158 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 55, NO. 1, JANUARY 2006

Closed-Form Expressions of Approximate ErrorRates for Optimum Combining With Multiple

Interferers in a Rayleigh Fading ChannelJin Sam Kwak, Member, IEEE, and Jae Hong Lee, Senior Member, IEEE

Abstract—This paper presents approximate error rates of M -ary phase shift keying (MPSK) for optimum combining (OC) withmultiple cochannel interferers in a flat Rayleigh fading channel.For the first-order approximation, we derive the closed-form ex-pression for ordered mean eigenvalues of the interference-plus-noise covariance matrix, which facilitates performance evaluationfor the OC with arbitrary numbers of interferers and antenna el-ements without Monte Carlo simulation and multiple numericalintegrals. We also derive the closed-form expressions for approx-imate error rates of MPSK for the OC in terms of the averageerror rate of MPSK for maximal ratio combining (MRC). Fromthe simple evaluation of ordered mean eigenvalues, we show thatthe first-order approximation gives a simple and accurate way toanalyze the performance of the OC.

Index Terms—Adaptive antennas, co-channel interference,M -ary phase shift keying (MPSK), optimum combining (OC).

I. INTRODUCTION

ADAPTIVE antenna arrays significantly improve both theperformance and capacity of wireless communication sys-

tems by mitigating multipath fading and suppressing interfer-ing signals [1]. The optimum combiner (OC) which maximizesthe signal-to-interference-plus-noise ratio (SINR) yields bet-ter performance than that of maximal ratio combiner (MRC)in the interference-limited system. In the absence of interfer-ence for an additive noise environment, the OC maximizes thesignal-to-noise ratio (SNR) and has the same performance asthe MRC [1], [2].

In a Rayleigh fading environment, the exact average bit errorrate (BER) performance of the OC for binary phase-shift-keying(BPSK) has been studied [2]–[4]. The closed-form expressionfor the average BER was derived for the OC in the presence ofthe arbitrary number of interferes [4]. For M -ary phase-shift-keying (MPSK), the average symbol error rate (SER) have beenexpressed as a multidimensional integral that can be a burden tobe evaluated numerically [5] or as a simple closed-form upperbound that may be loose for high SER [6].

On the other hand, approximation techniques have been stud-ied to simply evaluate the performance analysis of the OC

Manuscript received April 16, 2003; revised December 23, 2004. This workwas supported in part by the Brain Korea 21 Project and in part by the ITscholarship program from MIC and IITA, Korea. The review of this paper wascoordinated by Prof. H. Leib.

J. S. Kwak is with the Department of Electrical and Computer Engi-neering, the University of Texas at Austin, Austin, TX 78712-0240 USA(e-mail: samji@ece.utexas.edu).

J. H. Lee is with School of Electrical Engineering and Computer Science,Seoul National University, Seoul 151-742 Korea (e-mail: jhlee@snu.ac.kr).

Digital Object Identifier 10.1109/TVT.2005.861184

[7]–[10]. Villier proposed orthogonal approximation to deter-mine the average BER of several binary modulation schemes [7].However, the orthogonal approximation only takes into accountthe case where the number of antenna elements is larger thanthat of interferers, and the accuracy of approximation dependson the difference between the number of interferers and thenumber of antenna elements [7]. In [8], Pham and Balmainproposed a first-order approximation and derived a closed-formexpression of average BER for BPSK with an arbitrary num-ber of interferers. This approximation provides accurate resultsfor the performance of the OC, whereas Monte Carlo simu-lation is needed to evaluate the ordered mean eigenvalues ofthe interference-plus-noise covariance matrix [8]. Recently, theanalytical expressions for the ordered mean eigenvalues havebeen presented as the multidimensional integral, which can bederived as a closed-form solution in the simple cases such asdual-antenna reception or two cochannel interferers [9], [10].

In this paper, we derive the simple closed-form expression ofthe ordered mean eigenvalues to avoid Monte Carlo simulationand multiple numerical integrals in the presence of arbitrarynumbers of interferers and antenna elements. As the extendedresults of the performance analysis in [7] and [8], the approx-imate average SERs of MPSK for the OC are also derived interms of the average SER of MPSK for the MRC. The analyti-cal results of the approximation using ordered mean eigenvaluesprovide a simple and accurate way to assess the performance ofthe OC with multiple cochannel interferers.

The paper is organized as follows. Section II gives the systemmodel. In Section III, we derive the approximate error proba-bilities of MPSK for the OC and the closed-form expression forordered mean eigenvalues in the presence of arbitrary numbersof interferers. Section IV presents the numerical results, andSection V concludes the paper.

