Generic Approach to the Performance Analysis of Correlated Transmit/Receive Diversity MIMO Systems...

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 56, NO. 3, MARCH 2010 1147

Generic Approach to the Performance Analysisof Correlated Transmit/Receive Diversity MIMOSystems With/Without Co-Channel Interference

Dian-Wu Yue, Member, IEEE, and Q. T. Zhang, Fellow, IEEE

Abstract—Optimal transmit/receive diversity (TRD) is one ofthe most important configurations for wireless multiple-inputmultiple-output (MIMO) systems, due to its good performanceand ease of implementation. Though investigated intensively,the performance of optimal TRD in general correlated fadingwith cochannel interference is still not well understood. Sincethe optimal TRD’s output instantaneous signal-to-interfer-ence-plus-noise ratio (SINR) is equal to the largest sampleeigenvalue of a quadratic form involving signal and interferencechannel matrices, directly determining the probability densityfunction (pdf) of this eigenvalue has been a prevailing approachin the literature. Given the nonlinearity involved in the quadraticform, however, finding such a pdf is not simple except for some spe-cial channel conditions. In this paper, we formulate the problem,in a totally different framework, as testing the positive-definite-ness of a random matrix whereby the theory of matrix-variatedistributions can be invoked to obtain exact solutions in terms ofspecial functions. The solutions are very general including most ofexisting results as a special case and allowing for the correlationstructures of both signal and interferers to be arbitrary at bothtransmitter and receiver ends. Numerical results are presented tovalidate the theoretical analysis.

Index Terms—Matrix variate distributions, multiple-input mul-tiple-output systems, optimal transmit/receive diversity, outageperformance.

I. INTRODUCTION

W IRELESS transmission using multiple antennas has at-tracted much interest in recent years due to its capa-

bility to exploit the tremendous capacity inherent in multiple-input multiple-output (MIMO) channels [1], [2]. Various as-pects of wireless MIMO systems have been studied intensivelyin [1]–[27]. All theoretical analysis for MIMO systems in theliterature can be roughly divided into two categories: Capacity

Manuscript received August 05, 2005; revised February 02, 2009. Currentversion published March 10, 2010. This paper was presented in part at the IEEEInformation Theory Workshop, Chengdu, China, October 2006. The work ofD.-W. Yue was supported by the National Natural Science Foundation of Chinaby Grant 60672030. The work of Q. T. Zhang was supported by a grant fromthe Research Grants Council of the Hong Kong Special Administrative Region,China (Project CityU 123706).

D.-W. Yue is with the College of Information Science and Technology, DalianMaritime University, Dalian, Liaoning 116026, China (e-mail:[email protected]).

Q. T. Zhang is with the Department of Electronic Engineering, City Univer-sity of Hong Kong, Tat Chee Avenue, Kowloon Tong, Kowloon, Hong Kong(e-mail:[email protected]).

Communicated by B. S. Rajan, Associate Editor for Coding Theory.Color versions of Figures 1–3 in this paper are available online at http://iee-

explore.ieee.org.Digital Object Identifier 10.1109/TIT.2009.2039085

analysis for the system efficiency [1]–[7] and performance anal-ysis for the system reliability [8]–[27]. Although the capacityanalysis and performance analysis focus on two different as-pects of MIMO systems, both of them strongly rely on randommatrix theory and matrix variate distributions.

