Closed-Form Expressions of Approximate Error ... - Semantic Scholar
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Closed-Form Expressions of Approximate Error Rates for Optimum Combining With Multiple Interferers in a Rayleigh Fading Channel Jin 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) with multiple cochannel interferers in a flat Rayleigh fading channel. For the first-order approximation, we derive the closed-form expression for ordered mean eigenvalues of the interference-plusnoise covariance matrix, which facilitates performance evaluation for the OC with arbitrary numbers of interferers and antenna elements without Monte Carlo simulation and multiple numerical integrals. We also derive the closed-form expressions for approximate error rates of MPSK for the OC in terms of the average error rate of MPSK for maximal ratio combining (MRC). From the simple evaluation of ordered mean eigenvalues, we show that the first-order approximation gives a simple and accurate way to analyze the performance of the OC. Index Terms—Adaptive antennas, co-channel interference, M -ary phase shift keying (MPSK), optimum combining (OC).
I. INTRODUCTION DAPTIVE antenna arrays significantly improve both the performance and capacity of wireless communication systems by mitigating multipath fading and suppressing interfering signals [1]. The optimum combiner (OC) which maximizes the signal-to-interference-plus-noise ratio (SINR) yields better performance than that of maximal ratio combiner (MRC) in the interference-limited system. In the absence of interference for an additive noise environment, the OC maximizes the signal-to-noise ratio (SNR) and has the same performance as the MRC [1], [2]. In a Rayleigh fading environment, the exact average bit error rate (BER) performance of the OC for binary phase-shift-keying (BPSK) has been studied [2]–[4]. The closed-form expression for the average BER was derived for the OC in the presence of the arbitrary number of interferes [4]. For M -ary phase-shiftkeying (MPSK), the average symbol error rate (SER) have been expressed as a multidimensional integral that can be a burden to be evaluated numerically [5] or as a simple closed-form upper bound that may be loose for high SER [6]. On the other hand, approximation techniques have been studied to simply evaluate the performance analysis of the OC
A
Manuscript received April 16, 2003; revised December 23, 2004. This work was supported in part by the Brain Korea 21 Project and in part by the IT scholarship program from MIC and IITA, Korea. The review of this paper was coordinated by Prof. H. Leib. J. S. Kwak is with the Department of Electrical and Computer Engineering, the University of Texas at Austin, Austin, TX 78712-0240 USA (e-mail: [email protected]). J. H. Lee is with School of Electrical Engineering and Computer Science, Seoul National University, Seoul 151-742 Korea (e-mail: [email protected]). Digital Object Identifier 10.1109/TVT.2005.861184
[7]–[10]. Villier proposed orthogonal approximation to determine the average BER of several binary modulation schemes [7]. However, the orthogonal approximation only takes into account the case where the number of antenna elements is larger than that of interferers, and the accuracy of approximation depends on the difference between the number of interferers and the number of antenna elements [7]. In [8], Pham and Balmain proposed a first-order approximation and derived a closed-form expression of average BER for BPSK with an arbitrary number of interferers. This approximation provides accurate results for the performance of the OC, whereas Monte Carlo simulation is needed to evaluate the ordered mean eigenvalues of the interference-plus-noise covariance matrix [8]. Recently, the analytical expressions for the ordered mean eigenvalues have been presented as the multidimensional integral, which can be derived as a closed-form solution in the simple cases such as dual-antenna reception or two cochannel interferers [9], [10]. In this paper, we derive the simple closed-form expression of the ordered mean eigenvalues to avoid Monte Carlo simulation and multiple numerical integrals in the presence of arbitrary numbers of interferers and antenna elements. As the extended results of the performance analysis in [7] and [8], the approximate average SERs of MPSK for the OC are also derived in terms of the average SER of MPSK for the MRC. The analytical results of the approximation using ordered mean eigenvalues provide a simple and accurate way to assess the performance of the OC with multiple cochannel interferers. The paper is organized as follows. Section II gives the system model. In Section III, we derive the approximate error probabilities of MPSK for the OC and the closed-form expression for ordered mean eigenvalues in the presence of arbitrary numbers of interferers. Section IV presents the numerical results, and Section V concludes the paper. II. SYSTEM MODEL Consider an N -element antenna array and L cochannel interferers. The N -dimensional received signal vector is given by r(t) =
L PD uD sD (t) + Pk uk sk (t) + n
(1)
k =1
where sD (t) and sk (t) are the desired and the kth interfering signals using MPSK modulation with E[s2D (t)] = E[s2k (t)] = 1, respectively. uD and uk are the N × 1 propagation vectors with each component having unit mean power. n is the N × 1
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KWAK AND LEE: CLOSED-FORM EXPRESSIONS OF APPROXIMATE ERROR RATES FOR OC
additive noise vector with mean power σ 2 on each antenna element. The propagation vectors and noise vector are assumed to be mutually independent zero-mean complex Gaussian vectors. PD and Pk are the mean powers of the desired user and the kth interferer, respectively. Conditioned on the vectors uk , the N × N covariance matrix of the received interference-plus-noise can be written as R=
L
Pk u∗k uTk + σ 2 I
(2)
159
λi , i = 1, 2, . . . , m can be expressed as fλ (λ) = A0
n −m λi − σ 2 λi − σ 2 PIm exp − PI PI i=1 m
m
×
(λi − λj )2
(6)
j =i+1
−1 where A0 = ( m i=1 (n − i)!(m − i)!) .
k =1
where I is the N × N identity matrix, and the symbols ( · )∗ and (·)T denote conjugate and transpose, respectively. Note that R varies with the channel fading rate, which is assumed to be much slower than the symbol rate. The optimum weighting vector for the OC that maximizes the SINR at the array output is w = αR−1 u∗D , where α does not affect the output SINR as an arbitrary constant [1]. Then, the maximum output SINR per symbol is given by γs = PD uTD R−1 u∗D .
=
i=1
λi λ i − PD s
A. MGF-Based Approach The average SER for the OC conditioned on λ is given by [12], [13] 1 P¯sOC (λ) = π
(3)
Since R is a Hermitian matrix, R can be unitarily diagonalized as R = UT ΛU∗ , where U is a unitary matrix, and Λ = diag{λ1 , λ2 , . . . , λN } is a diagonal matrix of the ordered eigenvalues of R with λ1 ≥ λ2 ≥ · · · ≥ λN . Letting v = UuD = [vD ,1 vD ,2 · · · vD ,N ]T , the output SINR γs 2 can be expressed as γs = PD N i=1 vi /λi , where vi = |vD ,i | . Since v has the same statistical properties of uD by unitary transformation of U, vi follows a Chi-square distribution with two degrees of freedom, i.e., fv i (x) = exp(−x). Then, the moment generating function (MGF) of γs conditioned on the eigenvalues λi , i = 1, 2, . . . , N is given by [3]–[10] N P D vi OC Φγ s (s; λ1 , λ2 , . . . , λN ) = E exp s λi i=1 N
III. APPROXIMATE ERROR RATE OF MPSK FOR THE OC
.