II. SYSTEM MODEL

Consider an N -element antenna array and L cochannel in-terferers. The N -dimensional received signal vector is givenby

r(t) =√

PDuD sD (t) +L∑

k=1

√Pkuk sk (t) + n (1)

where sD (t) and sk (t) are the desired and the kth interfering sig-nals using MPSK modulation with E[s2

D (t)] = E[s2k (t)] = 1,

respectively. uD and uk are the N × 1 propagation vectorswith each component having unit mean power. n is the N × 1

0018-9545/$20.00 © 2006 IEEE

KWAK AND LEE: CLOSED-FORM EXPRESSIONS OF APPROXIMATE ERROR RATES FOR OC 159

additive noise vector with mean power σ2 on each antenna ele-ment. The propagation vectors and noise vector are assumed tobe mutually independent zero-mean complex Gaussian vectors.PD and Pk are the mean powers of the desired user and the kthinterferer, respectively.

Conditioned on the vectors uk , the N × N covariance matrixof the received interference-plus-noise can be written as

R =L∑

k=1

Pku∗ku

Tk + σ2I (2)

where I is the N × N identity matrix, and the symbols ( · )∗and (·)T denote conjugate and transpose, respectively. Note thatR varies with the channel fading rate, which is assumed tobe much slower than the symbol rate. The optimum weightingvector for the OC that maximizes the SINR at the array outputis w = αR−1u∗

D , where α does not affect the output SINR asan arbitrary constant [1]. Then, the maximum output SINR persymbol is given by

γs = PDuTDR−1u∗

D . (3)

Since R is a Hermitian matrix, R can be unitarily diago-nalized as R = UT ΛU∗, where U is a unitary matrix, andΛ = diagλ1, λ2, . . . , λN is a diagonal matrix of the or-dered eigenvalues of R with λ1 ≥ λ2 ≥ · · · ≥ λN . Lettingv = UuD = [vD,1 vD,2 · · · vD,N ]T , the output SINR γs

can be expressed as γs = PD

∑Ni=1 vi/λi , where vi = |vD,i |2.

Sincev has the same statistical properties ofuD by unitary trans-formation of U, vi follows a Chi-square distribution with twodegrees of freedom, i.e., fvi

(x) = exp(−x). Then, the momentgenerating function (MGF) of γs conditioned on the eigenvaluesλi, i = 1, 2, . . . , N is given by [3]–[10]

ΦOCγs

(s;λ1, λ2, . . . , λN ) = E

[exp

(N∑

i=1

PD vi

λis

)]

=N∏

i=1

(λi

λi − PD s

). (4)

Let m = minN,L and n = maxN,L. Since the N − msmallest eigenvalues of R are equal to the noise power σ2;i.e., λm+1 = λm+2 = · · · = λN = σ2, the conditional MGFbecomes

ΦOCγs

(s;λ) =(

σ2

σ2 − PD s

)N −m m∏i=1

(λi

λi − PD s

)(5)

where λ = [λ1λ2 · · ·λm ]T with λ1 ≥ λ2 ≥ · · · ≥ λm ≥ σ2.In order to derive the average SER of MPSK for the OC

with multiple cochannel interferers, it is required to evaluatethe exact joint distribution of eigenvalues in (5). However, itis difficult to have insight into the performance analysis ofthe OC with individual mean power Pk [6]. We assume thatmultiple cochannel interferers have equal average power PI ;then, the N × N random matrix PI

∑Lk=1 u∗

kuTk has the central

complex Wishart distribution [10], [11]. Consequently, the jointprobability density function (pdf) of the m distinct eigenvalues

λi, i = 1, 2, . . . ,m can be expressed as

fλ(λ) = A0

m∏i=1

PmI exp

(−λi − σ2

PI

)(λi − σ2

PI

)n−m

×m∏

j=i+1

(λi − λj )2 (6)

where A0 = (∏m

i=1 (n − i)!(m − i)!)−1.

III. APPROXIMATE ERROR RATE OF MPSK FOR THE OC

A. MGF-Based Approach

The average SER for the OC conditioned on λ is given by[12], [13]

POCs (λ) =

1π

∫ π−π/M

0

E[exp

(− gPSK

sin2 θγs

)∣∣∣λ] dθ

=1π

∫ π−π/M

0

ΦOCγs

(− gPSK

sin2 θ;λ)

dθ (7)

where gPSK = sin2(π/M). Then, the exact average SER ofMPSK for the OC can be obtained by averaging (7) with (6) andis given by

POCs,exact =

1π

∫ π−π/M

0

∫D

ΦOCγs

(− gPSK

sin2 θ;λ)

fλ(λ)(dλ)dθ

(8)

where D = σ2 ≤ λm ≤ λm−1 ≤ · · · ≤ λ1 ≤ ∞. In general,it is difficult to derive a closed-form expression for (8), and thenumerical integral, which usually requires excessive computa-tional time for large values of m, is required to evaluate theexact performance of MPSK for the OC [5].

