IEEE COMMUNICATIONS LETTERS, VOL. 5, NO. 9, SEPTEMBER 2001 369

Channel Capacity of MIMO Architecture Using theExponential Correlation Matrix

Sergey L. Loyka, Member, IEEE

Abstract—Multiple-input multiple output (MIMO) communi-cation architecture has recently emerged as a new paradigm forwireless communications in rich multipath environment, whichhas spectral efficiencies far beyond those offered by conventionaltechniques. Channel capacity of the MIMO architecture inindependent Rayleigh channels scales linearly as the numberof antennas. However, the correlation of a real-world wirelesschannel may result in a substantial degradation of the MIMOarchitecture performance. In this letter, we investigate the MIMOchannel capacity in correlated channels using the exponentialcorrelation matrix model. We prove that, for this model, increasein correlation is equivalent to decrease in signal-to-noise ratio(SNR). For example, = 0 7 is the same as 3-dB decrease in SNR.

Index Terms—Channel capacity, correlation, MIMO.

I. INTRODUCTION

M ULTIPLE-INPUT multiple-output (MIMO) communi-cation architecture has recently emerged as a new par-

adigm for wireless communications in rich multipath environ-ment [1]–[4]. Using multi-element antenna arrays (MEA) atboth transmitter and receiver, which effectively exploits the third(spatial) dimension in addition to time and frequency dimen-sions, this architecture achieves channel capacity far beyond thatof traditional techniques. In independent Rayleigh channels theMIMO capacity scales linearly as the number of antennas undersome conditions [1], [3], [4]. However, some impairments ofthe radio propagation channel may lead to a substantial degra-dation in MIMO performance. Some limitations on the MIMOcapacity are imposed by the number of multipath componentsor scatterers [4]. Another limitation on the MIMO channel ca-pacity, which is somewhat analogous to the multiple path limita-tion, is due to the correlation between individual sub-channelsof the matrix channel [1], [5]–[7]. Increase in the correlationcoefficient results in capacity decrease and, finally, when thecorrelation coefficient equals to unity, no advantage is providedby the MIMO architecture. The effect of fading correlation onthe MIMO channel capacity has been investigated in details in[5] using an abstract one-ring scattering model. However, thisapproach does not allow to study the effect of correlation in anexplicit way (i.e., capacity versus correlation).

Channel capacity of the two-antenna MIMO architecture hasbeen investigated in [6] as an explicit function of the correlationcoefficient. The general case of-antenna architecture has been

Manuscript received April 16, 2001. The associate editor coordinating thereview of this letter and approving it for publication was Prof. K. Kiasaleh..

The author is with the Department of Electrical Engineering, Ecolede Technologie Superieure, Montreal, QC H3C 1K3, Canada (e-mail:sergey.loyka@ieee.org).

Publisher Item Identifier S 1089-7798(01)08458-7.

considered in [7] using the uniform correlation matrix model(when all the correlation coefficients are equal). This model maybe used for the worst-case analysis or for some rough estima-tions using the average value of the correlation coefficient. How-ever, the uniform model is somewhat artificial—one expects thatthe correlation of neighboring subchannels is higher than thatof distant subchannels. In this way, we arrive to the exponentialcorrelation model, which has been successfully used for manycommunication problems [8]–[10]. In this letter, we study theMIMO channel capacity using the exponential correlation ma-trix model by analytical techniques and derive a simple formulafor the channel capacity indicating its validity range. We provethat, for this model, increase in correlation is equivalent to de-crease in the signal-to-noise ratio under some realistic condi-tions. For example, is equivalent to the 3 dB decreasein SNR (as compared to the case of ). Finally, we comparethis model with the uniform model and show that the exponen-tial model predicts better MIMO performance.

II. CHANNEL CAPACITY OF MIMO A RCHITECTURE

For fixed linear matrix channel with additive whiteGaussian noise and when the transmitted signal vector is com-posed of statistically independent equal power components eachwith a Gaussian distribution and the receiver knows the channel,its capacity is [1]

bits/s/Hz (1)

where is the number of transmit/receive antennas (for the sakeof simplicity we consider here the case when the number oftransmit and receive antennas are equal, but a general case canbe considered in a similar way);is the average signal-to-noiseratio (SNR); is identity matrix; is the normalizedchannel matrix, which is considered to be frequency indepen-dent over the signal bandwidth; and “” means transpose con-jugate. We adopt here the following normalization condition:

(2)

where denotes the components of. Hence, when(completely uncorrelated parallel sub-channels), is the

signal-to-noise ratio per receive branch. Some other kinds of thenormalization can also be used, but in this casewill have aslightly different meaning.

To study the effect of correlation in an explicit way andto separate it from the effect of unequal received powers, we

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370 IEEE COMMUNICATIONS LETTERS, VOL. 5, NO. 9, SEPTEMBER 2001

assume that all the received powers are equal. In this case,and (1) reduces to [7]:

(3)

where is the normalized channel correlation matrix, ,whose components are:

(4)

where “ ” denotes complex conjugate. The last equality in (4)holds due to the assumption of equal received powers. It imme-diately follows from (4) that .