(4)
Let m = min{N, L} and n = max{N, L}. Since the N − m smallest eigenvalues of R are equal to the noise power σ 2 ; i.e., λm +1 = λm +2 = · · · = λN = σ 2 , the conditional MGF becomes
N −m
m σ2 λi ΦOC (s; λ) = (5) γs σ 2 − PD s λ i − PD s i=1 where λ = [λ1 λ2 · · · λm ]T with λ1 ≥ λ2 ≥ · · · ≥ λm ≥ σ 2 . In order to derive the average SER of MPSK for the OC with multiple cochannel interferers, it is required to evaluate the exact joint distribution of eigenvalues in (5). However, it is difficult to have insight into the performance analysis of the OC with individual mean power Pk [6]. We assume that multiple cochannel interferers haveequal average power PI ; then, the N × N random matrix PI Lk=1 u∗k uTk has the central complex Wishart distribution [10], [11]. Consequently, the joint probability density function (pdf) of the m distinct eigenvalues
1 π
=
π −π /M
0
π −π /M
0
g PSK E exp − 2 γs λ dθ sin θ g PSK − 2 ; λ dθ (7) ΦOC γs sin θ
where gPSK = sin2 (π/M ). Then, the exact average SER of MPSK for the OC can be obtained by averaging (7) with (6) and is given by 1 OC P¯s,exact = π
π −π /M
0
D
g PSK − 2 ; λ fλ (λ)(dλ)dθ ΦOC γs sin θ (8)
where D = {σ 2 ≤ λm ≤ λm −1 ≤ · · · ≤ λ1 ≤ ∞}. In general, it is difficult to derive a closed-form expression for (8), and the numerical integral, which usually requires excessive computational time for large values of m, is required to evaluate the exact performance of MPSK for the OC [5]. To simplify the performance analysis of the OC, Pham and Balmain proposed a first-order approximation by using a Taylor series expansion [14, p. 156] in which each of m distinct eigenvalues is replaced by its mean value [8]. Then, the approximate MGF of maximum output SINR can be expressed as ∼ OC ¯ ¯ ¯ ΦOC γ s ,app (s) = Φγ s (s; λ1 , λ2 , . . . , λm )
(9)
¯ q = E[λq ], q = 1, 2, . . . , m. From the partial fraction where λ expansion in [15, (6.27.12)], the average SER of MPSK for the OC can be expressed in terms of the average SER of MPSK for the MRC, which has been studied in [12] and [13], and is given by OC P¯e,app =
N −m i=1
Bi P¯eMRC
m PD PD MRC ¯ P , i + C , 1 j e ¯j σ2 λ j =1 (10)
where P¯eMRC (γ, p) is the average SER of MPSK for the MRC with the average received SNR per symbol per branch γ, and the number of diversity branches p in the presence of no interfering
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signals [12], [13, p. 269]
where
d N −m −i (1 − PD s/σ 2 )N −m ΦOC lims→σ 2 /P D ds N −m −i γ s ,app (s) Bi = (−PD /σ 2 )N −m −i (N − m − i)!
H1 (x) = exp −
Cj =
lim
¯ j /P D s→λ
¯ j )ΦOC (s) . (1 − PD s/λ γ s ,app
H2 (x) = (12)
To analyze the error probability of MPSK for the OC in (10), it is required to obtain the individual mean values of m distinct eigenvalues. From the exact joint pdf in (6), the m distinct mean eigenvalues can be expressed as [9], [10] ∞ ∞ ∞ ¯ ··· λq · fλ (λ)dλ1 · · · dλm −1 dλm (13) λq = σ2
λm
∞
¯q = λ
∞
···
0
0
∞
σ 2 + PI
m
0
i=1
m −1
m
xl
k
=
ηa
m j =i+1
xl
.
(18)
l=i
ak ak ak g0 akj x1 1, j x2 2, j . . . xmm , j
k
=
ηa
g0 (akj )t x, akj
(19)
j =1
where akj = [ak1,j , ak2,j , . . . , akm ,j ]T is the jth column vector of all possible ηak distinct vectors for nonnegative integer aki,j , satisfying the conditions m
aki,j = k
(20)
i0
aki,j ≤ i0
for i0 = 1, 2, . . . , k
(21)
and
∞
xl fx (x) dx1 dx2 . . . dxm
m
xl
m
l=i
j −1
2
i=1
(14)
exp −
j −1
j =1
0≤
x ¯l
i=1
(17)
= xk (xk −1 + xk ) · · · (x1 + x2 + · · · + xk )
xl
where x = [x1 x2 · · · xm ]T with 0 ≤ x1 , x2 , . . . , xm ≤ ∞, x ¯l = E[xl ], and the joint pdf fx (x) is given by
×
l=i
l=i
l=q
fx (x) = A0
i=1
i=1
0
(16)
In order to represent the joint pdf in (15) as a simple integrable formula, we first define the function hk (x) for k ≤ m, i.e., k k hk (x) = xl
aki,j = 0 for i > k. ak
m
k · xk
n −m
i=1 j =i+1
l=q
0
l=q
= σ + PI
m
0
× fx (x) dx1 dx2 . . . dxm ∞ m ∞ 2 = σ + PI ···
2
H3 (x) =
λ2
for q = 1, 2, . . . , m. The closed-form solutions of (13) have been 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 simulation for large values of m. In this paper, we derive the closedform expression of (13) in order to simply evaluate the exact ordered mean eigenvalues in the presence of arbitrary numbers of interferers and antenna elements. From the change of variables λq = σ 2 + PI m l=q xl , q = ¯ q in (13) can be written as 1, 2, . . . , m, λ
m m
and
Using the finite binomial series and a variable substitution, we can derive the average SER of the MRC as a closed-form expression, which is given by [16, (16)] B. Closed-Form Expression for Ordered Mean Eigenvalues
k =1
(11) and
m
n −m xl
l=i
ak
In (19), t(x, akj ) = x1 1, j x2 2, j · · · xmm , j , and the coefficient g0 (akj ) of t(x, akj ) is given by −1 k
2 − ak1,j k− m 1 k i=1 ai,j g0 (aj ) = ··· ak1,j ak2,j akm ,j k k l − l−1 i=1 ai,j = . (23) akl,j ! l=1 Letting t(x, Ak )=[t(x, ak1 )t(x, ak2 ) · · · t(x, akη k )]T and g0 (Ak ) a = [g0 (ak1 )g0 (ak2 ) · · · g0 (akη k )]T , where Ak = [ak1 ak2 · · · akη k ], a a hk (x) can be expressed as
2 xl
l=i
= A0 H1 (x)H2 (x)H3 (x)
ak
(22)
(15)
hk (x) = g0 (Ak ), t(x, Ak )
(24)
KWAK AND LEE: CLOSED-FORM EXPRESSIONS OF APPROXIMATE ERROR RATES FOR OC
where ·, · denotes the inner product operation. Then, H2 (x) becomes
−1 m −1 m −1 −1 T for dm = and dj = [dm 1,j , d2,j , · · · , dm −1,j , 0] i,j l ai,e m −1 , i = 1, 2, · · · m − 1.
Let us define
=
g1 (bnj −m )
j =1
t
b nk ,−m j
(x, am k )
(25)
k =1
T where bnj −m = [bn1,j−m , bn2,j−m , . . . , bnη am−m is the jth column ,j ] n ,m vector of all possible ηb distinct vectors for nonnegative integer bni,j−m , satisfying the condition η am
bni,j−m
=n−m
for 0 ≤
bni,j−m
≤n−m
(26)
i=1
and the coefficient
g1 (bnj −m )
g1 bnj −m =
η am
l=1
is given by l−1
n − m − i=1 bni,j−m bnl,j−m
(n − m)! = η am n −m . l=1 bl,j !
(27)
If we let Bn ,m = [bn1 −m bn2 −m · · · bnη n−m , m ] and g1 (Bn ,m ) = b
T [g1 (bn1 −m )g1 (bn2 −m ) . . . , g1 (bnη n−m , m )] , then H2 (x) can be exb pressed as n ,m ηb η am n −m m b nl , −m j {g0 (al )} g1 bj t(x, cj ) H2 (x) = j =1 n ,m ηb
=
l=1
g1 bnj −m t g0 (Am ), bnj −m t(x, cj )
j =1
= g(Bn ,m , Am ), t(x, Cnm,m )
=
k =1 η em −1 m −1
g0
j =1 l=1
=
η em −1
gA
ale m −1 l,j
2 l t x, ae m −1 l,j
2 −1 em t(x, dj ) j
(29)
j =1 −1 −1 m −1 m −1 T where em = [em j 1,j e2,j · · · em −1,j ] is the jth column vector of all possible ηem −1 distinct vectors for positive integer −1 em i,j , satisfying the condition −1 1 ≤ em ≤ ηai i,j
(31)
Dm −1 = d1 d2 . . . dη em −1
(32)
and
T
−1 −1 −1 gA (Em −1 ) = gA em gA em . . . gA em m −1 1 2 η e
−1 where gA (em )= j to
m −1 l=1
(33) g0 (ale m −1 ); then, H3 (x) simplifies l,j
H3 (x) = gA (Em −1 , ), t(x, Dm −1 )2 .