To simplify the performance analysis of the OC, Pham andBalmain proposed a first-order approximation by using a Taylorseries expansion [14, p. 156] in which each of m distinct eigen-values is replaced by its mean value [8]. Then, the approximateMGF of maximum output SINR can be expressed as

ΦOCγs ,app(s) ∼= ΦOC

γs(s; λ1, λ2, . . . , λm ) (9)

where λq = E[λq ], q = 1, 2, . . . , m. From the partial fractionexpansion in [15, (6.27.12)], the average SER of MPSK for theOC can be expressed in terms of the average SER of MPSK forthe MRC, which has been studied in [12] and [13], and is givenby

POCe,app =

N −m∑i=1

BiPMRCe

(PD

σ2, i

)+

m∑j=1

Cj PMRCe

(PD

λj, 1)

(10)

where PMRCe (γ, p) is the average SER of MPSK for the MRC

with the average received SNR per symbol per branch γ, and thenumber of diversity branches p in the presence of no interfering

160 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 55, NO. 1, JANUARY 2006

signals [12], [13, p. 269]

Bi =lims→σ2/PD

dN −m −i

dsN −m −i

(1 − PD s/σ2)N −m ΦOC

γs ,app(s)

(−PD /σ2)N −m−i(N − m − i)!

(11)

and

Cj = lims→λ j /PD

(1 − PD s/λj )ΦOC

γs ,app(s)

. (12)

Using the finite binomial series and a variable substitution, wecan derive the average SER of the MRC as a closed-form ex-pression, which is given by [16, (16)]

B. Closed-Form Expression for Ordered Mean Eigenvalues

To analyze the error probability of MPSK for the OC in (10),it is required to obtain the individual mean values of m distincteigenvalues. From the exact joint pdf in (6), the m distinct meaneigenvalues can be expressed as [9], [10]

λq =∫ ∞

σ2

∫ ∞

λm

· · ·∫ ∞

λ2

λq · fλ(λ)dλ1 · · · dλm−1dλm (13)

for q = 1, 2, . . . ,m. The closed-form solutions of (13) havebeen presented in simple cases, such as m = 1 and 2 [9], [10].In general, the m-dimensional integral is evaluated numerically,and it requires more computational time than Monte Carlo simu-lation for large values of m. In this paper, we derive the closed-form expression of (13) in order to simply evaluate the exactordered mean eigenvalues in the presence of arbitrary numbersof interferers and antenna elements.

From the change of variables λq = σ2 + PI

∑ml=q xl , q =

1, 2, . . . ,m, λq in (13) can be written as

λq =∫ ∞

0

· · ·∫ ∞

0

∫ ∞

0

σ2 + PI

m∑l=q

xl

× fx(x) dx1 dx2 . . . dxm

= σ2 + PI

m∑l=q

∫ ∞

0

· · ·∫ ∞

0

∫ ∞

0

xlfx(x) dx1 dx2 . . . dxm

= σ2 + PI

m∑l=q

xl (14)

where x = [x1 x2 · · · xm ]T with 0 ≤ x1, x2, . . . , xm ≤∞, xl = E[xl ], and the joint pdf fx(x) is given by

fx(x) = A0

m∏i=1

exp

(−

m∑l=i

xl

)(m∑l=i

xl

)n−m

×m∏

j=i+1

(j−1∑l=i

xl

)2

= A0H1(x)H2(x)H3(x) (15)

where

H1(x) = exp

(−

m∑k=1

k · xk

)(16)

H2(x) =m∏

i=1

(m∑l=i

xl

)n−m

(17)

and

H3(x) =m−1∏i=1

m∏j=i+1

(j−1∑l=i

xl

)2

. (18)

In order to represent the joint pdf in (15) as a simple integrableformula, we first define the function hk (x) for k ≤ m, i.e.,

hk (x) =k∏

i=1

(k∑

l=i

xl

)

= xk (xk−1 + xk ) · · · (x1 + x2 + · · · + xk )

=ηka∑

j=1

g0

(ak

j

)x

ak1, j

1 xak2, j

2 . . . xak

m , jm

=ηka∑

j=1

g0(akj )t

(x,ak

j

)(19)

where akj = [ak

1,j , ak2,j , . . . , a

km,j ]

T is the jth column vector ofall possible ηk

a distinct vectors for nonnegative integer aki,j ,

satisfying the conditions

m∑i=1

aki,j = k (20)

0 ≤i0∑

i=1

aki,j ≤ i0 for i0 = 1, 2, . . . , k (21)

and

aki,j = 0 for i > k. (22)

In (19), t(x,akj ) = x

ak1, j

1 xak2, j

2 · · ·xakm , j

m , and the coefficientg0(ak

j ) of t(x,akj ) is given by

g0(akj ) =

(1

ak1,j

)(2 − ak

1,j

ak2,j

)· · ·

(k −

∑m−1i=1 ak

i,j

akm,j

)

=k∏

l=1

(l −

∑l−1i=1 ak

i,j

akl,j !

