On the Impact of Spatial Correlation on the FiniteDiversity-Multiplexing Tradeoff

Zouheir Rezki and David Haccoun Francois Gagnon Wessam AjibEcole Polytechnique de Montreal, Ecole de Technologie Superieure, Universite du Quebec a Montreal

Email: {zouheir.rezki,david.haccoun} @polymtl.ca Email: francois.gagnon@ etsmtl.ca Email: ajib.wessam@ uqam.ca

Abstract- We present a tight lower bound on the outageprobability of a spatially correlated multielement antenna (MEA)channel. Using this lower bound, an accurate finite-SNR estimateof the diversity-multiplexing tradeoff over a spatially correlatedRayleigh fading channel is derived. This estimate allows gaininginsight on the impact of spatial correlation on the diversity-multiplexing tradeoff at finite SNR. As expected, the diversitymultiplexing tradeoff is severely degraded as the spatial corre-lation increases. For example, a MIMO system operating at atransmission rate of R = rlog2 (1 + g 71) bps/Hz, where r is themultiplexing gain, g is the array gain and r, is the SNR at eachreceive antenna, and an SNR of 5 dB in a moderately correlatedchannel, achieves a better diversity gain than a system operatingat an SNR of 10 dB in a highly correlated channel, when r > 0.8.Another interesting point is that the maximum diversity gain isnot affected by the correlation, provided that the spatial channelcorrelation matrix is of full rank.

I. INTRODUCTION

Current research in Multiple Input Multiple Output(MIMO) communication systems focuses on either obtainingmaximum diversity gain or maximum spatial multiplexinggain. Recently, Zheng and Tse established the link betweenthese two criteria and showed that both gains can be achievedwith an asymptotic (high signal-to-noise (SNR) regime)optimal tradeoff [1]. At low to moderate SNR (typically 3dB-20 dB), this asymptotic tradeoff is in fact an optimisticupper bound on the finite-SNR diversity-multiplexingtradeoff. In [2], a framework was introduced to characterizethe diversity performance of rate-adaptive MIMO systemsat finite SNR. As expected, the achievable diversity gains atrealistic SNRs are significantly lower than the asymptoticvalues.

The above papers assume independent and identically dis-tributed i.i.d. fading channels. However, in real propagationenvironments, the fades are correlated due, for example, toinsufficient spacing between antenna elements [3]. The spatialcorrelation is known to be detrimental for the channel capacityas well as for the error performance of MEA communicationsystems as reported in [3]-[5]. Many other papers focus oncharacterizing the effect of correlation on the mutual infor-mation for specific space-time codes, mainly the orthogonalor quasi-orthogonal design [6], [7] over Rayleigh or Riceanfading channels. In this paper, we aim at establishing thepotential of MEA systems at finite SNR and in more realisticfading environments. More precisely, we address the impact

of spatial correlation on the finite SNR diversity-multiplexingtradeoff.When a quasi-static channel model is adopted and when

capacity-achieving codes are used, the block-error probabilityis equal to the outage probability of the channel which will beformally defined in the sequel. Therefore, in order to analyzethe finite-SNR diversity-multiplexing tradeoff over a spatiallycorrelated channel, an analytical expression of the outageprobability is needed. For this purpose, we derive a tight lowerbound on the outage probability over a spatially correlatedchannel, which is used to obtain an accurate estimate of thefinite SNR diversity-multiplexing tradeoff. This estimate canbe used to gain insight on the effect of spatial correlation onMEA communication systems performance.

The rest of this paper is organized as follows. Section IIpresents the system model and related definitions. In sectionIII, we derive lower bounds on the outage probability ofspatially uncorrelated and correlated channels. Finite-SNRdiversity-multiplexing tradeoff estimates, for both scenarios,are given in section IV. Numerical results are reported inSection V. Section VI contains the conclusion.

