derived forRayleigh fading with a possibly rank-deficient transmit covari-ance matrix . A rank-deficient transmit covariance matrixcan arise with pinhole channels in which the transmitted sig-nals have to reach the receiver through a small number of pin-holes [17]. For a rank-one covariance matrix, there is only onepath to link the scattering clusters at the transmitter and receiver.This situation can occur, for example, when signals have to gothrough windows in buildings. Other conditions that give riseto rank-one covariance matrices include zero angle spread ornonzero angle spread with all paths fading coherently [18].

Let denote the nonzeroeigenvalues of , where the rank of is . Fur-thermore, let a complex Wishart matrix with de-grees of freedom and covariance matrix be denoted by

. Note that if , whereis a matrix of zero-mean independent com-

plex Gaussian row vectors with covariance matrix [19], [20].With these definitions, the mutual information has the followingequivalent statistical representation.

Lemma 1: The mutual information for correlatedRayleigh fading is equivalent in distribution to the following:

(8)

where denotes “equivalent in distribution,”are the sorted eigenvalues of

the complex Wishart matrix , andis a diagonal matrix.

Proof: The proof is given in Appendix I.The outage probability of the rate-adaptive MIMO system can

be expressed using (6) and (8) as follows:

(9)

where . To derive a lower bound on , thefollowing lemma is used.


Lemma 2: For Rayleigh fading with transmit antenna corre-lation

(10)

where and are independentrandom variables. The variables can be written as

(11)

where are independent gamma random variables with pa-rameters and

(12)

(13)

Proof: The proof is given in Appendix II.Note that from [21], [22], the cdf of can be written as a singlegamma series

(14)

where is the incomplete gammafunction and

(15)

For the case of i.i.d. Rayleigh fading, . Thus, from(11)–(13), becomes a gamma random variable with parame-ters and has the cdf

(16)

A lower bound on the outage probability is given in the fol-lowing theorem.

Theorem 1: For Rayleigh fading,

(17)

where, and is given by (14) for the case of transmit

antenna correlation and by (16) for the case of no spatial corre-lation.

Proof: From (9) and the fact that, the lower bound on is obtained as follows:

(18)

(19)

(20)

(21)

(22)

Inequality (19) follows from the fact that for any positive setof numbers , we have

. Next, (20) follows fromLemma 2, and (21) is derived using the observation that

(23)

Finally, (22) follows from the independence of.

To obtain accurate diversity gains at finite SNR, the lowerbound (17) is maximized over the exponents for eachand . The feasible set for is determined by the factthat . This constraint leads to the requirement

. From the conditions andis given by

(24)

There exist efficient nonlinear programming techniques, suchas variations of Newton’s method [23] or sequential quadraticprogramming [24], to determine the vector


that maximizes the right hand side of (17). The initial con-dition for the optimization is chosen as follows:

(25)

The computation time for the optimization is much smaller thanMonte Carlo simulations for the exact outage probability. Fur-thermore, even without the maximization step, the analyticalform of the lower bound (17) provides useful insight that cannotbe obtained from simulations.

B. Lower Bound on Outage Probability: Rician Fading

In this section, a lower bound on the outage probability forRician fading is derived. The derivation is based on results fornoncentral Wishart matrices. In order to use these results, theRician channel matrix must modified by applying fixed unitarytransformations at the transmitter. Specifically, let the vector ofinformation symbols to be transmitted be denoted by . Letdenote the -point discrete Fourier transform (DFT) matrixwhose elements are

. Let be the permutation matrixgiven by

. . ....

(26)

With these definitions, the transmitted vector is given by

(27)

Under these unitary transformations, the equivalent channelmatrix is given by

(28)

(29)

(30)

where denotes the matrix of allzeros and is an i.i.d. complex Gaussian matrix equivalent indistribution to .

