Exact Bit Error Rate of MIMO MRC Systems withCochannel Interference and Rayleigh Fading

Amir Ali BasriCommunications Research Centre Canada

3701 Carling Avenue, Ottawa, Ontario, Canada K2H 8S2.abasri@crc.ca

Abstract—In this paper, we present a new method for derivingexact expressions for the average bit error rate (BER) of multiple-input multiple-output (MIMO) wireless systems with transmitbeamforming and maximal ratio combining (MRC) reception,known as MIMO MRC, in the presence of cochannel interference.It is assumed that both the desired user and interferers aresubject to independent flat Rayleigh fading. The conventionalmethod for deriving the average BER expressions of MIMOMRC systems relies on finding the probability density function ofthe output signal-to-interference-plus-noise ratio. The presentedmethod, however, is based on using the decision variable at theoutput of the maximal ratio combiner conditioned only on thefading channel of the desired user. An exact closed-form expres-sion is derived for the average BER of MIMO MRC systemswith two receive antennas and multiple transmit antennas. Theanalytical result is verified by Monte Carlo simulations.

I. INTRODUCTION

Multiple antennas available at both the transmitter andreceiver of multiple-input-multiple-output (MIMO) wirelesssystems promise significant improvement in systems perfor-mance by combatting the fading and suppressing cochannelinterference (CCI). In the absence of interference, when thechannel matrix is perfectly known at both the transmitter andreceiver of MIMO wireless systems, transmit beamformingwith maximal ratio combining (MRC) reception, commonlyreferred to as MIMO MRC1, can be applied to maximize thesignal-to-noise ratio (SNR) at the combiner output. MIMOMRC was first proposed in [1], and the joint transmit andreceive weight vectors were derived in [2]. It exploits thefull transmit-receive diversity of MIMO systems and providesan additional array gain compared with space-time coding[3]. The performance of MIMO MRC in systems with nointerference is studied in [4]-[8].

In the presence of interference, MIMO MRC is not optimalfrom the standpoint of maximizing the signal-to-interference-plus-noise ratio (SINR) at the combiner output as it ignores theinterference and maximizes the desired signal power only. Infact, the optimal strategy in systems with interference is theMIMO optimum combining technique which maximizes theoutput SINR [9]. However, to implement the MIMO optimumcombiner, the knowledge of the desired user’s channel matrixas well as the interferers’ channel matrix are needed at the

1MIMO MRC is also known in the literature as maximum ratio transmission[1], dominant eigenmode transmission [3], transmit-receive diversity [5], orMIMO transmit beamforming [8].

receiver while MIMO MRC requires only the desired user’schannel matrix. Therefore, MIMO MRC is a more practicaltechnique that can be implemented at the receiver even in thepresence of CCI.

The performance of MIMO MRC systems in the presenceof CCI is investigated in [10]–[15]. Specifically, the errorrate performance is studied in [13]-[15] over independentand identically distributed (i.i.d.) Rayleigh fading channels byusing the conventional method. The conventional method isbased on finding the probability density function (PDF) ofthe output SINR and is known as the PDF-based method. Inthis method, the error probability conditioned on the fadingchannels of both the desired user and interferers is expressedas the Gaussian Q-function with an argument which is afunction of only the output SINR. The average bit error rate(BER) is then derived by averaging the mentioned conditionalerror probability over the PDF of the output SINR or byusing the moment generating function (MGF) of the SINR.We will see that in practical wireless communication systems,average BER expressions derived by the PDF-based methodare approximate results.

In this paper, we present a method for deriving exactexpressions for the average BER of MIMO MRC systems inthe presence of CCI and noise for binary phase-shift keying(BPSK) signals over i.i.d. Rayleigh fading channels. Thismethod, referred to as the Exact method, is used in [16] toderive BER expressions for MRC in wireless systems with asingle transmit antenna and multiple receive antennas in thepresence of CCI. In this paper, we will generalize the results of[16] to MIMO MRC systems. The Exact method is based onusing the decision variable at the combiner output conditionedonly on the fading channel of the desired user. By using thismethod, we will derive an exact closed-from expression forthe average BER of MIMO MRC systems for the special caseof two receive antennas and multiple transmit antennas.

The remainder of this paper is organized as follows: SectionII describes the system model. In section III, first the con-ventional PDF-based method is reviewed and then the Exactmethod is presented. Numerical results are provided in sectionIV, and section V concludes the paper.