A MIMO system can be configured differently. One con-figuration is transmit/receive diversity (TRD) which has beenwidely used due to its simplicity and good performance. TheTRD systems is a kind of wireless systems including tradi-tional receive diversity as one of its special cases [8]–[10].The performance of optimal TRD systems depends on theiroperational environments without [11]–[20] or with [21]–[25]cochannel interferences, and the treatment is mainly focused onthe classical Rayleigh or Rician fading channels. In particular,performance analysis in a Rayleigh fading environment withoutcochannel interference was first treated by Dighe et al. [11]by assuming that the MIMO channels follow independent andidentical (i.i.d.) Rayleigh distribution. The resulting outageprobability is expressible in the form of a determinant. Dighe’sresult was subsequently extended by Kang and Alouini [12],[16] to a general case of independent, but not necessarily iden-tically distributed, Rician fading channels. The results, again,take the form of determinants. For the case with dual antennasat the transmitter or receiver end, they obtained [25] an explicitexpression for outage probability complementing the result ofDighe [11]. The performance of MIMO systems with optimalTRD in the presence of cochannel interference was tackled in[21]–[25] under various fading environments allowing for theMIMO fading channels of the intended user and interferersto be non-i.i.d. Rician/Rayleigh, i.i.d. Rician/Rayleigh, andRayleigh/Rayleigh. It should be mentioned that a more com-plicated MIMO multichannel beamforming system in Ricianfading has been recently investigated in [20].

All aforementioned studies focus on MIMO systems withuncorrelated or semicorrelated antennas. By semicorrelation,we mean that the spatial correlation exists only at one side,transmitter or receiver end, of the MIMO systems. Even forthe case with semicorrelation, it is usually assumed that the in-tended user and interferers have the same correlation structureto simplify the mathematical analysis. In fact, the system sat-isfying this assumption is reducible to the one with i.i.d. chan-nels. The optimal TRD system forms a transmit and a receivebeam-pattern such that the received instantaneous signal-to-in-terference-plus-noise ratio (SINR) is maximized. The instanta-neous SINR so obtained is equal to the largest sample eigen-value of a quadratic form in the channel matrix for the intended

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1148 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 56, NO. 3, MARCH 2010

user normalized by the one for the cochannel interferers. Thecentral issue to the performance analysis of the optimal MIMOsystem with TRD is to determine the probability density func-tion (pdf) of this sample eigenvalue; the predominant method-ology follows exactly the same line of thought. The difficultyis that given the nonlinearity of the quadratic form under con-sideration, finding its largest eigenvalue is generally intractabletherefore limiting the cases we can handle.

The i.i.d. or uncorrelated assumption is often invalid inmany practical applications. Significant correlation among theantennas exists in realistic environments due to, for example,limited spacing between antennas. Furthermore, the spatialstructure (and even the fading distribution) of the interferenceusually differ from its counterpart for intended user sincetheir signals propagate over different multipaths, suffer fromdifferent fading, and arrive at the receive antenna array withdifferent incident angles. To handle these general fading situa-tions, we must take different methodology.

In this paper, we tackle the performance issue of MIMOsystems with optimal TRD over general Rician/Rayleigh andRayleigh/Rayleigh fading channels in a unified framework.No attempt is made to find the distribution function of thelargest eigenvalue of the relevant quadratic form. But rather,we formulate the outage problem as testing the positive-defi-niteness of a random matrix. the philosophy behind this is thatdetermining the joint pdf of this random matrix is much easierthan its counterpart for the aforementioned largest sampleeigenvalue. The rest of the paper is organized as follows. Weformulate the problem in Section II by representing TRDsystem outage events in terms of a matrix inequality ratherthan the largest eigenvalue distribution; the idea traceable toRatnarajah [27] and Muirhead [28]. The outage performanceof MIMO systems with optimal TRD in the presence and ab-sence of cochannel interferers are analyzed in Sections III andIV, respectively. In our derivations, we need to represent theoutage probability by virtue of some matrix variate functionssuch as hypergeometric functions of matrix argument. Similarmeans was used by Kiessling in his capacity analysis of MIMOchannels [4]. Numerical results are presented in Section V toillustrate our theoretical results. Section VI finishes the paperwith concluding remarks. Throughout the paper, we adopt andcite relevant notations and results from multivariate statisticaltheory. In particular, we will cite various matrix-variate dis-tributions from [28], [29], and [30], so that all lengthy butunimportant proofs can be neglected to retain a smooth andlogical presentation of our main results. The necessary proofsof our main results are placed at the Appendix.