Let a complex noncentral Wishart matrix withdegrees of freedom , covariance matrix , and matrixof noncentrality parameters be denoted by

. Note that if

, where is a matrix of independent complexGaussian row vectors with covariance matrix and

. As in the case of correlated Rayleigh fading, it can be shownthat the mutual information in Rician fading is equivalentin distribution to the following:

(31)

where are the sortedeigenvalues of the complex noncentral Wishart matrix

.Similar to Lemma 2, the following lemma is introduced.Lemma 3: For Rician fading

(32)

where and are independentrandom variables. For are gammavariables with parameters ,and is a noncentral gamma variable with shape parameter

, noncentrality parameter , and scaleparameter . The cdf of is given by (33) shown atthe bottom of the page.

Proof: The proof is given in Appendix III.A lower bound on the outage probability is

derived using Lemma 3, as given in the following theorem. Theproof is similar to the Proof of Theorem 1 and, hence, is omitted.

Theorem 2: For Rician fading

(34)

where, and is given by (33).

As in Section III-A, the lower bound (34) is maximized usingnonlinear programming over for each and . The feasibleset for is similar to that for in Section III-A

(35)

C. Finite-SNR Diversity Gain

The lower bounds on outage probability given in (17) and(34) can be used to estimate the diversity gain as a functionof SNR and multiplexing gain. The result is summarized in thefollowing theorem.

(33)


Theorem 3: For SNR and multiplexing gain , an estimateof the diversity gain is given by (36) shown at the bottom of thepage, where and

are chosen to maximize the lower bounds (17) and (34),respectively, and .

Proof: The proof follows from substituting the lowerbounds in (17) and (34) into (5).

From the diversity gain (36), the high-SNR asymptotic be-havior can be determined, as given in the following.

Corollary 1: The high-SNR diversity gain for multiplexinggain is

correlated Rayleigh fading

Rician fading(37)

where

, and.

Proof: The proof is given in Appendix IV.From the comparison of (37) with the results in [4], the

high-SNR diversity gain is equal to the asymptotic diversitygain defined by (3) for both Rayleigh fading with afull-rank transmit covariance matrix and Ricianfading. Hence, the finite-SNR framework for diversity analysisintroduced in this paper can be viewed as a generalization ofprevious asymptotic analysis in the literature. Furthermore,since spatial correlation and LOS components do not affectthe high-SNR diversity gain, the finite-SNR analysis presentedhere is essential to understanding MIMO system performancein realistic propagation conditions.

IV. OUTAGE AND FINITE-SNR DIVERSITY PERFORMANCE:OSTBC

In this section, closed-form expressions for the outage prob-abilities of OSTBC are computed for both correlated Rayleighfading and Rician fading. These expressions are used to com-pute the exact finite-SNR diversity gains of OSTBC. The mu-tual information (conditioned on the channel realization) forOSTBC with spatial code rate is given by [15]

(38)

The spatial code rate is equal to the average number of in-dependent constellation symbols transmitted per channel usethrough the transmit antennas. For example, forthe Alamouti space–time code [25], and or

for the complex orthogonal designs given in [26]. Sincefor correlated Rayleigh fading and

for Rician fading, the following lemma is needed todetermine the outage probability of OSTBC.

Lemma 4: The cdfs of are given by

correlated Rayleigh fading (39)

Rician fading (40)

where

(41)

Proof: The proof is given in Appendix V.The outage probability of OSTBC can now be stated as follows.

Theorem 4: For OSTBC, the outage probability is given by

correlated Rayleigh fadingRician fading

(42)

where .Proof: The proof follows from the mutual information ex-

pression for OSTBC given by (38). For instance, the outageprobability for correlated Rayleigh fading is given by

(43)

(44)

(45)

The diversity gain is then obtained by substituting (45) into (5),and as summarized in the following.

Theorem 5: The diversity gain for OSTBC as a function ofSNR and multiplexing is given by


Rician fading(36)


for (49)

for (50)

for (51)


Rician fading

(46)

The high-SNR diversity gain of OSTBC can be determined from(46), as given in Corollary 2. The proof follows reasoning sim-ilar to that of Corollary 1.

Corollary 2: The high-SNR diversity gain of OSTBC formultiplexing gain is

(47)

where

correlated Rayleigh fadingRician fading

(48)

As before for general MIMO systems, the high-SNR asymp-totic diversity gain for OSTBC does not depend on spatial cor-relation or LOS components. Hence, it is necessary to considerfinite SNRs for the diversity–multiplexing tradeoff in order todetermine the effect of realistic propagation conditions.