II. SYSTEM MODEL

We consider a wireless communication system that employsnT transmit and nR receive antennas in presence of L inde-

978-1-4244-3574-6/10/$25.00 ©2010 Crown

pendent interferers and noise. The nR × 1 baseband receivedsignal vector r is given by

r =√

PDHDwT sD + HIP12I sI + n (1)

where sD is the transmitted symbol of the desired user andsI is the transmitted signal vector of interferers given bysI = (sI,1, sI,2, . . . , sI,L)T , where (.)T denotes the transposeoperator. We assume that the transmitted symbols of thedesired user and interferers are independent and equiprobablewith BPSK modulation, i.e sD, sI,1 . . . , sI,L ∈ {−1,+1}.HD is the nR×nT channel matrix of the desired user and HI

is the nR×L channel matrix of the interferers. Channel matri-ces HD and HI are assumed to be independent flat Rayleighfading in which the elements of HD and HI are i.i.d. circu-larly symmetric complex Gaussian random variables with zeromean and unit variance. The nT ×1 vector wT is the transmitbeamforming vector of the desired user with unit norm. PD isthe average received signal power of the desired user and P I isa diagonal matrix given by P I = diag {PI,1, . . . , PI,L} wherePI,j is the average received power of the jth interferer. ThenR×1 vector n is the additive white Gaussian noise (AWGN)distributed as CN

(0, σ2

nInR

), where CN(0,Φ) denotes a

circularly symmetric complex Gaussian distribution with zeromean and covariance matrix Φ, and InR

denotes the nR×nR

identity matrix. We assume that the desired user’s channelmatrix HD is perfectly known at both the transmitter andreceiver.

In MIMO MRC, the transmit beamforming vector wT andreceive combining vector wR are given as wT = umax andwR = HDumax where umax is a normalized eigenvector cor-responding to the largest eigenvalue λmax of HH

DHD, where(.)H denotes the conjugate transpose operator. Therefore, from(1), the decision variable at the output of the combiner can bewritten as

y = wHR r =

√PDuH

maxHHDHDumaxsD

+ uHmaxHH

DHIP12I sI + uH

maxHHDn. (2)

Since umax is the eigenvector of HHDHD corresponding to

the eigenvalue λmax, we have uHmaxHH

DHDumax = λmax,and hence, (2) can be rewritten as

y =√

PDλmaxsD + z (3)

where z is the interference-plus-noise component at the outputof the combiner and is equal to

z = uHmaxHH

DHIP12I sI + uH

maxHHDn. (4)

From (3) and (4), the output SINR γ can be expressed as

γ =PDλ2

max

uHmaxHH

D

(HIP IH

HI + σ2

nInR

)HDumax

. (5)

III. PERFORMANCE ANALYSIS

In this section, we investigate the average BER of BPSKsignals for MIMO MRC systems. First, we review the conven-

tional PDF-based method and then present the Exact method.

A. PDF-Based Method

The PDF-based method relies on the assumption that theinterference-plus-noise component in (4) conditioned on thefading channels of both the desired user and interferers iscircularly symmetric complex Gaussian distributed. From (4),this assumption is valid either when the transmitted signalvector of interferers is circularly symmetric complex Gaussiandistributed, i.e. sI ∼ CN (0, IL), or when the number ofinterferers goes to infinity and the central limit theorem canbe applied [15]. We will see that based on the Gaussianassumption, the BER conditioned on the fading channels ofboth the desired user and interferers can be expressed as theGaussian Q-function with an argument which is a function ofonly SINR. The average BER is then derived by averaging thementioned conditional BER over the PDF of the output SINR.

The conditional BER Pb(E|HD,HI) (the BER condi-tioned on the fading channels of both the desired user andinterferers) for the equiprobable transmitted signals sD = −1and sD = +1 is equal to the BER when sD = −1 istransmitted, and hence, from (3) we have

Pb (E|HD,HI) = Pb (E|HD,HI , sD = −1)= P (� (y) > 0|HD,HI , sD = −1)

= P(−√

PDλmax + �(z) > 0|HD,HI

)(6)

where �(z) denotes the real part of z.Now, if it is assumed that sI ∼ CN (0, IL), then z in (4)

conditioned on HD and HI is distributed as

z|HD,HI ∼ CN(0,uH

maxHHDRHDumax

)(7)

where the matrix R is given by

R = HIP IHHI + σ2

nInR. (8)

From (7), the distribution of �(z)|HD,HI is

�(z)|HD,HI ∼ N

(0,

12uH

maxHHDRHDumax

)(9)

where N(η, σ2) denotes a Gaussian distribution for a realrandom variable with mean η and variance σ2.