(34)
Substituting (28) and (34) into (15) yields the final expression for fx (x1 , x2 , . . . , xm ) in (35), shown at the bottom of the next −1 −1 )gA (em ) page, where Ωi,j,k = A0 g(bni −m , Am )gA (em j k i,j,k i,j,k T and pi,j,k = [p1 , p2 , . . . , pi,j,k ] = c + d + d . With i j k m the help of the identity [17, (3.351.3)], the ordered mean eigen¯ q , q = 1, 2, . . . , m, can be expressed as (36), shown at value λ the bottom of the next page. The closed-form solution in (36) gives a simple way to evaluate the exact ordered mean eigenvalue in the presence of arbitrary numbers of interferers and antenna elements without the 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 MPSK for the OC in (10). Appendix A shows that, as an example for m = 3 and n = 5, the coefficient Ωi,j,k and column vector pi,j,k in (36) can be explicitly computed, and some results of the ordered mean eigenvalues are given for different numbers of interferers and antenna elements.
(28)
where g(Bn ,m , Am ) = g1 (Bn ,m ), t(g0 (Am ), Bn ,m ) and cj is the jth column vector of Cnm,m = Am Bn ,m . Next, H3 (x) can be written as m −1 2 hk (x) H3 (x) =
l=1
−1 m −1 −1 Em −1 = em e2 . . . em m −1 1 η e
j =1 η am
m −1
l,j
H2 (x) = {hm (x)}n −m m n −m ηa m = g0 (am j )t(x, aj ) n ,m ηb
161
for i = 1, 2, . . . , m − 1
(30)
IV. NUMERICAL RESULTS In this section, we present some numerical results to illustrate the performance of the OC using first-order approximation by comparing with the results of orthogonal approximation in Appendix B, the upper bound in [10], and the Monte Carlo simulation. The performance of the MRC is also evaluated by using the exact average SER in [16, (16)] to present the effects of optimum combining on average SER in the presence of multiple cochannel interferers. For plotting the performance of the M -ary signal, the average SINR per bit and the average INR per bit are defined as ΓS,b = ΓS / log2 M , and ΓI ,b = ΓI / log2 M , respectively, where ΓI = PI /σ 2 is the average interference-to-noise ratio (INR) per symbol and ΓS = ΓD /(1 + LΓI ) is the average SINR per symbol with average SNR per symbol ΓD = PD /σ 2 . Fig. 1 shows the average SER of MPSK versus average SINR per bit for several values of M with N = 3, L = 2, and ΓI ,b = 0 dB. It is shown that the use of the OC considerably improves the system performance of MPSK, as shown for binary PSK in [1]. As compared with the average SERs using orthogonal
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Fig. 1. Average SER of MPSK versus average SINR per bit for several values of M with N = 3, L = 2, and ΓI , b = 0 dB.
Fig. 2. Average SER of MPSK versus average SINR per bit for several values of M with N = 3, L = 5, and ΓI , b = 0 dB.
approximation and upper bound in [10], the average SER using first-order approximation agrees with simulation results. Thus, from the simple evaluation of the individual mean eigenvalues in (36), the first-order approximation is an efficient technique to evaluate the performance of the OC. Note that the performance of the upper bound is close to that of orthogonal approximation as shown in [10]. However, since the upper bound in [10] requires to evaluate the numerical integrals, the orthogonal approximation is more proper than the upper bound in [10] to analyze the performance of the OC in the case of N > L. Fig. 2 shows the average SER of MPSK versus average SINR per bit for several values of M with N = 3, L = 5, and ΓI ,b = 0 dB. In Fig. 2, the average SER using the orthogonal
approximation is equal to the average SER of the MRC as shown in Appendix B. There is a good agreement between the average SER using first-order approximation, and the simulation results in the case of N < L. It is shown that the performance of upper bound is close to the performance of the MRC, so that the upper bound cannot reflect the effect of the OC when the degrees-of-freedom (DOFs) are insufficient to suppress all interferers. From this, the upper bound is also limited to the case of N > L. Fig. 3 shows the average SER of MPSK for the OC versus average 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 the average SER of the orthogonal approximation and simulation
fx (x1 , x2 , . . . , xm ) = A0 exp −
m
k · xk
g(B,n ,m Am ), t (x, Cnm,m ) gA (Em −1 , ), t(x, Dm −1 )2
k =1
= A0 exp −
η n , m η m −1 η m −1 b e e −1 −1 gA em T (x, ci + dj + dk ) k · xk g bni −m , Am gA em j k
m
i=1 j =1 k =1
k =1 n −m ηb
=
m r =q
Ωi,j,k
j =1 k =1 ∞
0 n −m
= σ 2 + PI ·
η em −1
i=1
¯ q = σ 2 + PI · λ
η em −1
∞
···
0
∞
= σ + PI ·
i=1 j =1 k =1
.
(35)
xl fx (x1 , x2 , . . . , xm ) dx1 dx2 · · · dxm
m −1 m −1
exp(−l · xl ) ·
0
ηe m η b e η
n −m m −1 m −1 ηb ηe ηe
pi , j , k xl l
l=1
Ωi,j,k
r =q i=1 j =1 k =1
2
m
Ωi,j,k
m
∞
exp(−l · xl ) ·
l=1,l =r 0 m m pi,j,k ! l i,j,k
r =q l=1
lp l
+1
pi,j,k + 1 · r r
pi , j , k xl l
dxl ·
∞
p ir , j , k
exp(−l · xr ) · xr
+1
dxl
0
(36)
KWAK AND LEE: CLOSED-FORM EXPRESSIONS OF APPROXIMATE ERROR RATES FOR OC
Fig. 3. Average SER of MPSK for OC versus average SINR per bit for several values of N with M = 4, L = 2, and ΓI , b = 0 dB.
163
Fig. 5. Diversity gain of OC over MRC versus the number of antenna elements N to achieve 0.1% average SER. M = 4, L = 3, and ΓI , b = 3 dB.
average SER for the OC, compared with orthogonal approximation and upper bound in [10]. Fig. 5 shows the diversity gain of the OC versus the number of antenna elements N with M = 4, L = 3, and ΓI ,b = 3 dB. The diversity gain is defined as the difference in the required SINR per bit to achieve a 0.1% average SER between the OC and the MRC. It is shown that the gain for the first-order approximation is very close to the gain for the exact analysis. It is also shown that the discrepancy of the diversity gain between the orthogonal approximation (or upper bound in [10]), and exact analysis is less than 0.8 dB for large numbers of antenna elements N ≥ 5. V. CONCLUSION
Fig. 4. Average SER of MPSK versus average SINR per bit for several values of ΓI , b with M = 4, N = 4, and L = 2.
results improves greatly as N − L increases. This is consistent with 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 SINR per 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 the average INR per bit increases. This is because the OC can more easily cancel out the effect of the strong interference than that of weak interference [1]. Table I shows the average SERs of MPSK using various analytical approaches for the OC and the MRC with M = 4, N = 3, and ΓI ,b = 0 dB. The exact error probabilities are evaluated from the numerical integration of (8) in order to investigate the accuracy of the approximate error probabilities in the OC. It is shown that the first-order approximation provides the accurate
In this paper, we investigated the performance of MPSK for the OC with an arbitrary number of interferers in a flat Rayleigh fading channel. We derived the closed-form expression for the ordered mean eigenvalues for the first-order approximation to avoid Monte Carlo simulation and multiple numerical integrals. The approximate average SER of MPSK for the OC was also derived by using the first-order approximation and orthogonal approximation in terms of the average SER of the MRC. The accuracy of the approximations was evaluated by comparing with the results of the upper bound in [10] and simulation. From the numerical examples, we showed that the first-order approximation is an efficient technique to assess the performance of the OC, and the simple closed-form expression for the ordered mean eigenvalues provides a simple and accurate approximation for the performance analysis of MPSK for the OC in the presence of 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 can be evaluated simply from (36) for arbitrary values of m and n.