V. NUMERICAL RESULTS

Numerical results are now presented for the finite-SNRoutage and diversity analysis given in Sections III and IV.Uniform linear arrays with half-wavelength antenna spacingare assumed at the transmitter and receiver. For correlatedRayleigh fading, a single transmit scattering cluster is assumedwith angle of departure of 30 from broadside and a uniformpower azimuth spectrum with full-width angle spread of 15 .With these parameters, the transmit covariance matrices for

are given in (49)–(51) at the top of the page.For Rician fading, K-factors of 5 and 10 dB are used. Theseparameters are similar to those used in the IEEE 802.11 WLANTask Group N [27].

Fig. 2 is a plot of the outage probability versus SNR for cor-related Rayleigh fading with . The lower

bound (17) is plotted as well as results of Monte Carlo simula-tions for the exact outage probability. Although there is a gapbetween the actual outage probability and the lower bound, itcan be seen that the slope of the lower bound is very similar tothat of the actual outage probability curve, in particular at lowSNRs. In fact, as shown in Fig. 3, the diversity–multiplexingtradeoff curves obtained from (36) are very close to the sim-ulated curves at SNRs of 5 and 10 dB. Note that (36) is typi-cally larger than the actual diversity gain since is derivedusing the lower bound (17) on outage probability. Nevertheless,the estimate agrees with the simulation value as .From Fig. 3, transmit correlation causes much lower diversitygain at realistic SNRs than the asymptotic result obtained in [4]for SNR .

Finite-SNR diversity–multiplexing tradeoff curves based ongiven in (36) are plotted in Fig. 4 for

and several different fading scenarios with SNR dB. Asexpected, the tradeoff curve for correlated Rayleigh fading isgreatly below the curve for i.i.d. Rayleigh fading. A qualitativelydifferent behavior is observed for Rician fading. At moderateSNRs and a fixed multiplexing gain satisfying

for some , the presence of a LOS component givesrise to a steep drop in outage probability, as seen in Fig. 5. As theSNR increases, the minimum eigenvalue of begins to af-fect the outage probability and causes a flatter outage probabilitycurve (lower diversity gain). Hence, for , thelog-log plot of outage probability versus SNR (Fig. 5) containsan inflection point, and the diversity gain at moderate SNRs islarger than the high-SNR asymptote. For , the im-pact of the LOS component on the outage probability slope isless significant since the sensitivity of the rate adaptation algo-rithm is quite low. Thus, diversity gain is less than the high-SNRasymptote for . From the numerical results, a typicalvalue of is around for dB and aroundfor dB . Also note thatall the curves for SNR dB converge at . Finally,for , the diversity gain in Rician fading approaches zerorapidly since the rank-one LOS matrix limits the effective de-grees of freedom in the channel. Note that as , theRician fading channel approaches a rank-one AWGN channelsuch that the outage probability is unity for and zero for

.From the diversity estimates given in (36) and the exact di-

versity gain for OSTBC given in (46), three-way tradeoff sur-faces of diversity gain, multiplexing gain, and SNR are plotted


Fig. 2. Outage probability for different multiplexing gains r; M = M =

g = 2, and correlated Rayleigh fading. The solid curves are exact simulationresults and the dashed curves are the lower bounds obtained using (17).

Fig. 3. Diversity gain versus multiplexing gain for M = M = g = 2 andcorrelated Rayleigh fading. The actual diversity gain from Monte Carlo simu-lations is plotted for SNRs of 5 and 10 dB.

in Figs. 6–8 for 1) with correlatedRayleigh fading, 2) with Rician fading( dB), and 3) with the Alam-outi space–time code and Rician fading ( dB). FromFig. 6, spatial correlation significantly reduces the achievablediversity gain and causes slow convergence with SNR of the di-versity–multiplexing tradeoff curve to the high-SNR asymptoticresult. As was observed in Fig. 4, the increase in diversity gaindue to Rician fading at moderate SNRs can be seen in Figs. 7and 8.