). (23)

Letting t(x,Ak )=[t(x,ak1)t(x,ak

2) · · · t(x,akηka)]T and g0(Ak )

= [g0(ak1)g0(ak

2) · · · g0(akηka)]T , where Ak = [ak

1ak2 · · · ak

ηka],

hk (x) can be expressed as

hk (x) = 〈g0(Ak ), t(x,Ak )〉 (24)

KWAK AND LEE: CLOSED-FORM EXPRESSIONS OF APPROXIMATE ERROR RATES FOR OC 161

where 〈·, ·〉 denotes the inner product operation. Then, H2(x)becomes

H2(x) = hm (x)n−m

=

ηm

a∑j=1

g0(amj )t(x,am

j )

n−m

=ηn , mb∑j=1

g1(bn−mj )

ηma∏

k=1

tbn −mk , j (x,am

k ) (25)

where bn−mj = [bn−m

1,j , bn−m2,j , . . . , bn−m

ηma ,j ]

T is the jth columnvector of all possible ηn,m

b distinct vectors for nonnegative in-teger bn−m

i,j , satisfying the condition

ηma∑

i=1

bn−mi,j = n − m for 0 ≤ bn−m

i,j ≤ n − m (26)

and the coefficient g1(bn−mj ) is given by

g1

(bn−m

j

)=

ηma∏

l=1

(n − m −

∑l−1i=1 bn−m

i,j

bn−ml,j

)

=(n − m)!∏ηm

a

l=1 bn−ml,j !

. (27)

If we let Bn,m = [bn−m1 bn−m

2 · · · bn−mηn , mb

] and g1(Bn,m ) =

[g1(bn−m1 )g1(bn−m

2 ) . . . , g1(bn−mηn , mb

)]T , then H2(x) can be ex-

pressed as

H2(x) =ηn , mb∑j=1

(g1

(bn−m

j

) ηma∏

l=1

g0 (aml )bn −m

l , j

)t(x, cj )

=ηn , mb∑j=1

g1

(bn−m

j

)t(g0(Am ),bn−m

j

)t(x, cj )

= 〈g(Bn,m ,Am ), t(x,Cn,mm )〉 (28)

where g(Bn,m ,Am ) = 〈g1(Bn,m ), t(g0(Am ),Bn,m )〉 and cj

is the jth column vector of Cn,mm = AmBn,m .

Next, H3(x) can be written as

H3(x) =

(m−1∏k=1

hk (x)

)2

=

ηm −1

e∑j=1

m−1∏l=1

g0

(al

em −1l , j

)t(x,al

em −1l , j

)2

=

ηm −1

e∑j=1

gA

(em−1

j

)t(x,dj )

2

(29)

where em−1j = [em−1

1,j em−12,j · · · em−1

m−1,j ]T is the jth column vec-

tor of all possible ηm−1e distinct vectors for positive integer

em−1i,j , satisfying the condition

1 ≤ em−1i,j ≤ ηi

a for i = 1, 2, . . . ,m − 1 (30)

and dj = [dm−11,j , dm−1

2,j , · · · , dm−1m−1,j , 0]T for dm−1

i,j =∑m−1

l=1

ali,em −1

l , j

, i = 1, 2, · · ·m − 1.

Let us define

Em−1 =[em−11 em−1

2 . . . em−1ηm −1e

](31)

Dm−1 =[d1d2 . . .dηm −1

e

](32)

and

gA(Em−1) =[gA

(em−11

)gA

(em−12

). . . gA

(em−1

ηm −1e

)]T

(33)

where gA(em−1j ) =

∏m−1l=1 g0(al

em −1l , j

); then, H3(x) simplifies

to

H3(x) = 〈gA(Em−1, ), t(x,Dm−1)〉2. (34)

Substituting (28) and (34) into (15) yields the final expressionfor fx(x1, x2, . . . , xm ) in (35), shown at the bottom of the nextpage, where Ωi,j,k = A0g(bn−m

i ,Am )gA(em−1j )gA(em−1

k )and pi,j,k = [pi,j,k

1 , pi,j,k2 , . . . , pi,j,k

m ]T = ci + dj + dk . Withthe help of the identity [17, (3.351.3)], the ordered mean eigen-value λq , q = 1, 2, . . . , m, can be expressed as (36), shown atthe bottom of the next page.