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TABLE I SYMBOL 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 with ηa1 = 1, ηa2 = 2, and ηa3 = 5 satisfying the conditions (20)–(22) can be obtained as 1 1 0 A1 = 0 , A2 = 1 2 , 0 0 0 0 0 1 0 1 A3 = 0 1 0 2 1 3 2 2 1 1
The coefficient column vector gA (E2 ) in (33) is expressed as T
gA (E2 ) = gA e21 gA e22 · · · gA e2η e2 = [1
(43)
and
Since Ωi,j,k and pi,j,k in (35) is easily obtained by g(B5,3 , A3 ), gA (E2 ), C5,3 3 , and D2 , the ordered mean eigenvalues can be exactly evaluated as
(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 ηb5,3 = 15 is given in (38) at the bottom of the page. The coefficient column vector g1 (B5,3 ) is also given in (39) at the bottom 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 "
1 1 E2 = 1 2
#
"
and
2 D2 = 1
B5,3
2 0 = 0 0 0
1 2
0 0
0 2 0 0 0
0 0 2 0 0
.
0 0 0 2 0
= [1
1
1
1 1 0 0 0
1 0 1 0 0
1 0 0 1 0
···
1
2
2
# 2885 PI . 2187
Using the similar approach for m = 3 and n = 5, Table II shows some results for ordered mean eigenvalues in other cases of m and n. From (36), the exact ordered mean eigenvalues can be simply evaluated without multiple numerical integrals and Monte Carlo simulation for arbitrary numbers of interferers and antenna elements.
g1 b22 1
582065 139968
(44)
(42)
0 0 0 0 2
" ¯ 3 ] = σ 2 + 1332815 λ 139968
¯2 λ
¯1 [λ
#T
g1 (B5,3 ) = g1 b21
1]T .
1 0 0 0 1
0 1 1 0 0
0 1 0 1 0
0 1 0 0 1
0 0 1 1 0
0 0 0. 1 1
0 0 1 0 1
(38)
T g1 b2η 5, 3 b
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 ¯ q − σ 2 )/P I , q = 1, 2, . . . , m FOR EXACT m DISTINCT MEAN EIGENVALUES (λ
APPENDIX B AVERAGE SER OF MPSK FOR THE OC USING ORTHOGONAL APPROXIMATION The first-order approximation can be used to evaluate the performance of the OC in the case of arbitrary N and L, while the orthogonal approximation, which has been proposed by Villier, can be used when the number of interferers is less than the number of antenna elements, i.e., N > L [7]. The orthogonal approximation is a simple method, wherein the L distinct random eigenvalues are replaced by a fixed λ = σ 2 + N PI . Then, the MGF of maximum SINR for the orthogonal approximation becomes ∼ OC ΦOC γ s ,OA (s) = Φγ s (s; λ1 , λ2 , . . . , λL )|λ q =λ, q =1,2,...,L
N −L
L σ2 λ = . (45) σ 2 − PD s λ − PD s
¯ k /L due to value of L distinct mean eigenvalues as λ = Lk=1 λ L ¯ 2 k =1 (λk − σ ) = LN PI [10]. As the extension of the results from orthogonal approximation to the case of N ≤ L, λ can be expressed as λ = σ 2 + nPI [10]. For N ≤ L, the approximate MGF is given by
N
N λ 1 ΦOC (s) = = γ s ,OA λ − PD s 1 − ΓS s = ΦMRC (s; ΓS , N ). γs
From (47), however, the performance of the OC using the orthogonal approximation is equal to the performance of the MRC so that the orthogonal approximation cannot represent the effect of interference rejection from the OC in case of N ≤ L. Thus, the orthogonal approximation in [7] has been applied to the performance analysis of the OC when the number of interferers is less than the number of antenna elements.
From a partial fraction expansion [18, (3.3)], the average SER of MPSK for the OC using the orthogonal approximation can be evaluated as OC P¯s,OA =
REFERENCES [1] J. H. Winters, “Optimum combining of signals in space-diversity reception,” 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 of optimum combining and maximal ratio combining in the presence of cochannel interference and channel correlation for wireless communication systems,” Mobile Netw. Appl., vol. 2, pp. 315–324, 1997. [3] V. A. Aalo and J. Zhang, “Performance of antenna array systems with optimum 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 biterror probability for optimum combining with a Rayleigh fading Gaussian cochannel 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 symbol error probability for optimum combining in the presence of multiple cochannel interferes and thermal noise,” Proc. IEEE GLOBECOM 2001, pp. 1182–1186, San Antonio, TX, Nov. 2001.
L (−1 − N ΓI )L −i N − i − 1 ¯ MRC ˜ Ps (Γd , i) L−i (−N ΓI )N −i i=1 +
N −L j =1
(−1 − N ΓI )L (−N ΓI )N −j
N −j−1 L−1
P¯sMRC (Γd , j)
(46) ˜ d = Γd /(1 + N ΓI ). From the results of the ordered where Γ mean eigenvalues in Appendix A, the orthogonal approximation uses the nonordered mean eigenvalue of R, which is the average C5,3 3 = A3 B5,3 0 0 2 = 0 2 0 6 4 4
0 4 2
2 2 2
0 1 5
1 0 5
(47)
0 2 4
1 1 4
1 1 4
0 3 3
1 2 3
1 2 3
2 1 3
1 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)
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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 55, NO. 1, JANUARY 2006
[6] A. Shah and A. M. Haimovich, “Performance analysis of optimum combining in wireless communications with Rayleigh fading and cochannel interference,” IEEE Trans. Commun., vol. 46, no. 4, pp. 473–479, Apr. 1998. [7] E. Villier, “Performance analysis of optimum combining with multiple interferers 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 antennas 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 combining for dual-antenna diversity with multiple interferers in a Rayleigh fading channel,” 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, “Bounds and approximations for optimum combining of signals in the presence of multiple 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 from normal samples,” Ann. Math. Statist., vol. 35, pp. 475–501, 1964. [12] A. Annamalai and C. Tellambura, “Error rates for Nakagami-m fading multichannel reception of binary and M -ary signals,” IEEE Trans. Commun., vol. 49, no. 1, pp. 58–68, Jan. 2001. [13] M. K. Simon and M.-S. Alouini, Digital Communication over Fading Channels: 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. Boca Raton, FL: CRC, 1996. [16] J. S. Kwak and J. H. Lee, “Approximate error probability of M -ary PSK for optimum combining with arbitrary number of interferers in a Rayleigh fading 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 of the 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 computer science from Seoul National University (SNU), Seoul, Korea, in 1998, 2000, and 2004, respectively. From October 2004 to September 2005, he was a post-doctoral research associate with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta. Currently he is with the University of Texas at Austin as a post-doctoral research fellow with the Department of Electrical and Computer Engineering. His research interests include most areas of wireless communication systems, especially MIMO with interference, adaptive antennas, space-time coding, and multicarrier transmission.
Jae Hong Lee (M’86–SM’03) received the B.S. and M.S. degrees in electronics engineering from Seoul National University (SNU), Seoul, Korea, in 1976 and 1978, respectively, and the Ph.D. degree in electrical engineering from the University of Michigan, Ann Arbor, in 1986. From 1978 to 1981 he was with the Department of Electronics Engineering, Republic of Korea Naval Academy, Jinhae, as an Instructor. In 1987, he joined the faculty of SNU. He was a member of technical staff at AT&T Bell Laboratories, Whippany, NJ, from 1991 to 1992. Currently, he is with SNU as a Professor in the School of Electrical Engineering. His current research interests include communication and coding theory, space-time code, multiple-input multiple-output (MIMO), and orthogonal frequency-division multiplexing (OFDM), and their application to wireless communications. Dr. Lee is a Vice President of the Institute of Electronics Engineers of Korea and the Korea Society of Broadcasting Engineers, and a member of Tau Beta Pi.