Much work has been done in this research topic in the pastthree years [31]–[33] since the submission of the present paper.Meanwhile, one of the main results without cochannel interfer-ence [see (48) in Section IV of this revised paper], pertaining tothe largest eigenvalue distribution for the case with two-sidedcorrelation, has appeared, in a different form, in [31] and [32].In this paper, we focus on the cases of Rician or Rayleigh sig-nals with or without Rayleigh faded interferers. The Rician/Ri-cian scenario is not considered in this paper due to the lack ofnecessary mathematical tools in the literature [29], [30].

II. MIMO SYSTEM MODEL AND FORMULATION

Throughout the paper, we will use the following notations. Bywe denote the identity matrix of size (the subscript will

be omitted wherever the size of the matrix is clear from the con-text), 0 signifies the all-zero matrix, denotesthe diagonal matrix with elements , the determinantof the quadratic matrix is denoted by oris a matrix with representing its th element and cor-respondingly, denotes its determinant. denotes thediagonal matrix of eigenvalues of . The symbol in-dicates that is positive definite; likewise, means

. We use notation to signify the trace of thesquare matrix to denote to mean theHermitian transposition. The symbol denotes the Kroneckerproduct of two matrices, means distributed as, isa complex Wishart distribution, is a complex vectorvariate Gaussian distribution, means a com-plex matrix variate Gaussian distribution and denotesexpected value with respect to .

A. System Model

Suppose the intended user employs antennas to receive sig-nals transmitted from antennas. The channels that link thetransmit and receive antennas are characterized by an ma-trix , which is assumed to follow the joint complex Gaussiandistribution with mean matrix and covariance matrix .Symbolically, we will write

(1)

where and define the correlation structure at the transmitand receive ends, respectively. It is assumed that the intendedsignal is corrupted by independent interferers, and the th in-terferer transmits its signal with antennas where .The desired information symbol is weighted by the transmitbeamformer before being feeded to the transmit antennas.The transmit beamformer is normalized to have a unit norm

so that the transmit energy equals . Thevector at the desired user’s receiver can, thus, be written

as

(2)

where is the the channel matrix characterizing the linksfrom the desired user’s receive antennas to the transmit an-tennas of interferer ; and is the symbols transmitted by in-terferer , such that with denoting the averagesymbol energy and denoting expectation. In the way similarto defining , we assume

(3)

We assume the additive noise vector to follow the com-plex Gaussian distribution of mean zero and covariance matrix

YUE AND ZHANG: CORRELATED TRANSMIT/RECEIVE DIVERSITY MIMO SYSTEMS 1149

. Conditioned on , the covariance matrix ofinterference-plus-noise component is given by

(4)

Now we take a closer look at the correlation structure of andin (2). The correlations of the matrices and are speci-

fied by and , respectively. Physically, andrepresent the correlation matrices of incoming signal andinterference at the receiver, respectively. Correspondingly, thetransmit-antenna correlations for the desired user is character-ized by the correlation matrix , whereas its counterpartfor interferer is specified by the correlation matrix .The structure of these correlation matrices depends on channel’sfading characteristics, geometry and polarization of antenna ar-rays, and signal/interferers angle of arrival and spread, as de-scribed in [3].

B. Formulation

The TRD system transmits one symbol at a time, and em-ploys a weighting vector to combine received vector to forma single decision variable. The transmit and receive weightingvectors, and , should be chosen to maximize the outputSINR at every time instant, as defined by

(5)

where denotes the expectation with respect to . The resultof expectation equals given in (4). Optimization of is theproblem of Rayleigh quotient. Given the channel-state informa-tion and conditional on , we optimize with respect to toobtain [24], [34]

(6)

where we have used the fact that to represent the resultin the form of Rayleigh quotient. Thus, we can upper bound

by

(7)

where are the nonzero eigenvalues of thematrix product

(8)

in the descending order, and are their corre-sponding eigenvectors. The nonordered eigenvalues and eigen-vectors will be denoted by and ,respectively.