Now, from (9), the conditional BER in (6) can be expressedas

Pb (E|HD,HI) = Q(√

2γ)

(10)

where γ is given in (5), and the Gaussian Q-function is definedas

Q(x) =∫ ∞

x

1√2π

exp

(− t2

2

)dt. (11)

We see that the assumption (7) has made it possible toexpress the conditional BER in (6) in the form of the GaussianQ-function with an argument proportional to the square rootof the output SINR, γ. Gaussian assumption for z|HD,HI

can also be made when L, the number of interferers, goes toinfinity.

To find the average BER, (10) should be averaged over thePDF of γ. So, the average BER can be written as

Pb(E) =∫ ∞

0

Q(√

2γ)fγ(γ)dγ =

1π

∫ π2

0

Mγ

(− 1

sin2θ

)dθ

(12)where fγ(γ) is the PDF of γ, and Mγ(s) is the MGF of γdefined as Mγ(s) =

∫∞0

esγfγ(γ)dγ [17].Therefore, in the PDF-based method deriving the expression

for the average BER starts with deriving the PDF of the outputSINR, and then, the average BER can be obtained either byaveraging the aforementioned Gaussian Q-function over thePDF of SINR or by using the MGF of the output SINR.

As mentioned above, assumption (7) is valid either whenthe interfering symbols are circularly symmetric complexGaussian distributed or when the number of interferers goesto infinity. However, in practice we deal with the case wherethe number of interferers is finite and the transmitted symbolsare not Gaussian distributed. Therefore, the average BERexpression derived by using (12) based on the assumption (7)can be viewed only as an approximate result.

B. Exact Method

In this section, we present the Exact method for derivingthe exact expression for the average BER of BPSK signalsin MIMO MRC systems. In this method, first the BERconditioned only on the fading channel of the desired user iscalculated. We will see that by conditioning only on the fadingchannel of the desired user, the conditional distribution of theinterference-plus-noise component in (4) is exactly Gaussian.Moreover, the BER conditioned only on the fading channel ofthe desired user will be equal to a Gaussian Q-function wherethe argument of the Q-function is a function of λmax ratherthan the output SINR γ obtained in the PDF-based method.The conditional BER will then be averaged to get an exactexpression for the average BER.

The conditional BER Pb(E|HD) (the BER conditionedonly on the fading channel of the desired user) for theequiprobable transmitted signals sD = −1 and sD = +1 isequal to the BER when sD = −1 is transmitted, and hence,from the decision variable in (3) we have

Pb (E|HD) = Pb (E|HD, sD = −1)= P (� (y) > 0|HD, sD = −1)

= P(−√

PDλmax + � (z) > 0|HD

). (13)

In order to determine the conditional BER in (13), first weshould find the distribution of � (z) conditioned on HD. Bybreaking the interference part of z in (4) into L components,z can be rewritten as

z =L∑

j=1

√PI,ju

HmaxHH

DhI,jsI,j + uHmaxHH

Dn (14)

where the vector hI,j is the jth column of the interferer’schannel matrix HI .

Note that since random vectors hI,j’s for j = 1, . . . , L aredistributed as hI,j ∼ CN (0, InR

) and HD is independentfrom hI,j’s, for j = 1, . . . , L we get√

PI,juHmaxHH

DhI,j |HD∼ CN(0, PI,ju

HmaxHH

DHDumax

)= CN (0, PI,jλmax) . (15)

For BPSK signals of sI,j ∈ {−1,+1}, we have|sI,j |2 = 1. Therefore, from (15), the distribution of√

PI,juHmaxHH

DhI,jsI,j |HD conditioned on sI,j can be writ-ten as√

PI,juHmaxHH

DhI,jsI,j |HD, sI,j ∼ CN(PI,jλmax |sI,j |2

)= CN (0, PI,jλmax) .

(16)

We can see from (16) that the PDF of√PI,ju

HmaxHH

DhI,jsI,j |HD conditioned on sI,j is nota function of sI,j . Therefore,

√PI,ju

HmaxHH

DhI,jsI,j |HD

is independent from sI,j , and thus, the distribution of√PI,ju

HmaxHH

DhI,jsI,j |HD conditioned on sI,j is equal tothe distribution of

√PI,ju

HmaxHH

DhI,jsI,j |HD itself. So,from (16) for j = 1, . . . , L we have√

PI,juHmaxHH

DhI,jsI,j |HD ∼ CN (0, PI,jλmax) . (17)