II. CHANNEL MODEL AND RELATED DEFINITIONS

Let a MIMO system consists of Nt transmit antennas andNr receive antennas. We restrict ourselves to a Rayleigh flat-fading scenario, where the entries of the Nr x Nt channelmatrix H are possibly correlated, circularly-symmetric zeromean complex Gaussian distributed. It is assumed that thechannel is quasi-static, unknown at the transmitter and com-pletely tracked at the receiver side. The effect of spatial fadingcorrelation is captured by modelling the MIMO channel H as:

H =R11/2HWR /2 (1)

where the matrix HW represents the Nr x Nt spatially whitechannel and the matrices Rr and Rt are positive definiteHermitian matrices that specify the receive and transmitcorrelations respectively [8].

For a SNR-dependent transmission rate R(SNR) in bps/Hz,the multiplexing and diversity gains are respectively defined

1-4244-0398-7/06/$20.00 ©2006 IEEE

as [2]:

rRlog2(1 + g ) (2)

d ln Pout (r, rj)d -T &t- (3)

where g is the array gain which is equal to Nr, r is themean SNR at each receive antenna and P, t(r, rj) is theoutage probability of the MEA channel. As defined by (2) and(3), the multiplexing gain r provides an indication of a rateadaptation strategy as the SNR changes, while the diversitygain can be used to estimate the additional SNR required todecrease the outage probability by a specific amount for agiven multiplexing gain. The outage probability Potl is definedas:

Po,,t = Prob(I < R),

where I is the mutual channel information given by:

I = log2 (det (INr + N HHH)) bps/Hz,

where INr is the identity matrix of dimension Nr andsuperscript H stands for conjugate transposition.

III. LOWER BOUND ON THE OUTAGE PROBABILITY

Direct computation of the mutual information distributcan be performed given the distribution of the eigenvaluesHHH. This appears to be quite complicated for a genmspatial fading channel. Instead, one can bound (5) using fadstatistics. We focus on a upper bound on (5) to derive a loibound on Po,t. Since Rt and Rr have the same impact onmutual information [9], we focus, without loss of generalon the spatial correlation at the transmitter (Rl/2 = rThe elements of Rt will be denoted Pk,l (k, 1, ..., Nt)

Denote the QR decomposition of Hw = QR, where <an unitary matrix and R is an upper triangular matrix.square magnitudes IRi,i 2 of the diagonal entries of Rchi-square distributed with 2(Nr - + 1) degrees of freed(DF). The off-diagonal elements of R are i.i.d. Gausswith zero mean and unit variance. Let the singular vadecomposition (SVD) of R' 2 UDVH, where UV satisfy UHU = VHV = IN, and D is a diagcmatrix whose diagonal elements are the singular valuesRt2 Using the above SVD decomposition and the factthe matrix HWUD2UHHHH has the same distributionHwD2HH and RD2 RH, imply that:

d log2 (det (INr + N R 2Hw RtHwRJ 2))

d log2 (det (INr + N RD2RH))

t

< 1°102(1 + Nt 1),1=1

'This is due to the fact that unitary transformations do not changestatistics of the random matrices.

where t = min(Nt,Nr), A1 ZD_1D2Rl,2 is the IthR 2RHandhsmodiagonal entry of RD R and the symbol-means equality

in distribution. Equation (8) holds because det(A) < H1 Al,,for any nonnegative definite matrix A. In order to derivea lower bound on the outage probability, the distributionfunction of A1 is needed. When all D2 (k = 1,..., Nt) areequal, this is actually the uncorrelated case, the constrainttrace(R1/2) Nt imposes D 2 1 (k = 1,..., Nt). That is,A1 is chi-square distributed with 2(Nr + Nt-21 + 1) d.f. Notethat A1 may be viewed as a generalized quadratic form of aGaussian vector and if all D2s (k = 1, ..., Nt) are not equal,the computation of A's distribution is not straightforward[10]. Assuming, without loss of generality, that all D2s(k = 1,..., Nt) are distinct, the distribution function of A1,(l = 1,..., t) is given by the following lemma.2

(4)Lemma 1: Assuming that all D2s (k

distinct, the distribution function of Ai (l(5) by:

fA, (x)