VI. DISCUSSION

The finite-SNR framework for the diversity–multiplexingtradeoff provides useful insight into MIMO performanceunder realistic propagation conditions. As demonstrated byCorollaries 1 and 2, the effects of spatial correlation and LOScomponents on the diversity–multiplexing tradeoff cannot be

Fig. 4. Finite-SNR diversity–multiplexing tradeoff curves obtained using (36)for M =M = g = 2 and different fading scenarios with SNR = 10 dB.

Fig. 5. Outage probability lower bounds (34) for different multiplexing gainsr, M = M = g = 2, and Rician fading with K = 10 dB.

Fig. 6. Diversity gain estimate (36) versus multiplexing gain and SNR forM = M = g = 4 and correlated Rayleigh fading.


Fig. 7. Diversity gain estimate (36) versus multiplexing gain and SNR forM = M = g = 4 and Rician fading with K = 5 dB.

Fig. 8. Diversity gain (46) versus multiplexing gain and SNR for Alamouticode withM = M = g = 2 and Rician fading (K = 5 dB).

observed at high-SNR. In contrast, the results given in Sec-tion V for finite SNR quantify the effect of these propagationconditions on the diversity–multiplexing tradeoff. The reasonfor the difference can be explained intuitively as follows.As the SNR approaches infinity, only the number of channeleigenmodes determines the performance. The relative strengthof these eigenmodes do not affect high-SNR behavior. Sincespatial correlation and LOS components primarily affect thecondition number of the channel matrix, the impact of suchnonideal propagation effects can be captured only by analyzingthe performance at finite SNR. Furthermore, the finite-SNRdiversity–multiplexing tradeoff exhibits qualitatively differentbehavior for Rician channels, as seen in Figs. 4, 7, and 8.From (37) and (47), this behavior is not observed in the limitSNR since for finite Rician K-factors, the overall MIMOchannel matrix is still full rank with probability one, althoughthe condition number is increased.

Finite-SNR diversity–multiplexing tradeoff curves are usefulto develop a systematic approach for space–time code designfor various propagation conditions. In the literature, the im-pact of channel correlation is often studied for the extremesof high and low SNRs. As aforementioned, nonideal channelconditions do not affect high-SNR performance. In [28], thelow-SNR region is studied by assuming a certain fractionof the channel eigenvalues are sufficiently small such that theydo not contribute to the mutual information (8); the remainingeigenvalues are assumed sufficiently large such that the term“ ” in (8) can be ignored. Based on these assumptions, thepairwise error probability (PEP), diversity gain, and coding gainof a space–time code are approximated. Code design criteriaare proposed for correlated channels by taking into account fi-nite SNR. Because the fraction is difficult to determine, sim-ulations are used to evaluate the impact of nonideal propaga-tion conditions. In this context, the finite-SNR diversity–mul-tiplexing tradeoff represents a useful analysis tool to evaluatethe performance of codes designed according to new criteriasuch as those given in [28]. The finite-SNR diversity gain fora given multiplexing gain quantifies the strength of the eigen-modes at each SNR without resorting to an ad-hoc determina-tion of the fraction . Furthermore, the finite-SNR frameworkcan be used to evaluate the performance of codes that are ap-proximately universal (at high SNR) [29]. Such codes achievethe asymptotic (high-SNR) diversity–multiplexing tradeoff forevery slow fading channel. However, different codes will in gen-eral have different performance at finite SNR. Detailed perfor-mance analysis of space–time codes and new code design cri-teria based on the finite-SNR diversity–multiplexing tradeoff areinteresting topics for future research.

An important property of the finite-SNR diversity gain is thelimiting behavior as the multiplexing gain approaches zero. Thisregion is most important for diversity gain since at low mul-tiplexing gains, the rate adaptation strategy is not very sensi-tive to changes in SNR. In other words, the data rate does notchange dramatically with SNR. Such a conservative rate adapta-tion strategy is usually employed to benefit from increased reli-ability using spatial diversity. With this motivation, the diversitygain for is given in the following theorem.

Theorem 6: The diversity gain as of a rate-adaptiveMIMO system for SNR and array gain is

(52)

where is given in (48). The result also holds for the case ofOSTBC transmission.