The closed-form solution in (36) gives a simple way toevaluate the exact ordered mean eigenvalue in the presence ofarbitrary numbers of interferers and antenna elements withoutthe Monte Carlo simulation and multiple numerical integrals.From the closed-form expression of ordered mean eigenvalues,we can simply obtain the approximate average SER of MPSKfor the OC in (10). Appendix A shows that, as an examplefor m = 3 and n = 5, the coefficient Ωi,j,k and column vectorpi,j,k in (36) can be explicitly computed, and some results ofthe ordered mean eigenvalues are given for different numbersof interferers and antenna elements.

IV. NUMERICAL RESULTS

In this section, we present some numerical results to illustratethe performance of the OC using first-order approximationby comparing with the results of orthogonal approximation inAppendix B, the upper bound in [10], and the Monte Carlosimulation. The performance of the MRC is also evaluated byusing the exact average SER in [16, (16)] to present the effects ofoptimum combining on average SER in the presence of multiplecochannel interferers. For plotting the performance of the M -arysignal, the average SINR per bit and the average INR per bit aredefined as ΓS,b = ΓS / log2 M , and ΓI ,b = ΓI / log2 M , respec-tively, where ΓI = PI /σ2 is the average interference-to-noiseratio (INR) per symbol and ΓS = ΓD /(1 + LΓI ) is the averageSINR per symbol with average SNR per symbol ΓD = PD /σ2.

Fig. 1 shows the average SER of MPSK versus average SINRper bit for several values of M with N = 3, L = 2, and ΓI ,b = 0dB. It is shown that the use of the OC considerably improvesthe system performance of MPSK, as shown for binary PSKin [1]. As compared with the average SERs using orthogonal

162 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 55, NO. 1, JANUARY 2006

Fig. 1. Average SER of MPSK versus average SINR per bit for several valuesof M with N = 3, L = 2, and ΓI ,b = 0 dB.

approximation and upper bound in [10], the average SER usingfirst-order approximation agrees with simulation results. Thus,from the simple evaluation of the individual mean eigenvaluesin (36), the first-order approximation is an efficient technique toevaluate the performance of the OC. Note that the performanceof the upper bound is close to that of orthogonal approxima-tion as shown in [10]. However, since the upper bound in [10]requires to evaluate the numerical integrals, the orthogonal ap-proximation is more proper than the upper bound in [10] toanalyze the performance of the OC in the case of N > L.

Fig. 2 shows the average SER of MPSK versus averageSINR per bit for several values of M with N = 3, L = 5, andΓI ,b = 0 dB. In Fig. 2, the average SER using the orthogonal

Fig. 2. Average SER of MPSK versus average SINR per bit for several valuesof M with N = 3, L = 5, and ΓI ,b = 0 dB.

approximation is equal to the average SER of the MRC as shownin Appendix B. There is a good agreement between the aver-age SER using first-order approximation, and the simulationresults in the case of N < L. It is shown that the performanceof upper bound is close to the performance of the MRC, so thatthe upper bound cannot reflect the effect of the OC when thedegrees-of-freedom (DOFs) are insufficient to suppress all in-terferers. From this, the upper bound is also limited to the case ofN > L.

Fig. 3 shows the average SER of MPSK for the OC versusaverage SINR per bit for several values of N with M = 4, L =2, and ΓI ,b = 0 dB. It is shown that the agreement between theaverage SER of the orthogonal approximation and simulation

fx(x1, x2, . . . , xm ) = A0 exp

(−

m∑k=1

k · xk

)⟨g(B,

n,mAm ), t (x,Cn,mm )

⟩〈gA(Em−1, ), t(x,Dm−1)〉2

= A0 exp

(−

m∑k=1

k · xk

) ηn , mb∑i=1

ηm −1e∑j=1

ηm −1e∑k=1

g(bn−m

i ,Am

)gA

(em−1

j

)gA

(em−1

k

)T (x, ci + dj + dk )

=ηn −mb∑i=1

ηm −1e∑j=1

ηm −1e∑k=1

(Ωi,j,k

m∏l=1

exp(−l · xl) · xpi , j , k

l

l

). (35)

λq = σ2 + PI ·m∑

r=q

∫ ∞

0

∫ ∞

0

· · ·∫ ∞

0

xlfx(x1, x2, . . . , xm ) dx1dx2 · · · dxm

= σ2 + PI ·m∑

r=q

ηn −mb∑i=1

ηm −1e∑j=1

ηm −1e∑k=1

Ωi,j,k

m∏l=1,l =r

∫ ∞

0

exp(−l · xl) · xpi , j , k

l

l dxl ·∫ ∞

0

exp(−l · xr ) · xpi , j , kr +1

r dxl

= σ2 + PI ·ηn −mb∑i=1

ηm −1e∑j=1

ηm −1e∑k=1

(Ωi,j,k

m∑r=q

m∏l=1

pi,j,kl !

lpi , j , kl

+1· pi,j,k

r + 1r

)(36)

KWAK AND LEE: CLOSED-FORM EXPRESSIONS OF APPROXIMATE ERROR RATES FOR OC 163

Fig. 3. Average SER of MPSK for OC versus average SINR per bit for severalvalues of N with M = 4, L = 2, and ΓI ,b = 0 dB.