= 1, ..., Nt) are1,...,t) is given

Nr-1+1 Nt-1I

,: a(l fG(k, D)(i) + E al fG(1,D7+k)(),k=l k=l

(9)where G(a, Q) is the Gamma random variable (RV) withdistribution function fG(O,a)(x) = -(a) exp(- ), x > 0,0> , 3 > 0. The coefficients a(l) and a(l+k) are given by:

a(')= (-D2 )-(N,-l+1-k)d(N,-1+1- k)

a1 [(1 ivD~±kyIf[(Iv)- vD, AlU'd(jV)(Nr-1+1_t)k jD ) + E\ (V) jv=D2

1(+) [1j OFA1@t (JV)]jv=D 1+k

where d(k ) stands for the kth derivative of f (x) anddx(k)TIA, (jv) is the characteristic function of Ai given by:

Nt-1I

'A1 (jjv) = (1-jvDz2)-(N,-1+1) 17 (1-ivD2 ))1 (10)k=1

Proof: The RV Ail is a weighted sum of chi-squaredistributed RVs. The DF of IRl is 2(Nr -I + 1) and theDF of Rl,k22 k :t l is 2. Computation of its characteristicfunction TA, (jv) leads to the expression given in lemma 1.Partial fraction decomposition of TFA, (jv) gives:

IA, (jv)Nr-1+1 Nt-1I

a()a()(l -jvD 2)-k+E a(l+k)_(1- vD2k)k=l k=l

(1 1)(7) Computing the inverse transform of each term leads to the

result in 9. This completes the proof of lemma 1. A

(8) Using the definition of P1t5, lemma 1 and the equation (8),a lower bound on Pot can be obtained. This is summarized

the 2Note that when all Dk (k = 1,..., Nt) are not distinct, the distribution

function of A1 (I = 1, ..., t) can be derived with the same mechanism.

1

in the following theorem. ~~~~lated spatial fading are respectively given by:

Theorem 1 (Lower Bound). Lower bounds on the outage duncorrr,r,pprobability for the uncorrelateId (D2 1,I k =1, ..., Nt) and

d (~T )

correlated spatial fading channels are respectively given by:

punicorr>f 7ii-i,rN 21 +1),1=1

rrt N,l±1+1pr>H(l S a(')Fi,,(Y k)1=1 k=li

+ S:a in (D2 J/I)k=11lk

dCorr (r,rTj,p):Till (I+gT)

Q1(=1xQiW

where Ql (~j) and Pi (~j) are given by:

where r Z= '1 bi, ~j = Vt((I + g91)bl 1) and

Fi, is the incomplete Gamma function defined as:

Fi, x a) = (a11!fX ta- tdt.

Proof:- (Sketch) We will give the proof for the correlated

case. The uncorrelated case follows along the same lines.

Using lemma 1, (8) and the definition of Po,t we have:

P"'t> Prob(H(1I+ TAl) < (I1+gl)'r)

> Prob (I1+ Al) < (l+ gTi)b,, I

tt7JProb(Al <i'1)

ajlFinic( k)1=1 k=1

Nt-l

+ a

k=1 1±k

In order to obtain tighter results, the lower bounds given in

Theorem are maximized over the set {b1, 1, t}

fur eachi miultiplexing gain r and SNR to obtain better

results. As was noted in [2], the computational time of the

this optimization problem is much smaller than Monte Carlo

simulations for the exact outage probability.

N, l1±1 )

ki=E a(k'V1 DQ

Nt-1l

+ 5:a c D1+kDlAjk,k=l

N, 1±1 ~

pi(w1 a(Q)Fin ( k)kiD

k=

(14)

(15)

Note that diuncorr(r,1,,p) and dcorr(r,r,,p) have similar

closed forms. The former can be obtained from the latter

by replacing P1 (i,), which is a weighted SUM Of Fin

functions, by the term Fic(~ Nr + Nt 21 + 1) and by

replacing Ql (ij), which is a weighted sum of correlation

related coefficients, by (1 (Nt + Nr

More interestingly, the maximum diversity dm, given

by dmax =1imro d(r,r,q, p), is unaffected by the spatial

correlation. This is summarized in the following corollary:

Corollary 2 (Maximum diversity). 'The maximum diversity

is the same for both correlated and uncorrelated spatial fading

scenarios and is given by:

IV. FINITE-SNR DIVERSITY AND CORRELATION

Using the lower bounds given in Theorem and the

definition given by (3), an estimate of the diversity

multiplexing tradeoff is derived. This is summarized in the

following corollary.