Proof: The proof is given in Appendix VI.Fig. 9 is a plot of the normalized limiting diversity gain,

, as a function of for . Fromthe plot, the diversity gain for SNRs between 3–20 dB is onlyaround 50%–83% of the asymptotic value for practical numbersof antennas. This result indicates that the high-SNR diversitygain of is often quite optimistic and is close to the actualdiversity gain only for SNR greater than 40 dB. The slow con-vergence of the diversity gain indicates that sacrificing data rateby using a low multiplexing gain does not provide the expectedasymptotic diversity gain at realistic SNR. The achievable


Fig. 9. Normalized limiting diversity gain as a function of SNR.

diversity gain predicted by Theorem 6 and Fig. 9 is useful todetermine the actual drop in PER as the SNR increases for aconservative rate adaptation policy . Such insight canbe used to select the rate adaptation algorithm at a particularSNR.

Note that the diversity gain as depends neither on co-variance matrix eigenvalues nor on Rician K-factors, as seen inFig. 4. As , the spectral efficiency approaches zero. Asstated in [30], the infinite-bandwidth limit of zero spectral ef-ficiency is not affected by fading. In fact, the limiting diversitygain expression (52) has the following interesting interpretationin terms of the wideband slope [30] and the high-SNR slope [31]of spectral efficiency. Let denote the capacity(in bits per second/Hertz) of an AWGN channel with array gain

, and let denote the energy-per-bit normalized to theone-sided noise spectral level. From (52), the normalized lim-iting diversity gain can be written as follows:

C(53)

where and C is the AWGN channelcapacity as a function of . From (53), the normalizedlimiting diversity gain measures the ratio of the slope of ca-pacity versus SNR to the slope of capacity versus . Inthis context, it is insightful to consider the wideband slope and

the high-SNR slope of spectral efficiency, respectively, definedby [30], [31]

C(54)

C

(55)

In AWGN channels, and . Furthermore, thefollowing inequalities hold:

C(56)

At low SNR, the numerator of (53) approaches zero while thedenominator approaches , and, hence, the normalized lim-iting diversity gain approaches zero. At high SNR, both the nu-merator and denominator approach , and the normalized lim-iting diversity gain approaches unity.

Another interesting interpretation of the normalized limitingdiversity gain arises from considering the relation betweenmutual information and minimum mean-square error (MMSE)[32]. Consider a continuous-time Gaussian channel andGaussian input with power spectral density given by

otherwise(57)

where is the array gain defined in Section II. The mutual infor-mation rate for this channel (in nats/s) is then

. Let mmse and cmmse denote the noncausal MMSEand causal MMSE, respectively, in estimating the input giventhe channel output. It is shown in [32] that

mmse (58)

cmmse (59)

Note that (58) and (59) hold under more general conditions.Hence, the normalized limiting diversity gain can be rewrittenas follows:

mmsecmmse

(60)

Thus, the normalized limiting diversity gain can be interpretedas difference between the causal MMSE and the noncausalMMSE relative to the causal MMSE of a continuous-timeGaussian channel. An interesting direction for future researchis to explore in greater detail the relations among the finite-SNRdiversity–multiplexing tradeoff, spectral efficiency in the wide-band and high-SNR regimes, and the link between mutualinformation and MMSE estimation.


VII. CONCLUSION

Finite SNRs are important in analyzing the diversity–multi-plexing tradeoff of MIMO systems in realistic environments.This paper presents a nonasymptotic framework of this tradeoffby introducing finite-SNR definitions of diversity and multi-plexing gains. Based on these definitions, diversity gains arecomputed as functions of the multiplexing gain and SNR forcorrelated Rayleigh fading and Rician fading. The finite-SNRdiversity gain allows accurate estimation of the additionalpower required to decrease the error probability by a specifiedvalue when rate adaptation is employed in the MIMO system.For a general MIMO system, the diversity gain is estimatedusing lower bounds on the outage probability for a fixed multi-plexing gain. The diversity estimate is typically slightly abovethe exact simulation value. Exact diversity gain expressions areobtained for OSTBC transmission. Spatial correlation signifi-cantly lowers the achievable diversity gain at finite SNR. Thepresence of LOS components in Rician fading yields diversitygains higher than the high-SNR asymptotic values at someSNRs and multiplexing gains; for multiplexing gains largerthan unity, the diversity gains are near zero.