Now, since hI,j’s for j = 1, . . . , L are independent, from(17) we get

L∑j=1

√PI,ju

HmaxHH

DhI,jsI,j |HD ∼ CN

⎛⎝0,

L∑j=1

PI,jλmax

⎞⎠

(18)and therefore, z in (14) conditioned on HD is distributed as

z|HD ∼ CN

⎛⎝0,

⎛⎝ L∑

j=1

PI,j + σ2n

⎞⎠λmax

⎞⎠ . (19)

From (19), the distribution of �(z) conditioned on HD is

�(z)|HD ∼ N

⎛⎝0,

12

⎛⎝ L∑

j=1

PI,j + σ2n

⎞⎠λmax

⎞⎠ . (20)

The result in (20) is a crucial outcome which shows thatthe distribution of �(z)|HD is exactly Gaussian. Therefore,from (20), the conditional BER in (13) can be written as

Pb (E|HD) = Q(√

2ξλmax

)(21)

where the Gaussian Q-function is defined in (11) and ξ is

ξ =PD

L∑j=1

PI,j + σ2n

(22)

which is the ratio of the desired user’s average received powerto the total average received power of the interferers plus noise.

Note that in contrast to the conditional BER expression ofthe PDF-based method in (10), which was derived based on

the assumption (7), the conditional BER expression in (21) isexact since (20) is exact.

Now, by averaging the conditional BER in (21) over thePDF of λmax, the average BER can be obtained as

Pb(E) =∫ ∞

0

Q(√

2ξλmax

)fλmax

(λmax) dλmax. (23)

Note that to calculate (23), only the PDF of λmax isrequired, like single-user systems, and there is no need tofind the PDF of the output SINR γ. In fact, (23) is similarto the average BER of BPSK signals in single-user MIMOMRC systems (with AWGN and no CCI) where the averageSNR per branch PD/σ2

n is replaced with ξ in (22). Therefore,when transmitted signals have BPSK modulation and fadingchannels of the desired user and interferers are independentwith Rayleigh distribution, the average BER expression forsingle-user MIMO MRC systems with AWGN can be usedas an exact expression to describe the average BER in thepresence of interference by replacing the average SNR perbranch with ξ in (22).

Analytical expressions for the average BER of single-userMIMO MRC systems are derived in [4]-[5] and can be usedto evaluate (23). Particularly, an exact closed-form expressionfor the average BER of single-user systems with BPSK signalsis derived in [4, eq. (8)] for the special case of two receiveantennas and multiple transmit antennas in a Rayleigh fadingenvironment. So, by using [4, eq. (8)] and replacing theaverage SNR per branch with ξ in (22), the average BER in(23) for nT ≥ 2 and nR = 2 can be computed as

Pb(E) = nT g(2ξ, nT + 1, 1) − 2(nT − 1)g(2ξ, nT , 1) + nT

× g(2ξ, nT − 1, 1) − nT

nT −2∑k=0

g(2ξ, k + nT + 1, 2)

×(

k + nT

nT

)+ 2(nT − 1)

nT −1∑k=0

g(2ξ, k + nT , 2)

×(

k + nT − 1nT − 1

)− nT

nT∑k=0

g(2ξ, k + nT − 1, 2)

×(

k + nT − 2nT − 2

)(24)

where

g(c, n, a) =1

2an

(1 − μ

n−1∑k=0

(2k

k

)(1 − μ2

4

)k)

(25)

and μ =√

c/ (c + 2a).The result in (24) is precise and no approximation is

involved while the result of the PDF-based method is onlyexact when the assumption (7) is valid which is not the casefor a system with finite L and BPSK modulated signals.

The results of the Exact method can be easily generalizedfor M-ary phase-shift keying (M-PSK) modulation. In M-PSK, the transmitted symbols are chosen from the set of{

exp(

i2πmM

) |m = 0, 1, . . . , M − 1}

where M is the numberof points in the signal constellation. So, for these types of

0 5 10 1510

−5

10−4

10−3

10−2

10−1

100

PD/σ

n2 (dB)

Ave

rage

BE

R

nT=2 (Analytical)

nT=2 (Simulation)

nT=3 (Analytical)

nT=3 (Simulation)

nT=4 (Analytical)

nT=4 (Simulation)

Fig. 1. Average BER versus PD/σ2n of MIMO MRC systems with

BPSK modulation and two receive antennas for different numbers of transmitantennas nT in the presence of equal-power interferers when PI,j = σ2

n forj = 1, . . . , L and L = 8.

signals the magnitude of the transmitted symbols is also unity,i.e. |sI,j |2 = 1 for j = 1, . . . , L, and hence, (16) and(17) can be applied again. Therefore, the distribution of theinterference-plus-noise component conditioned on the fadingchannel of the desired user would be Gaussian as well, andexact expressions for the conditional BER and average BERcan be similarly obtained.