Corollary (Diversity estimate): An estimate of the diver-

sity for the uncorrelated (D2 1,I k= 1, ..., Nt) and corre-

dTmax =lim d(r,Tj, p)r--O

NtNr gr9i1(I+g, ~(16)

Proof: (see Appendix).

A similar remark concerning the maximum diversity was

pointed out in [8], but for a high SNR regime.

In establishing the last result, we assumed that Rt has full

rank. When the matrix Rt is rank-deficient, the channel matrix

H is also rank-deficient and we expect that the maximum

diversity gain is lower than that given by (16). We will not

further investigate this point in the present paper.

Ni S: ((I + gr,)bl- bigTj(1 + gT,)bl 1

1ill

Finc(~li,Nr +Nt -21 +1)

1)

(12)

1)bigTI(1 + gT,)b, 1

(13)

in the following theorem.

7

100

10*

=10

.0j

10

10 4 Uncorrelated, Narasimhan lower bound-- - Uncorrelated, our lower bound

Correlated, p=0.5, our lower bound10 -Uncorrelated, exact (simulation)

-ACorrelated, p=0.5, exact (simulation)--Correlated, p=0.9, our lower bound

-6 Correlated, p=0.9, exact (simulation)10n 1 NA r r a in

2.5

*dexact (simulation), SNR=15dB, uncorrelated- exact (simulation), SNR=1 5dB, p=0.5adexact (simulation), SNR=1 5dB, p=0.9

est-d , SNR=15dB, uncorrelatedest-d SNR=15dB, p=0.5est-d , SNR=15dB, p=0.9

._

a) 1.50)

0.5-

18 20 0 0.2 0.4 0.6 0.8 1 1.2 1.4

Multiplexing Gain

Fig. 1. Comparison of lower bounds on P,ou>t and the exact simulation forNt = N, = 2

V. NUMERICAL RESULTS

In this section, simulation results for an MEA system withNt = Nr = 2 are given. The transmit correlation matrix ischosen according to [8]:

Rt (=I

). (17)

A more general spatial correlation model can be found in [11].The lower bound given by Theorem is plotted in fig.l as wellas the exact Po,t, given by simulation, for r = 0.5, 1. Figure 1

also shows the lower bound found by Narasimhan in [2], whichconsiders the uncorrelated case only. As can be observed, our

bound, for the uncorrelated case, is tighter especially at lowSNR. More importantly, our framework can capture the effectof spatial correlation and as can be seen in fig. 1, our lowerbounds follow the same shape as the exact curves for boththe uncorrelated (p = 0) and the correlated (p = 0.5, 0.9)scenarios.

The exact diversity, obtained with Monte-Carlo simulationsusing (3) and an estimated diversity computed using corollary1 are plotted in fig.2 for an SNR of 15 dB. As can be noticed,the diversity multiplexing tradeoff estimate is a good fit tothe exact simulation tradeoff curve. Therefore, the estimateddiversity can be used to gain insight on the the diversity-multiplexing tradeoff over spatially correlated and uncorrelatedchannels while avoiding time consuming simulations. Interest-ingly, it can be noticed that with a correlation of p = 0.5,the diversity gain is only slightly degraded and one expectto achieve a quasi-equal uncorrelated diversity gain as shownin fig.2. However, the diversity is severely degraded whenthe spatial correlation increases to p = 0.9. For example, as

illustrated in fig.3, an MEA system operating at a multiplexinggain r > 0.8 and an SNR of 5 dB in a moderately correlatedchannel (p = 0.5), achieves a better diversity gain than a

Fig. 2. The impact of spatial correlation on the diversity estimates for NtN, = 2

system operating at the same multiplexing gain and an SNRof 10 dB in a highly correlated channel (p = 0.9). Theseobservations are confirmed when the exact diversity plots are

considered.