As the multiplexing gain approaches zero, the normalizedlimiting diversity gain as a function of SNR is the same for dif-ferent MIMO systems and propagation environments. The lim-iting diversity gain can be interpreted in terms of the widebandslope and the high-SNR slope of spectral efficiency. In addition,the normalized limiting diversity gain is equal to the differencebetween the causal MMSE and noncausal MMSE relative tothe causal MMSE in estimating the input of a continuous-timeGaussian channel given the channel output. The limiting diver-sity gain is much lower than the high-SNR asymptotic value,a fact that has implications for MIMO systems with conserva-tive rate adaptation strategies. The insights gained from the fi-nite-SNR diversity–multiplexing tradeoff analysis are useful todesign MIMO systems for realistic SNRs and propagation en-vironments.

APPENDIX IPROOF OF LEMMA 1

Let the eigenvalue decomposition of be, where is a unitary matrix,

, and de-

notes block diagonal concatenation. From (7), ,and the identity [33], the mutualinformation is given by

(61)

where (61) follows from the fact that sinceis unitary [19]. Now, define the following matrix partition:

, where is a i.i.d. com-plex Gaussian matrix. Thus,

(62)

where (62) follows by letting.

APPENDIX IIPROOF OF LEMMA 2

Consider , the th smallest eigenvalue of theWishart matrix and define the matrix

. By definition, is full rank (rank). For a Hermitian matrix , let

denote the eigenvalues sorted in ascending order. By Poincare’sseparation theorem [33]

(63)

From the relations and, (63) can be

rewritten as

(64)

From (64) with

(65)

since , the maximum eigenvalue of thematrix , is bounded above by the trace.

The matrix can be computed from the fol-lowing partitions of :

(66)

(67)


(70)

(71)

From (69)

(68)

(69)

where is the Schur complement of in . Sim-ilarly, define . From [19, Th.3.2.10], and

.From the inverse of partitioned matrices, one can show (70)and (71) given at the top of the page, where is avector and is a scalar. If

(72)

then

(73)

where is independent of [19]. By applying (73)to and to successively smaller upper left submatrices of

, it can be seen that

(74)

where the are independent. It remains to determine the dis-tribution of . Since and areWishart matrices, the Laplace transforms of the probability den-sity functions (pdfs) of and are given by

(75)

(76)

From (73) and the independence of and , the Laplacetransform of the pdf of is

(77)

By inspection of (77), can be written as the sum of indepen-dent gamma variables, as given in (11). From (65), (69), and(74),

(78)

and, hence,

(79)

where .

APPENDIX IIIPROOF OF LEMMA 3

The proof is divided into two parts. First, supposeand consider the following partitions of :

(80)

(81)

As in (69), let be the Schurcomplement of in . It can be deduced from Problem10.11 of [19] that and

are central Wishartmatrices. Proceeding as in Appendix II, one obtains

(82)

where the random variable is independent of .Thus,

(83)

where are independent. Since the Laplace transforms of thepdfs of and for are

(84)

(85)

the Laplace transform of the pdf of is

(86)


Thus, are gamma variables with pa-rameters .

Now, suppose . From the partitioned matrix givenin (81), the Schur complementhas the central Wishart distribution

. In this case

(87)

where is independent of [19]. From the prop-erties of noncentral Wishart matrices, it can be shown that theLaplace transform of the pdf of is given by

(88)

Thus, the Laplace transform of the pdf of is

(89)

From (89) and [34], is a noncentral gamma random vari-able with shape parameter , noncentrality param-eter , and scale parameter . The cdf ofis given by (33) with . Thus, similar to (78)

(90)

and, hence

(91)

where .