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 55, NO. 1, JANUARY 2006
Closed-Form Expressions of Approximate Error Rates for Optimum Combining With Multiple Interferers in a Rayleigh Fading Channel Jin 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) with multiple cochannel interferers in a flat Rayleigh fading channel. For the first-order approximation, we derive the closed-form expression for ordered mean eigenvalues of the interference-plusnoise covariance matrix, which facilitates performance evaluation for the OC with arbitrary numbers of interferers and antenna elements without Monte Carlo simulation and multiple numerical integrals. We also derive the closed-form expressions for approximate error rates of MPSK for the OC in terms of the average error rate of MPSK for maximal ratio combining (MRC). From the simple evaluation of ordered mean eigenvalues, we show that the first-order approximation gives a simple and accurate way to analyze the performance of the OC. Index Terms—Adaptive antennas, co-channel interference, M -ary phase shift keying (MPSK), optimum combining (OC).
I. INTRODUCTION DAPTIVE antenna arrays significantly improve both the performance and capacity of wireless communication systems by mitigating multipath fading and suppressing interfering signals [1]. The optimum combiner (OC) which maximizes the signal-to-interference-plus-noise ratio (SINR) yields better performance than that of maximal ratio combiner (MRC) in the interference-limited system. In the absence of interference for an additive noise environment, the OC maximizes the signal-to-noise ratio (SNR) and has the same performance as the MRC [1], [2]. In a Rayleigh fading environment, the exact average bit error rate (BER) performance of the OC for binary phase-shift-keying (BPSK) has been studied [2]–[4]. The closed-form expression for the average BER was derived for the OC in the presence of the arbitrary number of interferes [4]. For M -ary phase-shiftkeying (MPSK), the average symbol error rate (SER) have been expressed as a multidimensional integral that can be a burden to be evaluated numerically [5] or as a simple closed-form upper bound that may be loose for high SER [6]. On the other hand, approximation techniques have been studied to simply evaluate the performance analysis of the OC
A
Manuscript received April 16, 2003; revised December 23, 2004. This work was supported in part by the Brain Korea 21 Project and in part by the IT scholarship program from MIC and IITA, Korea. The review of this paper was coordinated by Prof. H. Leib. J. S. Kwak is with the Department of Electrical and Computer Engineering, the University of Texas at Austin, Austin, TX 78712-0240 USA (e-mail: [email protected]). J. H. Lee is with School of Electrical Engineering and Computer Science, Seoul National University, Seoul 151-742 Korea (e-mail: [email protected]). Digital Object Identifier 10.1109/TVT.2005.861184
[7]–[10]. Villier proposed orthogonal approximation to determine the average BER of several binary modulation schemes [7]. However, the orthogonal approximation only takes into account the case where the number of antenna elements is larger than that of interferers, and the accuracy of approximation depends on the difference between the number of interferers and the number of antenna elements [7]. In [8], Pham and Balmain proposed a first-order approximation and derived a closed-form expression of average BER for BPSK with an arbitrary number of interferers. This approximation provides accurate results for the performance of the OC, whereas Monte Carlo simulation is needed to evaluate the ordered mean eigenvalues of the interference-plus-noise covariance matrix [8]. Recently, the analytical expressions for the ordered mean eigenvalues have been presented as the multidimensional integral, which can be derived as a closed-form solution in the simple cases such as dual-antenna reception or two cochannel interferers [9], [10]. In this paper, we derive the simple closed-form expression of the ordered mean eigenvalues to avoid Monte Carlo simulation and multiple numerical integrals in the presence of arbitrary numbers of interferers and antenna elements. As the extended results of the performance analysis in [7] and [8], the approximate average SERs of MPSK for the OC are also derived in terms of the average SER of MPSK for the MRC. The analytical results of the approximation using ordered mean eigenvalues provide a simple and accurate way to assess the performance of the OC with multiple cochannel interferers. The paper is organized as follows. Section II gives the system model. In Section III, we derive the approximate error probabilities of MPSK for the OC and the closed-form expression for ordered mean eigenvalues in the presence of arbitrary numbers of interferers. Section IV presents the numerical results, and Section V concludes the paper. II. SYSTEM MODEL Consider an N -element antenna array and L cochannel interferers. The N -dimensional received signal vector is given by r(t) =
L PD uD sD (t) + Pk uk sk (t) + n
(1)
k =1
where sD (t) and sk (t) are the desired and the kth interfering signals using MPSK modulation with E[s2D (t)] = E[s2k (t)] = 1, respectively. uD and uk are the N × 1 propagation vectors with 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
additive noise vector with mean power σ 2 on each antenna element. The propagation vectors and noise vector are assumed to be mutually independent zero-mean complex Gaussian vectors. PD and Pk are the mean powers of the desired user and the kth interferer, respectively. Conditioned on the vectors uk , the N × N covariance matrix of the received interference-plus-noise can be written as R=
L
Pk u∗k uTk + σ 2 I
(2)
159
λi , i = 1, 2, . . . , m can be expressed as fλ (λ) = A0
n −m λi − σ 2 λi − σ 2 PIm exp − PI PI i=1 m
m
×
(λi − λj )2
(6)
j =i+1
−1 where A0 = ( m i=1 (n − i)!(m − i)!) .
k =1
where I is the N × N identity matrix, and the symbols ( · )∗ and (·)T denote conjugate and transpose, respectively. Note that R varies with the channel fading rate, which is assumed to be much slower than the symbol rate. The optimum weighting vector for the OC that maximizes the SINR at the array output is w = αR−1 u∗D , where α does not affect the output SINR as an arbitrary constant [1]. Then, the maximum output SINR per symbol is given by γs = PD uTD R−1 u∗D .
=
i=1
λi λ i − PD s
A. MGF-Based Approach The average SER for the OC conditioned on λ is given by [12], [13] 1 P¯sOC (λ) = π
(3)
Since R is a Hermitian matrix, R can be unitarily diagonalized as R = UT ΛU∗ , where U is a unitary matrix, and Λ = diag{λ1 , λ2 , . . . , λN } is a diagonal matrix of the ordered eigenvalues of R with λ1 ≥ λ2 ≥ · · · ≥ λN . Letting v = UuD = [vD ,1 vD ,2 · · · vD ,N ]T , the output SINR γs 2 can be expressed as γs = PD N i=1 vi /λi , where vi = |vD ,i | . Since v has the same statistical properties of uD by unitary transformation of U, vi follows a Chi-square distribution with two degrees of freedom, i.e., fv i (x) = exp(−x). Then, the moment generating function (MGF) of γs conditioned on the eigenvalues λi , i = 1, 2, . . . , N is given by [3]–[10] N P D vi OC Φγ s (s; λ1 , λ2 , . . . , λN ) = E exp s λi i=1 N
III. APPROXIMATE ERROR RATE OF MPSK FOR THE OC
.