The outage probability of TRD systems can be defined di-rectly in terms of the instantaneous SINR or bychannel capacity [12], [23]

(9)

Both leads to the same expression for an outage event: ,but with the protection ratio defined differently as shown by

outage in terms ofoutage in terms of

(10)

In either case, we can write the outage probability as

(11)

To determine the outage performance, the central issue is to de-termine the pdf of or equivalently, its cumulative densityfunction (cdf).

Determination of the cdf of the principal eigenvalueof a rank- nonnegative definite matrix of the form

has been addressed intensively in theliterature [28]. The predominant methodology is to arrange thesample eigenvalues in a descending order and then to determinethe pdf of the largest one. The methodology is also prevailingin the area of communications [24]. Such methodology, how-ever, often leads to mathematically intractability except forsome simple cases. In this paper, we, therefore, consider thenonordered sample eigenvalues instead. The key step is torepresent the outage event , alternatively, by virtue ofnonordered eigenvalues. To this end, we write the sample space

(12)

The right-hand side is further expressible in matrix form. Hence,

(13)

where means that is a positive definitematrix. The equivalence of the two expressions is obvious, inmuch the same way as what we do in selection combining.Let denote the matrix of eigenvectors of . Namely,

. Hence we can write

(14)

The positive definiteness of implies that all of eigen-values are positive, and vice versa, thus showing the cor-rectness of (13). This equivalence was previously used in [27]and [28, Chap. 9].

We use it here to represent the outage probability yielding

(15)

The matrix representation of outage event, though simple inprinciple, provides a novel framework to tackle the outage issueof the optimal TRD system. The key to success along this di-rection is to find the joint cumulative distribution function ofmatrix .


For ease of presentation, we define variable

(16)

(17)

and the complex matrix

(18)

III. OUTAGE PERFORMANCE WITH COCHANNEL INTERFERENCE

We first proceed to operational environments with cochannelinterference. For mathematical tractability, let us first simplifythe interference covariance matrix given in (4). We assume thatthe operating environment is interference-dominated, so that thenoise component is negligible. Hence, we can rewrite (4) as

(19)

where . For the case withand , it is easy to

use [28, Th. 3.2.4] to assert that , up to a factor of , followsthe Wishart distribution, as shown by

(20)

where . Clearly, this is the extension of the clo-sure property of chi-square distribution. For the general settingof ’s, we can accurately approximate by using a singleWishart-distributed matrix, say , in much the same as whatwe do for a sum of chi-square variables [35]. The resulting ma-trix has the following distribution:

(21)

for which the parameters and can be determined byequating the first two moments of and ; for details, see[29, Ch. 3]. From the above analysis, it follows that we can usea single a Wishart-distributed matrix, say , to replaceto simplify the analysis. It also follows that is usually muchgreater than the number of antennas of the intended user. Thus,without loss of the generality, we can write the decision matrix(8) as

(22)

whereby, for a given power protection ratio , the outage prob-ability can be written as

(23)

where and is defined in terms of randomchannel matrices and , as shown by

(24)

A. Rician/Rayleigh Fading Scenario

We assume the signal suffers from Rician fading so thatthe corresponding channel matrix .Suppose that the interferer employs transmit antennas suchthat . We also assume that the channel-gain vectorsfor the interferer that link each transmit antenna to the re-ceive antennas are independent and identically distributed as

. Then, we can assert that .Under these assumptions and by introducing the followingmatrix notations:

(25)

and

(26)

we can explicitly work out the outage probability defined in(23), obtaining results summarized in the following theorem.The proof of this theorem is placed in Appendix B.

Theorem 1: The outage probability of the optimal TRDsystem with cochannel interference is given by

(27)

where

Here denotes the operator , andis the complex multivariate Gamma function. The notation

denotes a generalized Hermite polynomial with threecomplex matrix arguments, whose definition and properties aregiven in [29] and [30]. This kind of functions, though difficultin numerical calculation [29], serve as a fundamental tool inthe study of the distribution of some quadratic forms. Equation(27) is a general formula, providing a solid foundation forfurther study.