Fig. 4. Average SER of MPSK versus average SINR per bit for several valuesof ΓI ,b with M = 4, N = 4, and L = 2.

results improves greatly as N − L increases. This is consistentwith the results in [7] for binary PSK, and the upper bound in [10]also has the same tendency of the orthogonal approximation.

Fig. 4 shows the average SER of MPSK versus average SINRper bit for several values of ΓI ,b with M = 4, N = 4, and L = 2.It is shown that the performance of the OC improves as theaverage INR per bit increases. This is because the OC can moreeasily cancel out the effect of the strong interference than thatof weak interference [1].

Table I shows the average SERs of MPSK using various ana-lytical approaches for the OC and the MRC with M = 4, N = 3,and ΓI ,b = 0 dB. The exact error probabilities are evaluatedfrom the numerical integration of (8) in order to investigate theaccuracy of the approximate error probabilities in the OC. It isshown that the first-order approximation provides the accurate

Fig. 5. Diversity gain of OC over MRC versus the number of antenna elementsN to achieve 0.1% average SER. M = 4, L = 3, and ΓI ,b = 3 dB.

average SER for the OC, compared with orthogonal approxima-tion and upper bound in [10].

Fig. 5 shows the diversity gain of the OC versus the numberof antenna elements N with M = 4, L = 3, and ΓI ,b = 3 dB.The diversity gain is defined as the difference in the requiredSINR per bit to achieve a 0.1% average SER between the OCand the MRC. It is shown that the gain for the first-order ap-proximation is very close to the gain for the exact analysis. Itis also shown that the discrepancy of the diversity gain betweenthe orthogonal approximation (or upper bound in [10]), and ex-act analysis is less than 0.8 dB for large numbers of antennaelements N ≥ 5.

V. CONCLUSION

In this paper, we investigated the performance of MPSK forthe OC with an arbitrary number of interferers in a flat Rayleighfading channel. We derived the closed-form expression for theordered mean eigenvalues for the first-order approximation toavoid Monte Carlo simulation and multiple numerical integrals.The approximate average SER of MPSK for the OC was alsoderived by using the first-order approximation and orthogonalapproximation in terms of the average SER of the MRC. Theaccuracy of the approximations was evaluated by comparingwith the results of the upper bound in [10] and simulation. Fromthe numerical examples, we showed that the first-order approxi-mation is an efficient technique to assess the performance of theOC, and the simple closed-form expression for the ordered meaneigenvalues provides a simple and accurate approximation forthe performance analysis of MPSK for the OC in the presenceof the arbitrary numbers of interferers and antenna elements.

APPENDIX A

EVALUATION OF ORDERED MEAN EIGENVALUES IN (36)

This Appendix shows that the ordered mean eigenvalues canbe evaluated simply from (36) for arbitrary values of m and n.

164 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 55, NO. 1, JANUARY 2006

TABLE ISYMBOL ERROR RATES USING VARIOUS ANALYTICAL METHODS FOR OC AND MRC WITH M = 4, N = 3, AND ΓI ,b = 0 dB

In the case of m = 3 and n = 5, e.g., A1,A2, and A3 withη1a = 1, η2

a = 2, and η3a = 5 satisfying the conditions (20)–(22)

can be obtained as

A1 =

1

00

, A2 =

1 0

1 20 0

, and

A3 =

0 0 1 0 1

0 1 0 2 13 2 2 1 1

(37)

respectively. The coefficient column vectors g0(Ak ) for k =1, 2, 3 are easily calculated as g0(A1) = 1,g0(A2) = [1 1]T ,and g0(A3) = [1 2 1 1 1]T . From the condition in (26), B5,3

with η5,3b = 15 is given in (38) at the bottom of the page. The

coefficient column vector g1(B5,3) is also given in (39) at thebottom of the page. Then, for H2(x) in (28), we obtain (40) and(41), shown at the bottom of the next page. For H3(x) in (34),E2 and D2 in (31) and (32) are given by

E2 =[

1 11 2

]and D2 =

[2 1 01 2 0

]T

. (42)

The coefficient column vector gA(E2) in (33) is expressed as

gA(E2) =[gA

(e21

)gA

(e22

)· · · gA

(e2

η2e

)]T

= [1 1]T .

(43)

Since Ωi,j,k and pi,j,k in (35) is easily obtained byg(B5,3,A3),gA(E2),C

5,33 , and D2, the ordered mean eigen-

values can be exactly evaluated as

[λ1 λ2 λ3] = σ2 +[1332815139968

582065139968

28852187

]PI .