(4)
Let m = min{N, L} and n = max{N, L}. Since the N − m smallest eigenvalues of R are equal to the noise power σ 2 ; i.e., λm +1 = λm +2 = · · · = λN = σ 2 , the conditional MGF becomes
N −m
m σ2 λi ΦOC (s; λ) = (5) γs σ 2 − PD s λ i − PD s i=1 where λ = [λ1 λ2 · · · λm ]T with λ1 ≥ λ2 ≥ · · · ≥ λm ≥ σ 2 . In order to derive the average SER of MPSK for the OC with multiple cochannel interferers, it is required to evaluate the exact joint distribution of eigenvalues in (5). However, it is difficult to have insight into the performance analysis of the OC with individual mean power Pk [6]. We assume that multiple cochannel interferers haveequal average power PI ; then, the N × N random matrix PI Lk=1 u∗k uTk has the central complex Wishart distribution [10], [11]. Consequently, the joint probability density function (pdf) of the m distinct eigenvalues
1 π
=
π −π /M
0
π −π /M
0
g PSK E exp − 2 γs λ dθ sin θ g PSK − 2 ; λ dθ (7) ΦOC γs sin θ
where gPSK = sin2 (π/M ). Then, the exact average SER of MPSK for the OC can be obtained by averaging (7) with (6) and is given by 1 OC P¯s,exact = π
π −π /M
0
D
g PSK − 2 ; λ fλ (λ)(dλ)dθ ΦOC γs sin θ (8)
where D = {σ 2 ≤ λm ≤ λm −1 ≤ · · · ≤ λ1 ≤ ∞}. In general, it is difficult to derive a closed-form expression for (8), and the numerical integral, which usually requires excessive computational time for large values of m, is required to evaluate the exact performance of MPSK for the OC [5]. To simplify the performance analysis of the OC, Pham and Balmain proposed a first-order approximation by using a Taylor series expansion [14, p. 156] in which each of m distinct eigenvalues is replaced by its mean value [8]. Then, the approximate MGF of maximum output SINR can be expressed as ∼ OC ¯ ¯ ¯ ΦOC γ s ,app (s) = Φγ s (s; λ1 , λ2 , . . . , λm )
(9)
¯ q = E[λq ], q = 1, 2, . . . , m. From the partial fraction where λ expansion in [15, (6.27.12)], the average SER of MPSK for the OC can be expressed in terms of the average SER of MPSK for the MRC, which has been studied in [12] and [13], and is given by OC P¯e,app =
N −m i=1
Bi P¯eMRC
m PD PD MRC ¯ P , i + C , 1 j e ¯j σ2 λ j =1 (10)
where P¯eMRC (γ, p) is the average SER of MPSK for the MRC with the average received SNR per symbol per branch γ, and the number of diversity branches p in the presence of no interfering
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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 55, NO. 1, JANUARY 2006
signals [12], [13, p. 269]
where
d N −m −i (1 − PD s/σ 2 )N −m ΦOC lims→σ 2 /P D ds N −m −i γ s ,app (s) Bi = (−PD /σ 2 )N −m −i (N − m − i)!
H1 (x) = exp −
Cj =
lim
¯ j /P D s→λ
¯ j )ΦOC (s) . (1 − PD s/λ γ s ,app
H2 (x) = (12)
To analyze the error probability of MPSK for the OC in (10), it is required to obtain the individual mean values of m distinct eigenvalues. From the exact joint pdf in (6), the m distinct mean eigenvalues can be expressed as [9], [10] ∞ ∞ ∞ ¯ ··· λq · fλ (λ)dλ1 · · · dλm −1 dλm (13) λq = σ2
λm
∞
¯q = λ
∞
···
0
0
∞
σ 2 + PI
m
0
i=1
m −1
m
xl
k
=
ηa
m j =i+1
xl
.
(18)
l=i
ak ak ak g0 akj x1 1, j x2 2, j . . . xmm , j
k
=
ηa
g0 (akj )t x, akj
(19)
j =1
where akj = [ak1,j , ak2,j , . . . , akm ,j ]T is the jth column vector of all possible ηak distinct vectors for nonnegative integer aki,j , satisfying the conditions m
aki,j = k
(20)
i0
aki,j ≤ i0
for i0 = 1, 2, . . . , k
(21)
and
∞
xl fx (x) dx1 dx2 . . . dxm
m
xl
m
l=i
j −1
2
i=1
(14)
exp −
j −1
j =1
0≤
x ¯l
i=1
(17)
= xk (xk −1 + xk ) · · · (x1 + x2 + · · · + xk )
xl
where x = [x1 x2 · · · xm ]T with 0 ≤ x1 , x2 , . . . , xm ≤ ∞, x ¯l = E[xl ], and the joint pdf fx (x) is given by
×
l=i
l=i
l=q
fx (x) = A0
i=1
i=1
0
(16)
In order to represent the joint pdf in (15) as a simple integrable formula, we first define the function hk (x) for k ≤ m, i.e., k k hk (x) = xl
aki,j = 0 for i > k. ak
m
k · xk
n −m
i=1 j =i+1
l=q
0
l=q
= σ + PI
m
0
× fx (x) dx1 dx2 . . . dxm ∞ m ∞ 2 = σ + PI ···
2
H3 (x) =
λ2
for q = 1, 2, . . . , m. The closed-form solutions of (13) have been 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 simulation for large values of m. In this paper, we derive the closedform expression of (13) in order to simply evaluate the exact ordered mean eigenvalues in the presence of arbitrary numbers of interferers and antenna elements. From the change of variables λq = σ 2 + PI m l=q xl , q = ¯ q in (13) can be written as 1, 2, . . . , m, λ
m m
and
Using the finite binomial series and a variable substitution, we can derive the average SER of the MRC as a closed-form expression, which is given by [16, (16)] B. Closed-Form Expression for Ordered Mean Eigenvalues
k =1
(11) and
m
n −m xl
l=i
ak
In (19), t(x, akj ) = x1 1, j x2 2, j · · · xmm , j , and the coefficient g0 (akj ) of t(x, akj ) is given by −1 k
2 − ak1,j k− m 1 k i=1 ai,j g0 (aj ) = ··· ak1,j ak2,j akm ,j k k l − l−1 i=1 ai,j = . (23) akl,j ! l=1 Letting t(x, Ak )=[t(x, ak1 )t(x, ak2 ) · · · t(x, akη k )]T and g0 (Ak ) a = [g0 (ak1 )g0 (ak2 ) · · · g0 (akη k )]T , where Ak = [ak1 ak2 · · · akη k ], a a hk (x) can be expressed as
2 xl
l=i
= A0 H1 (x)H2 (x)H3 (x)
ak
(22)
(15)
hk (x) = g0 (Ak ), t(x, Ak )
(24)
KWAK AND LEE: CLOSED-FORM EXPRESSIONS OF APPROXIMATE ERROR RATES FOR OC
where ·, · denotes the inner product operation. Then, H2 (x) becomes
−1 m −1 m −1 −1 T for dm = and dj = [dm 1,j , d2,j , · · · , dm −1,j , 0] i,j l ai,e m −1 , i = 1, 2, · · · m − 1.
Let us define
=
g1 (bnj −m )
j =1
t
b nk ,−m j
(x, am k )
(25)
k =1
T where bnj −m = [bn1,j−m , bn2,j−m , . . . , bnη am−m is the jth column ,j ] n ,m vector of all possible ηb distinct vectors for nonnegative integer bni,j−m , satisfying the condition η am
bni,j−m
=n−m
for 0 ≤
bni,j−m
≤n−m
(26)
i=1
and the coefficient
g1 (bnj −m )
g1 bnj −m =
η am
l=1
is given by l−1
n − m − i=1 bni,j−m bnl,j−m
(n − m)! = η am n −m . l=1 bl,j !
(27)
If we let Bn ,m = [bn1 −m bn2 −m · · · bnη n−m , m ] and g1 (Bn ,m ) = b
T [g1 (bn1 −m )g1 (bn2 −m ) . . . , g1 (bnη n−m , m )] , then H2 (x) can be exb pressed as n ,m ηb η am n −m m b nl , −m j {g0 (al )} g1 bj t(x, cj ) H2 (x) = j =1 n ,m ηb
=
l=1
g1 bnj −m t g0 (Am ), bnj −m t(x, cj )
j =1
= g(Bn ,m , Am ), t(x, Cnm,m )
=
k =1 η em −1 m −1
g0
j =1 l=1
=
η em −1
gA
ale m −1 l,j
2 l t x, ae m −1 l,j
2 −1 em t(x, dj ) j
(29)
j =1 −1 −1 m −1 m −1 T where em = [em j 1,j e2,j · · · em −1,j ] is the jth column vector of all possible ηem −1 distinct vectors for positive integer −1 em i,j , satisfying the condition −1 1 ≤ em ≤ ηai i,j
(31)
Dm −1 = d1 d2 . . . dη em −1
(32)
and
T
−1 −1 −1 gA (Em −1 ) = gA em gA em . . . gA em m −1 1 2 η e
−1 where gA (em )= j to
m −1 l=1
(33) g0 (ale m −1 ); then, H3 (x) simplifies l,j
H3 (x) = gA (Em −1 , ), t(x, Dm −1 )2 .