When applied to the Rician/Rayleigh case with semicorrela-tion, (27) leads to a simple expression, as shown below.

Corollary 1: Let . Assuming thathave nonzero distinct eigenvalues

, and denoting

then the outage probability is given by

(28)


where is a matrix function of the scalarwith entries

and

This is the major result of [24]. Its proof is lengthy and notneeded, and thus is omitted.

B. Rayleigh/Rayleigh Fading Scenario

This combination can be treated as a special caseof Section III-A by setting . Namely,

. With the condition, Theorem 1 leadsto the following corollary.

Corollary 2: Let . Then

(29)

where

(30)

The corollary is made by inserting into (27) and in-voking a property of generalized Hermite polynomial (i.e., thecomplex counterpart of [29, Expression (1.8.3)]).

Note that the notation denote a hypergeometric func-tion of two matrix arguments. Matrix variate distributions, es-pecially central quadratic form distributions, can be written interms of hypergeometric functions with matrix argument. Hy-pergeometric functions of matrix arguments is a natural gener-alization of their (generalized) counterparts of scalar arguments,which have been used widely in the field of sciences and engi-neering. For a more detailed study of hypergeometric functionsof matrix arguments, the reader is referred to [29], [36], and [37].

Recently, the exact ergodic capacity of arbitrarily correlatedMIMO channels in Rayleigh fading (with full channel state in-formation at the receiver and no channel state information at thetransmitter) has been presented by Kiessling [4] in the form ofhypergeometric function of two matrix arguments. Surprisingly,the outage probability of the optimal TRD system in Rayleighfading is also expressible in the form of a hypergeometric func-tion of two matrix arguments, as clearly indicated in (29).

Our concern is whether (29) can be further simplified. Tothis end, we note that when , the hypergeometric function

involved in (29) can be converted to scalar hypergeo-metric functions which are much easier to calculate by usingfor example, the built-in functions in Matlab, Mathematica,and Maple. The simplification can be done by invoking thefollowing lemma (see [4, Lemma 2 ]).

Lemma 1: Let andwith and

. Furthermore, define

(31)

(32)

and

(33)

for . Then

(34)

where with

(35)

for .

When some of the ’s or ’s are equal, we obtain the resultsas limiting case on the right of (34) via L’Hospital’s rule (see [4]for a detail process.)

Let us return to the general case with . There is a simplemethod to convert this problem into the corresponding one with

. The basic skill is to obtain the exact outage probability asthe result of a limiting process. The interested reader is referredto Kiessling [4] for details. By the same token, we can simplify(29) to obtain an alternative expression which is much easier innumerical calculation.

Corollary 3: Let andwith and

. Then

(36)

where is defined as follows:

(37)

and the entries of matrix are given by

(38)

The expression in (27) is a general result. Its correctness canbe examined by showing that the main result of [22] is one ofits special cases.


Corollary 4: Let and . Then

(39)

where is an matrix function of the scalar with entries

The function is called the incomplete beta function (see[38, eq. [8.391]]).

This result is exactly the same as [22, eq. (20)]. The proof isa little complicated, yet not important to us, and thus is omitted.

For the general case, (27) cannot be further simplified. Weshow, however, that it is related to (36) in a very simple manner,as stated in the following corollary.

Corollary 5: When , we have that

(40)

where is defined in Corollary 1.The proof is very straightforward by invoking an inequality forfunction as shown in [29, Ch. 1]. It is interestingto note that the outage probability for the Rician/Rayleigh caseis upper bounded by its Rayleigh/Rayleigh counterpart.

IV. OUTAGE PERFORMANCE WITHOUT COCHANNEL

INTERFERENCE

When cochannel interference is absent, we can setto rewrite (4) as

(41)

where has been normalized to signify the branch noise cor-relation matrix whereas denotes the noise variance at eachbranch. Theoretically, the Rayleigh/Rayleigh case is directlyderivable from the results obtained in the previous section. Thisis not the case, however, since the results obtained there arebased on the interference-dominated assumption that implies

.We, therefore, need a difference treatment due to the replace-

ment of a random matrix summation with a constant matrixin the quadratic form . Nevertheless, the procedure is

parallel.