(44)

Using the similar approach for m = 3 and n = 5, Table IIshows some results for ordered mean eigenvalues in other casesof m and n. From (36), the exact ordered mean eigenvalues canbe simply evaluated without multiple numerical integrals andMonte Carlo simulation for arbitrary numbers of interferers andantenna elements.

B5,3 =

2 0 0 0 0 1 1 1 1 0 0 0 0 0 00 2 0 0 0 1 0 0 0 1 1 1 0 0 00 0 2 0 0 0 1 0 0 1 0 0 1 1 00 0 0 2 0 0 0 1 0 0 1 0 1 0 10 0 0 0 2 0 0 0 1 0 0 1 0 1 1

. (38)

g1(B5,3) =[g1

(b2

1

)g1

(b2

2

)· · · g1

(b2

η5,3b

)]T

= [1 1 1 1 1 2 2 2 2 2 2 2 2 2 2]T . (39)

KWAK AND LEE: CLOSED-FORM EXPRESSIONS OF APPROXIMATE ERROR RATES FOR OC 165

TABLE II(λq − σ2)/PI , q = 1, 2, . . . , m FOR EXACT m DISTINCT MEAN EIGENVALUES

APPENDIX B

AVERAGE SER OF MPSK FOR THE OC USING

ORTHOGONAL APPROXIMATION

The first-order approximation can be used to evaluate the per-formance of the OC in the case of arbitrary N and L, while theorthogonal approximation, which has been proposed by Villier,can be used when the number of interferers is less than thenumber of antenna elements, i.e., N > L [7]. The orthogonalapproximation is a simple method, wherein the L distinct ran-dom eigenvalues are replaced by a fixed λ = σ2 + NPI . Then,the MGF of maximum SINR for the orthogonal approximationbecomes

ΦOCγs ,OA(s) ∼= ΦOC

γs(s;λ1, λ2, . . . , λL )|λq =λ, q=1,2,...,L

=(

σ2

σ2 − PD s

)N −L (λ

λ − PD s

)L

. (45)

From a partial fraction expansion [18, (3.3)], the average SERof MPSK for the OC using the orthogonal approximation canbe evaluated as

POCs,OA

=L∑

i=1

(−1 − NΓI )L−i

(−NΓI )N −i

(N − i − 1

L − i

)PMRC

s (Γd , i)

+N −L∑j=1

(−1 − NΓI )L

(−NΓI )N −j

(N − j − 1

L − 1

)PMRC

s (Γd , j)

(46)

where Γd = Γd/(1 + NΓI ). From the results of the orderedmean eigenvalues in Appendix A, the orthogonal approximationuses the nonordered mean eigenvalue of R, which is the average

value of L distinct mean eigenvalues as λ =∑L

k=1 λk /L due to∑Lk=1(λk − σ2) = LNPI [10]. As the extension of the results

from orthogonal approximation to the case of N ≤ L, λ can beexpressed as λ = σ2 + nPI [10]. For N ≤ L, the approximateMGF is given by

ΦOCγs ,OA(s) =

(λ

λ − PD s

)N

=(

11 − ΓS s

)N

= ΦMRCγs

(s; ΓS ,N). (47)

From (47), however, the performance of the OC using the or-thogonal approximation is equal to the performance of the MRCso that the orthogonal approximation cannot represent the effectof interference rejection from the OC in case of N ≤ L. Thus,the orthogonal approximation in [7] has been applied to the per-formance analysis of the OC when the number of interferers isless than the number of antenna elements.

REFERENCES

[1] J. H. Winters, “Optimum combining of signals in space-diversity recep-tion,” IEEE J. Select. Areas Commun., vol. 2, no. 4, pp. 528–539, Jul.1984.

[2] J. Cui, D. D. Falconer, and A. U. Sheikh, “Performance evaluation ofoptimum combining and maximal ratio combining in the presence of co-channel interference and channel correlation for wireless communicationsystems,” Mobile Netw. Appl., vol. 2, pp. 315–324, 1997.

[3] V. A. Aalo and J. Zhang, “Performance of antenna array systems withoptimum combining in a Rayleigh fading environment,” IEEE Commun.Lett., vol. 4, no. 12, pp. 387–389, Dec. 2000.

[4] A. Shah, A. M. Haimovich, M. K. Simon, and M. Alouini, “Exact bit-error probability for optimum combining with a Rayleigh fading Gaussiancochannel interferer,” IEEE Trans. Commun., vol. 48, no. 6, pp. 908–912,Jun. 2000.

[5] M. Chiani, M. Z. Win, A. Zanella, and J. H. Winters, “Exact symbolerror probability for optimum combining in the presence of multiple co-channel interferes and thermal noise,” Proc. IEEE GLOBECOM 2001,pp. 1182–1186, San Antonio, TX, Nov. 2001.