APPENDIX IVPROOF OF COROLLARY 1

First, consider the case of correlated Rayleigh fading. Since, we have as

. Furthermore, if as .Thus, it suffices to consider the case . In this case

(92)

In addition

(93)

since and as for. From (36), (92), and (93),

(94)

where . Note that

(95)

since if . Now, as , maxi-mizing the lower bound (17) on the outage probability is equiv-alent to minimizing (94) with respect to . From the fea-sible set for and (95), the feasible set for is

. Thus, the high-SNR diversity gain for correlatedRayleigh fading is

(96)

For Rician fading, sinceas . Furthermore, if

as . From (33), it can be shown in amanner similar to (92) and (93) that for ,

(97)


(98)

Thus, from (36), (97), and (98)

(99)

where . Similar to (95), . Max-imizing the lower bound (34) on the outage probability is equiv-alent to minimizing (99) with respect to as . Thefeasible set for is

. Thus, the high-SNRdiversity gain for Rician fading is

(100)

APPENDIX VPROOF OF LEMMA 4

For correlated Rayleigh fading, , whereis the Schur complement defined in (69) with . From(76), the Laplace transform of the pdf of is

(101)

Thus, from [21], [22], the cdf of can be written as

(102)

where the are defined in (41).For Rician fading, the Laplace transform of the pdf of

is given in (88). Thus, from [34], is anoncentral gamma random variable with shape parameter

, noncentrality parameter , and scale param-eter . The cdf of is given by (40).

APPENDIX VIPROOF OF THEOREM 6

Note that as and , for correlatedRayleigh fading and Rician fading, respectively. Since and

are functions of , the following expansion can be written:

(103)

(104)

where and are the coefficients of expansion. Similarly

(105)

(106)

Thus, for correlated Rayleigh fading

(107)

since as implies

using an analysis similar to (93). For Rician fading, the sameprocedure is used to give the result

(108)

Finally, for OSTBC, note that .Furthermore, one can show that

(109)

Thus

(110)

REFERENCES

[1] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time codes forhigh data rate wireless communication: Performance criterion and codeconstruction,” IEEE Trans. Inf. Theory, vol. 44, pp. 744–765, Mar.1998.

[2] A. Paulraj and T. Kailath, “Increasing Capacity in Wireless Broad-cast Systems Using Distributed Transmission/Directional Reception(DTDR),” U.S. Pat. 5 345 599, 1993.

[3] P. W. Wolniansky, G. J. Foschini, G. D. Golden, and R. A. Valenzuela,“V-BLAST: An architecture for realizing very high data rates over therich-scattering wireless channel,” in Proc. URSI ISSSE-98, Pisa, Italy,Sep. 1998, pp. 295–300.

[4] L. Zheng and D. N. C. Tse, “Diversity and multiplexing: A fundamentaltradeoff in multiple antenna channels,” IEEE Trans. Inf. Theory, vol.49, pp. 1073–1096, May 2003.

[5] IEEE 802.11: Wireless LAN Medium Access Control (MAC) and Phys-ical Layer (PHY) Specifications: High-Speed Physical Layer in the 5GHz Band, Supplement to IEEE 802.11 Standard, 802.11, Sep. 1999.

[6] H. El Gamal, “On the robustness of space-time coding,” IEEE Trans.Signal Process., vol. 50, pp. 2417–2428, Oct. 2002.


[7] S. Siwamogsatham and M. P. Fitz, “Robust space-time codes for cor-related Rayleigh fading channels,” IEEE Trans. Signal Process., vol.50, pp. 2408–2416, Oct. 2002.

[8] B. Clerckx, L. Vandendope, D. Vanhoenacker-Janvier, and A. J.Paulraj, “On the “high SNR” assumption in space-time codes de-signs,” in Proc. IEEE ICC 2004, Paris, France, Jun. 2004.

[9] M. Vajapeyam, J. Geng, and U. Mitra, “Low SNR design of space-timeblock codes based on union bound and indecomposable error patterns,”in Proc. IEEE ICC 2004, Paris, France, Jun. 2004.

[10] C. Berrou and A. Glavieux, “Near optimum error correcting codingand decoding: Turbo-codes,” IEEE Trans. Commun., vol. 44, no. 10,pp. 1261–1271, Oct. 1996.

[11] R. G. Gallager, “Low density parity check codes,” IRE Trans. Inf.Theory, vol. 8, pp. 21–28, Jan. 1962.