A. Rician Faded Signals

Given the change in covariance matrix , we need to modifyand accordingly, as shown by

(42)

Correspondingly, matrices and are modified to

.(43)

and

.(44)

With these notations, we can write which,after some manipulations as shown in Appendix A, leads to thefollowing result.

Theorem 2: The outage probability of the optimal TRDsystem without cochannel interference is given by

(45)

where

(46)

Consider its simplest case where , and. Under these conditions, (45) reduces to a simple

form as follows.

Corollary 6: Let , and . Andlet denote the eigenvalues ofand let denote . Further denote

. Then we have

(47)

where is a matrix function with its th entrygiven by

for . The expression (47) is just the main resultderived in [12] and [16]. Since the proof is not important to ourdiscussion, it is omitted.

B. Rayleigh Faded Signals

Another important case is Rayleigh faded signals for whichand (45) can be simplified.

Corollary 7: when , we have that

(48)

where

(49)

This corollary’s proof is similar to that of Corollary 2 and thusis omitted.

Similar to , the hypergeometric function in-volved in (48) can be also easily calculated by representing it interms of scalar hypergeometric functions for ease of calculation.Specifically, by using the same techniques as used by Kiessling[4], we can obtain the following corollary.


Corollary 8: Let andwith and

.

(50)

where is given by

(51)

and the entry of the matrix is given by

.(52)

Equation (50) can be further simplified to [31, eq. (1)], one ofthe main results presented therein. To examine the correctnessof our results given in (48), let us consider the special case ofindependent noise and i.i.d. fading Rayleigh channels such that

and . These conditions, when inserted into(48) and simplified, leads to (53) shown later.

Corollary 9: Let and . Then

(53)

where is a matrix function with its th entriesgiven by

for .This result is identical to the corresponding one in [11] and [16].If we further set , then (53) can be rewritten as

(54)

which is exactly the same as the known result described in [25].Its proof is not difficult but not important and thus, is omitted.

Just as for the case with cochannel interference, there exists asimple relationship between the outage performance for generalRician faded signals and for their Rayleigh counterparts shownin (50), as summarized in the following assertion.

Corollary 10:

(55)

where is defined in (49).

V. NUMERICAL RESULTS

The validity of Theorem 1 and Theorem 2 has been rigorouslyexamined by showing that they include most of existing resultsin the literature as special cases. In this section, we examinethe correctness of Corollary 2 and Corollary 8 with numericalresults. For simplicity, we assume that the spatial correlationamong antennas follows the exponential model with correlation

between antennas and given by. Physically, denotes the correlation magnitude,

and stands for the correlation coefficient.We assume that the receiver is equipped with antennas for

the reception of Rayleigh faded signals from intended transmitantennas. The received signals are corrupted by Rayleigh fadedinterference from interferers. Thus, Corollaries 3 and 9 areapplicable in theoretical evaluation. Simulation results are alsoincluded for comparison. Each point in the simulated curves isproduced by averaging over at least 100 000 independent com-puter runs. Throughout this section, we set and , andassume that the correlation at the intended transmit and receiveends is characterized by and , respectively.

We first investigate the case with cochannel interference. Forease of illustration, assume the presence of only one cochannelinterferer (i.e., ) which employs antennas for transmis-sion. Further assume that the correlation structure at the bothsides of the interference channel matrix is the same, char-acterized by .

Fig. 1 shows the variation of outage probability with thenumber of the interferer’s transmit antennas. The parametersetting is: , and . The curves in thefigure are for , respectively. As expected,the outage performance becomes worse as increases, butthe decrease magnitude becomes smaller and smaller. It isalso observed that the simulated results coincide with theirtheoretical counterparts.