IV. NUMERICAL RESULTS

In this section, we present a set of numerical results for theaverage BER of BPSK signals for MIMO MRC systems inthe presence of CCI. The fading channels of the desired userand interferers are assumed to be independent with Rayleighdistribution.

The average BER is studied in Fig. 1 by the analyticalexpression (24) and Monte Carlo simulations for MIMO MRCsystems with two receive antennas and equal-power interfererswhen the power of each interferer is equal to the power ofthe noise, i.e. PI,j = σ2

n for j = 1, . . . , L. Fig. 1 displays theaverage BER versus PD/σ2

n (the average SNR per branch) fordifferent numbers of transmit antennas nT when the numberof interferers is L = 8. We can see that as nT increasesthe average BER decreases, as expected. The Monte Carlosimulations are performed until 104 samples of errors areobserved for each data point. Analytical results in Fig. 1match precisely the simulation results verifying the derivedexpression (24).

In Fig. 2, we compare the average BER expression derivedby the conventional PDF-based method in [14, eq. (28)]2

2Note that the typo in [14, eq. (28)] should be corrected as Pe = a2−

a√

b2√

π

min(R,T )∑m=1

(R+T )m−2m2∑n=|R−T |

n∑r=0

dm,nΓ(L + r)Γ(r + 0.5)b2L−3

4 u2L−1

40

r!Γ(L)m2L−1

4 e−bu02m

× W 1−4r−2L4 , 2L−1

4

(bu0

m

).

0 5 10 15 2010

−10

10−8

10−6

10−4

10−2

100

PD/P

I (dB)

Ave

rage

BE

R

L=1 (Exact)L=1 (PDF−based)L=3 (Exact)L=3 (PDF−based)L=6 (Exact)L=6 (PDF−based)L=12 (Exact)L=12 (PDF−based)

Fig. 2. Average BER versus PD/PI of interference-limited MIMO MRCsystems with BPSK modulation, two transmit antennas and two receiveantennas for different numbers of equal-power interferers L when PI,j = PI

for j = 1, . . . , L.

with expression (24). The results in [14] are obtained forinterference-limited systems, where the effect of the noisecan be neglected, with equal-power interferers. Therefore,we consider the case where σ2

n = 0 and PI,j = PI forj = 1, . . . , L, where the scalar PI is a constant. Fig. 2demonstrates the average BER versus PD/PI for differentnumbers of interferers L in MIMO MRC systems with twotransmit antennas and two receive antennas. As can be seen,for L = 1 there is a large gap between the approximate resultof the PDF-based method in [14, eq. (28)] and the exactexpression (24). However, it is apparent that as the numberof interferers increases the difference between the exact andapproximate results decreases. This is because of this fact thatfrom the central limit theorem the assumption (7) in the PDF-based method is more sensible in the presence of a largernumber of interferers.

V. CONCLUSION

In this paper, we presented the Exact method for derivingexact expressions for the average BER of MIMO MRC sys-tems with BPSK signals in the presence of CCI and AWGN.The fading channels of the desired user and interferers areassumed to be independent with Rayleigh distribution. TheExact method is based on using the decision variable at thecombiner output conditioned only on the fading channel ofthe desired user. We found that by conditioning only on thefading channel of the desired user, the distribution of theinterference-plus-noise component is exactly Gaussian, whichresults in exact expressions for the conditional BER andaverage BER. The Exact method is much less complicatedthan the conventional PDF-based method since there is noneed for finding the PDF of the output SINR. We found thatthe exact expression of the average BER of MIMO MRCsystems in the presence of CCI can be obtained from the

expression for the average BER of single-user MIMO MRCsystems by replacing the average SNR per branch with theratio of the desired signal’s average received power to thetotal average received power of the interferers plus noise.An exact closed-from expression for the average BER ofMIMO MRC systems with two receive antennas and multipletransmit antennas was derived and verified by Monte Carlosimulations. We studied the difference between the resultsof the PDF-based method and the Exact method and foundthat this difference is substantial in the presence of a smallernumber of interferers. The results of the Exact method canbe easily generalized to M-PSK modulation as for this typeof modulation the distribution of the interference-plus-noisecomponent conditioned on the fading channel of the desireduser is also exactly Gaussian.

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