(34)
Substituting (28) and (34) into (15) yields the final expression for fx (x1 , x2 , . . . , xm ) in (35), shown at the bottom of the next −1 −1 )gA (em ) page, where Ωi,j,k = A0 g(bni −m , Am )gA (em j k i,j,k i,j,k T and pi,j,k = [p1 , p2 , . . . , pi,j,k ] = c + d + d . With i j k m the help of the identity [17, (3.351.3)], the ordered mean eigen¯ q , q = 1, 2, . . . , m, can be expressed as (36), shown at value λ the bottom of the next page. The closed-form solution in (36) gives a simple way to evaluate the exact ordered mean eigenvalue in the presence of arbitrary numbers of interferers and antenna elements without the 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 MPSK for the OC in (10). Appendix A shows that, as an example for m = 3 and n = 5, the coefficient Ωi,j,k and column vector pi,j,k in (36) can be explicitly computed, and some results of the ordered mean eigenvalues are given for different numbers of interferers and antenna elements.
(28)
where g(Bn ,m , Am ) = g1 (Bn ,m ), t(g0 (Am ), Bn ,m ) and cj is the jth column vector of Cnm,m = Am Bn ,m . Next, H3 (x) can be written as m −1 2 hk (x) H3 (x) =
l=1
−1 m −1 −1 Em −1 = em e2 . . . em m −1 1 η e
j =1 η am
m −1
l,j
H2 (x) = {hm (x)}n −m m n −m ηa m = g0 (am j )t(x, aj ) n ,m ηb
161
for i = 1, 2, . . . , m − 1
(30)
IV. NUMERICAL RESULTS In this section, we present some numerical results to illustrate the performance of the OC using first-order approximation by comparing with the results of orthogonal approximation in Appendix B, the upper bound in [10], and the Monte Carlo simulation. The performance of the MRC is also evaluated by using the exact average SER in [16, (16)] to present the effects of optimum combining on average SER in the presence of multiple cochannel interferers. For plotting the performance of the M -ary signal, the average SINR per bit and the average INR per bit are defined as ΓS,b = ΓS / log2 M , and ΓI ,b = ΓI / log2 M , respectively, where ΓI = PI /σ 2 is the average interference-to-noise ratio (INR) per symbol and ΓS = ΓD /(1 + LΓI ) is the average SINR per symbol with average SNR per symbol ΓD = PD /σ 2 . Fig. 1 shows the average SER of MPSK versus average SINR per bit for several values of M with N = 3, L = 2, and ΓI ,b = 0 dB. It is shown that the use of the OC considerably improves the system performance of MPSK, as shown for binary PSK in [1]. As compared with the average SERs using orthogonal
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Fig. 1. Average SER of MPSK versus average SINR per bit for several values of M with N = 3, L = 2, and ΓI , b = 0 dB.
Fig. 2. Average SER of MPSK versus average SINR per bit for several values of M with N = 3, L = 5, and ΓI , b = 0 dB.
approximation and upper bound in [10], the average SER using first-order approximation agrees with simulation results. Thus, from the simple evaluation of the individual mean eigenvalues in (36), the first-order approximation is an efficient technique to evaluate the performance of the OC. Note that the performance of the upper bound is close to that of orthogonal approximation as shown in [10]. However, since the upper bound in [10] requires to evaluate the numerical integrals, the orthogonal approximation is more proper than the upper bound in [10] to analyze the performance of the OC in the case of N > L. Fig. 2 shows the average SER of MPSK versus average SINR per bit for several values of M with N = 3, L = 5, and ΓI ,b = 0 dB. In Fig. 2, the average SER using the orthogonal
approximation is equal to the average SER of the MRC as shown in Appendix B. There is a good agreement between the average SER using first-order approximation, and the simulation results in the case of N < L. It is shown that the performance of upper bound is close to the performance of the MRC, so that the upper bound cannot reflect the effect of the OC when the degrees-of-freedom (DOFs) are insufficient to suppress all interferers. From this, the upper bound is also limited to the case of N > L. Fig. 3 shows the average SER of MPSK for the OC versus average 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 the average SER of the orthogonal approximation and simulation
fx (x1 , x2 , . . . , xm ) = A0 exp −
m
k · xk
g(B,n ,m Am ), t (x, Cnm,m ) gA (Em −1 , ), t(x, Dm −1 )2
k =1
= A0 exp −
η n , m η m −1 η m −1 b e e −1 −1 gA em T (x, ci + dj + dk ) k · xk g bni −m , Am gA em j k
m
i=1 j =1 k =1
k =1 n −m ηb
=
m r =q
Ωi,j,k
j =1 k =1 ∞
0 n −m
= σ 2 + PI ·
η em −1
i=1
¯ q = σ 2 + PI · λ
η em −1
∞
···
0
∞
= σ + PI ·
i=1 j =1 k =1
.
(35)
xl fx (x1 , x2 , . . . , xm ) dx1 dx2 · · · dxm
m −1 m −1
exp(−l · xl ) ·
0
ηe m η b e η
n −m m −1 m −1 ηb ηe ηe
pi , j , k xl l
l=1
Ωi,j,k
r =q i=1 j =1 k =1
2
m
Ωi,j,k
m
∞
exp(−l · xl ) ·
l=1,l =r 0 m m pi,j,k ! l i,j,k
r =q l=1
lp l
+1
pi,j,k + 1 · r r
pi , j , k xl l
dxl ·
∞
p ir , j , k
exp(−l · xr ) · xr
+1
dxl
0
(36)
KWAK AND LEE: CLOSED-FORM EXPRESSIONS OF APPROXIMATE ERROR RATES FOR OC
Fig. 3. Average SER of MPSK for OC versus average SINR per bit for several values of N with M = 4, L = 2, and ΓI , b = 0 dB.
163
Fig. 5. Diversity gain of OC over MRC versus the number of antenna elements N to achieve 0.1% average SER. M = 4, L = 3, and ΓI , b = 3 dB.
average SER for the OC, compared with orthogonal approximation and upper bound in [10]. Fig. 5 shows the diversity gain of the OC versus the number of antenna elements N with M = 4, L = 3, and ΓI ,b = 3 dB. The diversity gain is defined as the difference in the required SINR per bit to achieve a 0.1% average SER between the OC and the MRC. It is shown that the gain for the first-order approximation is very close to the gain for the exact analysis. It is also shown that the discrepancy of the diversity gain between the orthogonal approximation (or upper bound in [10]), and exact analysis is less than 0.8 dB for large numbers of antenna elements N ≥ 5. V. CONCLUSION
Fig. 4. Average SER of MPSK versus average SINR per bit for several values of ΓI , b with M = 4, N = 4, and L = 2.
results improves greatly as N − L increases. This is consistent with 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 SINR per 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 the average INR per bit increases. This is because the OC can more easily cancel out the effect of the strong interference than that of weak interference [1]. Table I shows the average SERs of MPSK using various analytical approaches for the OC and the MRC with M = 4, N = 3, and ΓI ,b = 0 dB. The exact error probabilities are evaluated from the numerical integration of (8) in order to investigate the accuracy of the approximate error probabilities in the OC. It is shown that the first-order approximation provides the accurate
In this paper, we investigated the performance of MPSK for the OC with an arbitrary number of interferers in a flat Rayleigh fading channel. We derived the closed-form expression for the ordered mean eigenvalues for the first-order approximation to avoid Monte Carlo simulation and multiple numerical integrals. The approximate average SER of MPSK for the OC was also derived by using the first-order approximation and orthogonal approximation in terms of the average SER of the MRC. The accuracy of the approximations was evaluated by comparing with the results of the upper bound in [10] and simulation. From the numerical examples, we showed that the first-order approximation is an efficient technique to assess the performance of the OC, and the simple closed-form expression for the ordered mean eigenvalues provides a simple and accurate approximation for the performance analysis of MPSK for the OC in the presence of 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 can be evaluated simply from (36) for arbitrary values of m and n.