The influence of the interferer’s correlation coefficient on theoutage probability is shown in Fig. 2 where is set to 3 and thethree curves are shown for and 0.9, respectively.Other parameters are set to be and . Weobserve that over the region of moderate and high signal-to-in-terference ratio (SIR), the outage performance improves withincreased . This is is easy to understand since a higher in-terference correlation implies a sharper directional beam whichis easier to be nullified by using interference-covariance ma-trix inversion involved in our quadratic form. Clearly, unlike theeffect of the intended user’s correlation, the spatial correlationof cochannel interference is an advantage to the outage perfor-mance of TRD systems. From these curves, we can see, again, anearly perfect agreement between the theoretical and simulatedresults.

We next consider the case without cochannel interference.Fig. 3 shows the outage probability as a function of SIR for dif-ferent values of . Here we set . The three curvesare for and 0.9, respectively. It is clear that theoutage performance drops with increased transmit correlationcoefficient . This is quite intuitive since high transmit correla-tion means the lose of more degrees of freedom in transmit di-versity. A perfect agreement between simulation and theoreticresults are observed again.

VI. CONCLUSION

In this paper, the outage problem for optimal receive anddiversity MIMO systems was formulated as the one of testing


Fig. 1. Variation of outage probability with the number of interfering antennas.

Fig. 2. Influence of interference correlation � on the outage performance.

the positive-definiteness of a quadratic form in random ma-trices. By representing the quadratic form as a series of Hermitepolynomials with complex matrix arguments and by utilizingthe relevant properties of the latter, we obtain two exact outageformulas for general cases of Rician faded signals with orwithout Rayleigh faded interference, respectively, as summa-rized in Theorem 1 and Theorem 2. The new results are verygeneral allowing for different correlation structures for signaland interference and allowing for correlation at both transmitand receive ends. We have shown that these general results,indeed, include most of existing ones as their special cases, andare therefore likely to be used as a solid tool for future studies.

When the desired signals degenerate to be Rayleigh faded, thetwo formulas reduce to the form containing only simple andcomputationally manageable hypergeometric functions withscalar arguments. We examined the theory by comparing itwith Monte Carlo simulations based on the Rayleigh/Rayleighscenario, and a nearly perfect agreement between the two wasobserved, demonstrating the validity of our theoretic analysis.

APPENDIX

Throughout this appendix, we use many relevant notationsand results from multivariate statistical theory, in particular, var-ious matrix-variate distributions. Although relevant results are


Fig. 3. Influence of signal transmit correlation on the outage probability.

available in the statistical literature [28], [29], and [30], they aregiven only for real variables. The extension of these results totheir complex counterparts, as required in this paper, is straight-forward.

A. Proof of Theorem 1

The Distributions of quadratic forms in matrix argument havebeen investigated extensively by many authors. For more de-tails, the reader is referred to [29] and [30]. In order to proveTheorem 1, we first extend a lemma for real data to its complexcounterpart to obtain the following.

Lemma 2: Let andlet be a Hermite positive definite matrix. Then the PDFof quadratic form is given by

(56)

where and

(57)

Note that is an arbitrary scalar constant. The PDF foris called the Wishart type representation, and for is calledthe power series type representation.

To prove Theorem 1, we also need two properties of the gen-eralized Hermite polynomial with three complex matrix argu-ments, as described here.

Lemma 3:

(58)

where

(59)

For denotes a partition of into parts to form a-tupple such that

.

Lemma 4:

(60)

where is an arbitrary Hermite positive definite matrix.

Proof of Theorem 1: We begin with the case of anddetermine the PDF of the quadratic form in (24). Under thecondition of given matrix , by plugging into (56) ofLemma 2, the conditional PDF of can be expressed as

(61)


where

(62)

Then by applying Lemma 3 we carry on the expectation ofwith respect to yielding

(63)

where

(64)

The desired outage probability is nothing but the integration ofover . The integral, however, involves matrix ar-

guments and needs to be simplified. To this end, we invoke aproperty of the generalized Hermite polynomial, i.e., Lemma 4.By applying this property, setting , and using the defini-tions of and , we complete the proof for this case of .