C5,33 = A3B5,3

=

0 0 2 0 2 0 1 0 1 1 0 1 1 2 1

0 2 0 4 2 1 0 2 1 1 3 2 2 1 36 4 4 2 2 5 5 4 4 4 3 3 3 3 2

(40)

and

g(B5,3,A3) = 〈g1(B5,3), t(g0(A3),B5,3)〉= [1 4 1 1 1 4 2 2 2 4 4 4 2 2 2]T . (41)

166 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 55, NO. 1, JANUARY 2006

[6] A. Shah and A. M. Haimovich, “Performance analysis of optimum com-bining in wireless communications with Rayleigh fading and cochannelinterference,” IEEE Trans. Commun., vol. 46, no. 4, pp. 473–479, Apr.1998.

[7] E. Villier, “Performance analysis of optimum combining with multipleinterferers in flat Rayleigh fading,” IEEE Trans. Commun., vol. 47, no. 10,pp. 1503–1510, Oct. 1999.

[8] T. D. Pham and K. G. Balmain, “Multipath performance of adaptive an-tennas with multiple interferers and correlated fadings,” IEEE Trans. Veh.Technol., vol. 48, no. 2, pp. 342–352, Mar. 1999.

[9] J. S. Kwak and J. H. Lee, “Performance analysis of optimum combiningfor dual-antenna diversity with multiple interferers in a Rayleigh fadingchannel,” IEEE Commun. Lett., vol. 6, no. 12, pp. 541–543, Dec. 2002.

[10] M. Chiani, M. Z. Win, A. Zanella, R. K. Mallik, and J. H. Winters, “Boundsand approximations for optimum combining of signals in the presence ofmultiple cochannel interferers and thermal noise,” IEEE Trans. Commun.,vol. 51, no. 2, pp. 296–307, Feb. 2003.

[11] A. T. James, “Distribution of matrix variates and latent roots derived fromnormal samples,” Ann. Math. Statist., vol. 35, pp. 475–501, 1964.

[12] A. Annamalai and C. Tellambura, “Error rates for Nakagami-m fadingmultichannel reception of binary and M -ary signals,” IEEE Trans. Com-mun., vol. 49, no. 1, pp. 58–68, Jan. 2001.

[13] M. K. Simon and M.-S. Alouini, Digital Communication over FadingChannels: A Unified Approach to Performance Analysis. New York:Wiley, 2000.

[14] A. Papoulis, Probability, Random Variables, and Stochastic Processes.New York: McGraw-Hill, 1991.

[15] D. Zwillinger, CRC Standard Mathematical Tables and Formulae. BocaRaton, FL: CRC, 1996.

[16] J. S. Kwak and J. H. Lee, “Approximate error probability of M -ary PSKfor optimum combining with arbitrary number of interferers in a Rayleighfading channel,” IEICE Trans. Commun., vol. E86-B, pp. 3544–3550,Dec. 2003.

[17] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products.New York: Academic, 2000.

[18] A. Cuyt, K. Driver, J. Tan, and B. Verdonk, “A finite sum representation ofthe Appell series f1(a, b, b′; c;x, y),” J. Comput. Appl. Math., vol. 105,pp. 213–219, 1999.

Jin Sam Kwak (S’98–M’04) received the B.S., M.S.,and Ph.D. degrees in electrical engineering and com-puter science from Seoul National University (SNU),Seoul, Korea, in 1998, 2000, and 2004, respectively.

From October 2004 to September 2005, he wasa post-doctoral research associate with the School ofElectrical and Computer Engineering, Georgia Insti-tute of Technology, Atlanta. Currently he is with theUniversity of Texas at Austin as a post-doctoral re-search fellow with the Department of Electrical andComputer Engineering. His research interests include

most areas of wireless communication systems, especially MIMO with interfer-ence, adaptive antennas, space-time coding, and multicarrier transmission.

Jae Hong Lee (M’86–SM’03) received the B.S. andM.S. degrees in electronics engineering from SeoulNational University (SNU), Seoul, Korea, in 1976and 1978, respectively, and the Ph.D. degree in elec-trical engineering from the University of Michigan,Ann Arbor, in 1986.

From 1978 to 1981 he was with the Departmentof Electronics Engineering, Republic of Korea NavalAcademy, Jinhae, as an Instructor. In 1987, he joinedthe faculty of SNU. He was a member of technicalstaff at AT&T Bell Laboratories, Whippany, NJ, from

1991 to 1992. Currently, he is with SNU as a Professor in the School of Elec-trical Engineering. His current research interests include communication andcoding theory, space-time code, multiple-input multiple-output (MIMO), andorthogonal frequency-division multiplexing (OFDM), and their application towireless communications.

Dr. Lee is a Vice President of the Institute of Electronics Engineers of Koreaand the Korea Society of Broadcasting Engineers, and a member of Tau Beta Pi.