It should be noted that (1)–(4) hold for a deterministicchannel. When the channel is random (stochastic), the capacityis random, too. The mean (ergodic) capacity can be defined inthis case [3]. Using the expectation over the channel matrix in(1), (2) and (4), we obtain the following upper bound on themean (ergodic) capacity:

(5)

where is the expectation over the channel matrix. The in-equality in (5) holds due to Jensen’s inequality and concavity of

function [13]. Thus, our method will provide an upperbound on the mean capacity. For the sake simplicity, we assumefurther that the channel is deterministic, keeping in mind thatthe same results hold true for as well.

Denoting the matrix under the determinant in (3) by, weobtain:

(6)

where

(7)

and

(8)

Thus, we may present the MIMO capacity in the followingform:

(9)

The first term is the MIMO capacity of parallel completely in-dependent sub-channels [1] ( in this case) and the secondterm is the contribution of the sub-channel correlation. Note thatthe second term is negative (or zero) because . Forhigh SNR ( ), and depends on the corre-lation matrix only. In fact, is the Gram’s determinant [11].Thus, in order to estimate the MIMO capacity in a correlatedchannel, we need to evaluate .

III. EXPONENTIAL CORRELATION MATRIX MODEL

For this model, the components ofare given by

(10)

where is the (complex) correlation coefficient of neighboringreceive branches (the correlation coefficient of signals in thesereceive branches). Obviously, (10) may be not an accuratemodel for some real-world scenarios but this is a simplesingle-parameter model which allows one to study the effect ofcorrelation on the MIMO capacity in an explicit way and to getsome insight. It remains to be investigated whether this modelis applicable or not to some specific scenarios. Note, however,that this model is physically reasonable in the sense that thecorrelation decreases with increasing distance between receiveantennas and it also corresponds to some realistic physicalconfigurations (e.g., [9, p. 26]). Comparison with the recentmeasurement results [14] shows also that it provides reasonableconclusions when applied to a MIMO system.

Substituting (10) into (7) and after some equivalent transfor-mations of the determinant (the second row multiplied byissubtracted from the first row, the third row multiplied byissubtracted from the second row etc.), we arrive to the following:

(11)

A recurrent formula can be derived for from (11) and be fur-ther used for its numerical evaluation (in this way, we reduce thecomputational time substantially as compared to the full matrixcomputation for large ). Unfortunately, it is not possible to de-rive a closed-form analytical expression from (11). But we canstill obtain a simple closed-form expression in a practically-im-portant case of high SNR ( ), when . In thiscase, the main diagonal components only give the primary con-tribution to :

(12)

and

(13)

In the case of and , we finally obtain:

(14)

In the limiting case of , one obtains from (14):

(15)

LOYKA: CHANNEL CAPACITY OF MIMO ARCHITECTURE USING THE EXPONENTIAL CORRELATION MATRIX 371

Fig. 1. MIMO channel capacity versus correlation coefficient,� = 30 dB.

When , the last two equations reduce to the well-knownformulas (in this case, ) [1]:

and (16)

Comparison of (14) and (15) with (16) clearly indicates thatthe effect of the channel correlation is equivalent to the loss inthe SNR. Hence, for example, is equivalent to 3-dB re-duction in the SNR. Another possible interpretation of (14) and(15) is that the correlation of individual subchannels gives an in-crease in the noise level because for each particular sub-channelall the other subchannels are just the sources of interference (see,for example, the detection algorithm in [2]). Note also that thechannel capacity does not depend on the correlation coefficientphase.

Fig. 1 shows the MIMO capacity of a deterministic channelversus the correlation coefficient evaluated by the full matrixcomputation [see (9) and (10)] and by approximate formulas(13) and (14) for and 50, and dB. The MIMOchannel capacity evaluated using the uniform correlation ma-trix model [7] is also shown for comparison. As one may seefrom this figure, the accuracy of approximate formulas is quitegood. It should be noted that the accuracy decreases asand

decreases. The uniform model predicts lower capacity, as itshould be (because it is the worst case model—the correlationbetween distant receive branches is the same as between neigh-boring ones). We also see that the MIMO capacity decreasessignificantly for to , that agrees well with the knownresults on the spatial diversity techniques [12] and with recentmeasurements of the MIMO channel [14]. Detailed statisticalanalysis using Monte Carlo simulations of a correlated Rayleighchannel shows that its mean (ergodic) capacity is approximately20%– 40% smaller than that in Fig. 1 and it decreases with cor-relation in the same way as in the deterministic case.

IV. CONCLUSION

The MIMO channel capacity depends substantially on thecorrelation between individual subchannels of the matrixchannel, reaching its maximum value for zero correlation(completely independent parallel subchannels). In this letter,we have developed a method of MIMO capacity estimationusing the correlation matrix approach. In particular, using theexponential correlation matrix model, we derived a simpleformula for the MIMO channel capacity. The main conclusionis that the effect of channel correlation is the same as decreasein the signal-to-noise ratio under some realistic conditions.For example, is the same as 3-dB decrease in SNR.The exponential correlation matrix model predicts higherchannel capacity than the uniform correlation matrix model,as it intuitively should be. While developed for a deterministicchannel, our method provides an upper bound on the meanergodic capacity of a stochastic channel as well.

ACKNOWLEDGMENT

The author wishes to thank Profs. Mosig and Rimoldi for dis-cussions which inspired this work. The insightful comments ofthe reviewers are also appreciated.

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