We next consider the case of . Let

(65)

where . Due to the fact

(66)

then in this case the proof is quite similar to the proof given forthe case where , and so is omitted.

Finally, we need the identity,

, to give the unified representation of (27).

B. Proof of Theorem 2

The following property of the generalized Hermite polyno-mial with three complex matrix arguments is useful in the proof.

Lemma 5: For a random matrix ,

(67)

where .In [1], Telatar gave the following useful limiting result for a

Wishart-distributed matrix sequence.

Lemma 6: Let . When , then

(68)

Proof of Theorem 2: Without loss of generality, we canassume from (43) and (44) that . Under the conditionof Theorem 1, we first let be a variable, and furtherlet . Then, according to Lemma 6, wecan assert that when , the TRD system with cochannel

interference will reduce to the TRD without cochannel inter-ference. Correspondingly, the outage probability of the optimalTRD system with cochannel interference (27) will reduce tothe outage probability of the optimal TRD system withoutcochannel interference, which is just (45) in Theorem 2. Letus verify this assertion. By inserting into (27) andcomparing the two expressions of (27) and (45), we only needto prove (69) and (70) shown here.

a) For , when , then

(69)

b) For , when , then

(70)

Here, we have used the fact that

(71)

Based on Lemma 5, the proof of (69) and (70) can be doneby showing the validity of the following assertion. Namely, foran arbitrary Hermite matrix and , we have

(72)

To this end, we invoke a simple property of zonal polynomialsthat for scalar to simplify (72). It re-mains to show

(73)

whose validity can be checked by directly using the definition

of given in (59).

REFERENCES

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[2] G. J. Foschini and M. J. Gans, “On limits of wireless communicationin a fading environment when using multiple antennas,” Wireless Pers.Commun., vol. 6, pp. 311–335, Mar. 1998.

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Dian-Wu Yue (M’09) received the B.S. and M.S. degrees in mathematics fromNankai University, Tianjin, China, in 1986 and 1989, respectively, and the Ph.D.degree in electrical engineering from Beijing University of Posts and Telecom-munications, Beijing, China, in 1996.

From 1989 to 1993, he was a Research Assistant of Applied Mathematics atDalian University of Technology, Dalian, Liaoning, China. From 1996 to 2003,he was an Associate Professor of Electrical Engineering with Nanjing Univer-sity of Posts and Telecommunications, Nanjing, Jiangsu, China. He is currentlya Professor of Electrical Engineering with Dalian Maritime University, Dalian.During 2000–2001, he was a Visiting Scholar with the University of Manitoba,Winnipeg, MB, Canada. During 2001–2002, he was a Postdoctoral Fellow withthe University of Waterloo, Waterloo, ON, Canada. His current research inter-ests include cooperative relay communications, cognitive radio, cross-layer de-signs, space–time codes and MIMO wireless communications, turbo codes, anditerative decoding.

Q. T. Zhang (S’84–M’85–SM’95–F’09) received the B.Eng. degree from Ts-inghua University, Beijing, China, and the M.Eng. degree from South ChinaUniversity of Technology, Guangzhou, China, both in wireless communica-tions, and the Ph.D. degree in electrical engineering from McMaster University,Hamilton, ON, Canada.

After graduation from McMaster University in 1986, he held a research po-sition and Adjunct Assistant Professorship at the same institution. In January1992, he joined Spar Aerospace Ltd., Satellite and Communication Systems Di-vision, Montreal, Canada, as a Senior Member of Technical Staff, participatingin the project of Radar Satellite. He joined Ryerson University, Toronto, Canada,in 1993 and became a Professor in 1999. Since October 1999, he has been withthe City University of Hong Kong, where he is now a Chair Professor of Infor-mation Engineering. His research interest is on wireless communications withcurrent focus on wireless multiple-antenna systems, cooperative systems, andcognitive radio.

Dr. Zhang was an Associate Editor for the IEEE COMMUNICATIONS LETTERS

from 2000 to 2007.

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