VI. CONCLUSION

In this paper, we addressed the finite SNR diversity mul-tiplexing tradeoff over spatially correlated channels. We firstderived tight lower bounds on the outage probability for spa-

tially correlated and uncorrelated channels. Then, using thesesbounds, estimates of the corresponding diversity-multiplexingtradeoff were determined. Simulation results have shown thatour diversity estimates follow closely the exact diversity-multiplexing tradeoff. The framework presented in this contri-bution subsumes the uncorrelated channel case. To the best ofour knowledge, this is the first characterization of the diversitymultiplexing tradeoff at finite SNR over spatially correlatedchannels.

APPENDIXPROOF OF COROLLARY 2

We derive the maximum diversity for the uncorrelated andcorrelated cases separately.

1) Uncorrelated spatial fading:

duncorr(r p)r--~O

lim t E (4j)K (4j), (18)1=1

where J1 ((l) and K1 ((l) are given by:

J ((l)(wNt+Nr-21e-4, )(Nt + Nr-

Frinc(,Nr + Nt -21 + 1)

21)!-(19)

((1 + g9)bt b1gTj(1 + gT)bl 1_0)

1.6 1.8 2u 2 4 b ts I u I 2 1 4 1 bSNR per receive antenna:,q (dB)

113 IA 1r

I

K ((lj)

When r -> 0, all b, -> 0,1 ,=l..., t. Note that J1((,)f

I ()) where

(21)

and f'((l) denotes the derivative of f ((l). Let Al () =

Ji ( )x 41 and B1 (1,) = K1 (1,) /41. Using Taylorexpansion of f((j) and f'((j) around 0, we obtain:

1/(Nt + Nr 21 + 1)! Nt+N -21+1 (22)

which leads to:

-dexact (simulation), SNR=5dB, p=0.5est-d, SNR=5dB, p=0.5estd SNR=1OdB,p=0.9

Xdexact (simulation), SNR=1 OdB, p=O.92.5

._

a) 1.5a)0

lim Al (1,) = lim Ji(Qj) x 4j = Nt + Nr -21+1. (23)Onteotebanl () ar-s:

On the other hand, writing Bi (ij) as

(24)Bi (4j) = '0 (1 _ bjgTj(l + g{q) bI-1Ntn(1 +cgTp)btin

and computing

0.5

&0) 0.2 0.4 0.6 0.8 1 1.2 1.4Multiplexing Gain

1.6 1.8 2

Fig. 3. Comparison of our diversity estimate and the exact (simulation) forp = 0.5, SNR= 5 dB and p = 0.9, SNR= 10 dB, when Nt = N, = 2

lir bigT(1+ g)bl- 1

bj-lO (l + gT)b, -I

we obtain that:

91'

(1 +grj) log (1 +grj)'

from which (16) follows immediately.

lim J, (,))K, (41,)= lim Al (,))B, (41,)

=_ r1 1-grNtj( (1 +grj)log(I +grg))x(Nt +Nr -21 + 1),

from which (16) follows immediately.2) Correlated spatial fading:

t

lim dco (r, T1, p) = limr--~0 r--~10=1

l) Kl (1j), (27)

where P1 (4,), Qi (i,) and K1 (i,) are respectively givenby (15), (14) and (20). Note that Ql (4j) = fA, (4,). Thus,

Q,(0) = fAl()= 2w d (28)

Since the Fourier Transform of the nth derivative offA, (x) is (jv) 'I'A, (jv), then it can be shown, usingthe Residue Theorem, that fA, (0) = 0 and so are allthe nth derivatives of fA, (0) for n < Nt + Nr -21 + 1.3Observing again that Qi (i,) is the derivative of P1 (i,)and using Taylor expansion to the (Nt + Nr- 21 + 1)thterm, we find that:

lrn Ql(( Nt + Nr -21 +1bl-m =l( (l (29)

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3This is because Nt+Nr-21+1 is the degree of ,1'Al (jv)'s denominator.

(25)

Tlh N (I + gq) log (I + gq)Thus,

Therefore,

bl -O Pi (4l )(26)

Nt+N(I +g1),log(I +gq)x (Nt + Nr -21 + 1),

f(W = Finc (.l, Nr + Nt 21 + 1) .