[12] D. J. C. MacKay, “Good error-correcting codes based on very sparsematrices,” IEEE Trans. Inf. Theory, vol. 45, no. 2, pp. 399–431, Mar.1999.

[13] Z. Wang and G. B. Giannakis, “Outage mutual information of space-time MIMO channels,” IEEE Trans. Inf. Theory, vol. 50, no. 4, pp.657–662, Apr. 2004.

[14] M. Chiani, M. Z. Win, and A. Zanella, “On the capacity of spatiallycorrelated MIMO Rayleigh-fading channels,” IEEE Trans. Inf. Theory,vol. 49, no. 10, pp. 2363–2371, Oct. 2003.

[15] A. Paulraj, R. Nabar, and D. Gore, Introduction to Space-Time WirelessCommunications. Cambridge, U.K.: Cambridge Univ. Press, 2003.

[16] E. Telatar, “Capacity of multi-antenna Gaussian channels,” EuropeanTrans. Telecomm. (ETT), vol. 10, no. 6, pp. 585–596, Nov. 1999.

[17] D. Gesbert, H. Bölcskei, D. A. Gore, and A. J. Paulraj, “Outdoor MIMOwireless channels: Models and performance prediction,” IEEE Trans.Commun., vol. 50, no. 12, pp. 1926–1934, Dec. 2002.

[18] A. Paulraj, private communication.[19] R. J. Muirhead, Aspects of Multivariate Statistical Theory. New

York: Wiley, 1982.[20] A. M. Tulino and S. Verdú, “Random matrix theory and wireless com-

munications,” Found. Trends Commun. Inf. Theory, vol. 1, no. 1, Jun.2004.

[21] P. G. Moschopoulos, “The distribution of the sum of independentgamma random variables,” Ann. Inst. Statist. Math. (Part A), vol. 37,pp. 541–544, 1985.

[22] M.-S. Alouini, A. Abdi, and M. Kaveh, “Sum of gamma variatesand performance of wireless communication systems over Nak-agami-fading channels,” IEEE Trans. Veh. Technol., vol. 50, no. 6, pp.1471–1480, Nov. 2001.

[23] T. F. Coleman and Y. Li, “On the convergence of reflective Newtonmethods for large-scale nonlinear minimization subject to bounds,”Math. Progr., vol. 67, no. 2, pp. 189–224, 1994.

[24] P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization.London, U.K.: Academic, 1981.

[25] S. Alamouti, “A simple transmit diversity technique for wireless com-munications,” IEEE J. Sel. Areas Commun., vol. 16, pp. 1451–1458,Oct. 1998.

[26] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time blockcodes from orthogonal designs,” IEEE Trans. Inf. Theory, vol. 45, pp.1456–1467, July 1999.

[27] TGn Channel Models, IEEE 802.11 standard contribution 802.11-03/940r4.

[28] B. Clerckx, “Space-Time Signaling for Real-World MIMO Channels,”Ph.D. Dissertation, Université catholique de Louvain, Louvain, France,Sep. 2005.

[29] S. Tavildar and P. Viswanath, “Approximately universal codes overslow fading channels,” IEEE Trans. Inf. Theory, to be published.

[30] S. Verdú, “Spectral efficiency in the wideband regime,” IEEE Trans.Inf. Theory, vol. 48, no. 6, pp. 1319–1343, Jun. 2002.

[31] S. Shamai and S. Verdú, “The impact of frequency-flat fading on thespectral efficiency of CDMA,” IEEE Trans. Inf. Theory, vol. 47, no. 4,pp. 1302–1327, May 2001.

[32] D. Guo, S. Shamai, and S. Verdú, “Mutual information and minimummean-square error in Gaussian channels,” IEEE Trans. Inf. Theory, vol.51, no. 4, pp. 1261–1282, Apr. 2005.

[33] H. Lutkepohl, Handbook of Matrices. Chichester, U.K.: Wiley, 1996.[34] G. Letac and H. Massam, A Tutorial on Noncentral Wishart Distri-

butions 2004 [Online]. Available: http://www.lsp.ups-tlse.fr/Fp/Letac/Wishartnoncentrales.pdf

finite-snr diversity-multiplexing tradeoff for correlated rayleigh and rician mimo channels

Documents