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TABLE I SYMBOL 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 with ηa1 = 1, ηa2 = 2, and ηa3 = 5 satisfying the conditions (20)–(22) can be obtained as 1 1 0 A1 = 0 , A2 = 1 2 , 0 0 0 0 0 1 0 1 A3 = 0 1 0 2 1 3 2 2 1 1
The coefficient column vector gA (E2 ) in (33) is expressed as T
gA (E2 ) = gA e21 gA e22 · · · gA e2η e2 = [1
(43)
and
Since Ωi,j,k and pi,j,k in (35) is easily obtained by g(B5,3 , A3 ), gA (E2 ), C5,3 3 , and D2 , the ordered mean eigenvalues can be exactly evaluated as
(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 ηb5,3 = 15 is given in (38) at the bottom of the page. The coefficient column vector g1 (B5,3 ) is also given in (39) at the bottom 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 "
1 1 E2 = 1 2
#
"
and
2 D2 = 1
B5,3
2 0 = 0 0 0
1 2
0 0
0 2 0 0 0
0 0 2 0 0
.
0 0 0 2 0
= [1
1
1
1 1 0 0 0
1 0 1 0 0
1 0 0 1 0
···
1
2
2
# 2885 PI . 2187
Using the similar approach for m = 3 and n = 5, Table II shows some results for ordered mean eigenvalues in other cases of m and n. From (36), the exact ordered mean eigenvalues can be simply evaluated without multiple numerical integrals and Monte Carlo simulation for arbitrary numbers of interferers and antenna elements.
g1 b22 1
582065 139968
(44)
(42)
0 0 0 0 2
" ¯ 3 ] = σ 2 + 1332815 λ 139968
¯2 λ
¯1 [λ
#T
g1 (B5,3 ) = g1 b21
1]T .
1 0 0 0 1
0 1 1 0 0
0 1 0 1 0
0 1 0 0 1
0 0 1 1 0
0 0 0. 1 1
0 0 1 0 1
(38)
T g1 b2η 5, 3 b
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 ¯ q − σ 2 )/P I , q = 1, 2, . . . , m FOR EXACT m DISTINCT MEAN EIGENVALUES (λ
APPENDIX B AVERAGE SER OF MPSK FOR THE OC USING ORTHOGONAL APPROXIMATION The first-order approximation can be used to evaluate the performance of the OC in the case of arbitrary N and L, while the orthogonal approximation, which has been proposed by Villier, can be used when the number of interferers is less than the number of antenna elements, i.e., N > L [7]. The orthogonal approximation is a simple method, wherein the L distinct random eigenvalues are replaced by a fixed λ = σ 2 + N PI . Then, the MGF of maximum SINR for the orthogonal approximation becomes ∼ OC ΦOC γ s ,OA (s) = Φγ s (s; λ1 , λ2 , . . . , λL )|λ q =λ, q =1,2,...,L
N −L
L σ2 λ = . (45) σ 2 − PD s λ − PD s
¯ k /L due to value of L distinct mean eigenvalues as λ = Lk=1 λ L ¯ 2 k =1 (λk − σ ) = LN PI [10]. As the extension of the results from orthogonal approximation to the case of N ≤ L, λ can be expressed as λ = σ 2 + nPI [10]. For N ≤ L, the approximate MGF is given by
N
N λ 1 ΦOC (s) = = γ s ,OA λ − PD s 1 − ΓS s = ΦMRC (s; ΓS , N ). γs
From (47), however, the performance of the OC using the orthogonal approximation is equal to the performance of the MRC so that the orthogonal approximation cannot represent the effect of interference rejection from the OC in case of N ≤ L. Thus, the orthogonal approximation in [7] has been applied to the performance analysis of the OC when the number of interferers is less than the number of antenna elements.
From a partial fraction expansion [18, (3.3)], the average SER of MPSK for the OC using the orthogonal approximation can be evaluated as OC P¯s,OA =
REFERENCES [1] J. H. Winters, “Optimum combining of signals in space-diversity reception,” 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 of optimum combining and maximal ratio combining in the presence of cochannel interference and channel correlation for wireless communication systems,” Mobile Netw. Appl., vol. 2, pp. 315–324, 1997. [3] V. A. Aalo and J. Zhang, “Performance of antenna array systems with optimum 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 biterror probability for optimum combining with a Rayleigh fading Gaussian cochannel 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 symbol error probability for optimum combining in the presence of multiple cochannel interferes and thermal noise,” Proc. IEEE GLOBECOM 2001, pp. 1182–1186, San Antonio, TX, Nov. 2001.
L (−1 − N ΓI )L −i N − i − 1 ¯ MRC ˜ Ps (Γd , i) L−i (−N ΓI )N −i i=1 +
N −L j =1
(−1 − N ΓI )L (−N ΓI )N −j
N −j−1 L−1
P¯sMRC (Γd , j)
(46) ˜ d = Γd /(1 + N ΓI ). From the results of the ordered where Γ mean eigenvalues in Appendix A, the orthogonal approximation uses the nonordered mean eigenvalue of R, which is the average C5,3 3 = A3 B5,3 0 0 2 = 0 2 0 6 4 4
0 4 2
2 2 2
0 1 5
1 0 5
(47)
0 2 4
1 1 4
1 1 4
0 3 3
1 2 3
1 2 3
2 1 3
1 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)
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[6] A. Shah and A. M. Haimovich, “Performance analysis of optimum combining in wireless communications with Rayleigh fading and cochannel interference,” IEEE Trans. Commun., vol. 46, no. 4, pp. 473–479, Apr. 1998. [7] E. Villier, “Performance analysis of optimum combining with multiple interferers 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 antennas 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 combining for dual-antenna diversity with multiple interferers in a Rayleigh fading channel,” 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, “Bounds and approximations for optimum combining of signals in the presence of multiple 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 from normal samples,” Ann. Math. Statist., vol. 35, pp. 475–501, 1964. [12] A. Annamalai and C. Tellambura, “Error rates for Nakagami-m fading multichannel reception of binary and M -ary signals,” IEEE Trans. Commun., vol. 49, no. 1, pp. 58–68, Jan. 2001. [13] M. K. Simon and M.-S. Alouini, Digital Communication over Fading Channels: 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. Boca Raton, FL: CRC, 1996. [16] J. S. Kwak and J. H. Lee, “Approximate error probability of M -ary PSK for optimum combining with arbitrary number of interferers in a Rayleigh fading 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 of the 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 computer science from Seoul National University (SNU), Seoul, Korea, in 1998, 2000, and 2004, respectively. From October 2004 to September 2005, he was a post-doctoral research associate with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta. Currently he is with the University of Texas at Austin as a post-doctoral research fellow with the Department of Electrical and Computer Engineering. His research interests include most areas of wireless communication systems, especially MIMO with interference, adaptive antennas, space-time coding, and multicarrier transmission.
Jae Hong Lee (M’86–SM’03) received the B.S. and M.S. degrees in electronics engineering from Seoul National University (SNU), Seoul, Korea, in 1976 and 1978, respectively, and the Ph.D. degree in electrical engineering from the University of Michigan, Ann Arbor, in 1986. From 1978 to 1981 he was with the Department of Electronics Engineering, Republic of Korea Naval Academy, Jinhae, as an Instructor. In 1987, he joined the faculty of SNU. He was a member of technical staff at AT&T Bell Laboratories, Whippany, NJ, from 1991 to 1992. Currently, he is with SNU as a Professor in the School of Electrical Engineering. His current research interests include communication and coding theory, space-time code, multiple-input multiple-output (MIMO), and orthogonal frequency-division multiplexing (OFDM), and their application to wireless communications. Dr. Lee is a Vice President of the Institute of Electronics Engineers of Korea and the Korea Society of Broadcasting Engineers, and a member of Tau Beta Pi.
