Bit Error Rate Analysis of Maximal-Ratio Combining ... - IEEE Xplore
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Bit Error Rate Analysis of Maximal-Ratio Combining over Correlated Gaussian Vector Channels Siamak Sorooshyari and David G. Daut, Senior Member, IEEE
Abstract— Analytical results are derived for the performance of maximal-ratio combining (MRC) over correlated Gaussian vector channels. Generality is maintained by assuming arbitrarypower users and no specific form for the covariance matrices of the received faded signals. The scenarios of Rayleigh fading and statistical independence among diversity branches are considered as special cases of the general analysis. The analysis is presented within the context of a system with receive diversity and cochannel interference. The results obtained are applicable to antipodal signaling over a multiuser single-input multiple-output (SIMO) channel. Index Terms— Bit error rate, cochannel interference, maximal ratio combining, diversity, fading channels.
the channel matrix. Aside from providing a framework for performance analysis in the realistic scenario of correlated fading, such analysis provides quantitative insight into the degradation brought on by various degrees of statistical dependence among diversity branches. The remainder of this paper is organized as follows. Section II presents the statistical foundations used in the derivation of system performance. In Section III a lower bound is derived for the performance of MRC over arbitrarily correlated Gaussian vector channels. Several exact BER expressions are also presented for more specific fading conditions. Section IV presents simulation results illustrating the tightness of the lower bound, and quantifying the performance degradation incurred due to correlated fading.
I. I NTRODUCTION
T
HE prevalence of diversity has been instrumental in the advancement of reliable wireless communication characterized by a low bit error rate (BER). Vector channels typically arise in a diversity system where cochannel interferers are present. The vector output of such systems is mathematically described as the product of a channel matrix and an information-bearing vector. Prior to detection, the output vector of the channel may be processed via maximal-ratio combining (MRC) to form a decision statistic. The benefits of MRC arise from providing diversity gain in a dispersive channel at a reasonable complexity. The efficient implementation of MRC has been realized for multicarrier communication systems [1], spread spectrum systems with RAKE receivers [2], and ultra-wide bandwidth (UWB) systems [3]. While the performance of MRC over correlated fading channels has been extensively studied [4] [5] [6] [7]; few results exist for the BER analysis of MRC over vector channels subject to correlated fading. In fact, the only existing work neglects thermal noise in the analysis [8]. The importance of this research stems from its application to various wireless systems employing spatial, frequency, or temporal diversity. Correlated fading in diversity systems is often an unavoidable consequence of system characteristics and the physics of the radio propagation channel. The analysis presented in this paper applies to variable-power users and correlated fading with arbitrary covariance matrices for the vectors comprising
Manuscript received October 16, 2006; revised March 27, 2007; accepted April 17, 2007. The associate editor coordinating the review of this paper and approving it for publication was A. Conti. This work was presented in part at the Conference on Information Sciences and Systems (CISS 2006), Princeton, New Jersey, March 2006. S. Sorooshyari is with Alcatel-Lucent, Whippany, NJ 07981 USA (e-mail: [email protected]). D. G. Daut is with the Department of Electrical and Computer Engineering, Rutgers University, Piscataway, NJ 08854 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TWC.2008.060835.
II. S TATISTICAL P RELIMINARIES The calculation of moments of ratios frequently arises in the performance analysis of communication systems. The utility of the moment generating function (MGF) for the calculation of moments of random variables is well known. Within the context of performance analysis of communication systems, the MGF approach has been well documented in [4] with an emphasis on faded diversity channels. In [9], Meng has elegantly addressed the calculation of noninteger moments of ratios. Incorporating the notation of Meng, we designate x as the smallest integer exceeding x, define x x − x, and (n,m) allow MX ,Y (s1 , s2 ) to represent the nth and mth partial derivatives of the bivariate MGF MX ,Y (s1 , s2 ) with respect to the variables s1 and s2 , respectively. A primary finding of [9] states that for 0 ≤ a, 0 < b, and P [X ≥ 0, Y > 0] = 1 a ∞ ∞ X 1 (a,0 ) E MX ,Y (−s1 , −s2 ) = Yb Γ(a)Γ(b) 0 0 a−1 b−1 s2
× s1
ds1 ds2
(1)
where Γ(.) denotes the standard Gamma function [10]. The proof of this result is given in Appendix A of [9], and is dependent upon noting that E[g(X, Y )] =
∞ 0
∞ 0
h(s1 , s2 )MX ,Y (−s1 , −s2 ) ds1 ds2
(2)
∞∞ with h(s1 , s2 ) 0 0 g(x, y)es1 x+s2 y dxdy. Two limiting cases of (1) are given for a nonnegative integer k, 0 ≤ a, 0 < b, and P [X > 0] = 1 as k ∞ X 1 (k ,0 ) MX ,Y (0 , −s) sb−1 ds (3) E = Yb Γ(b) 0 and E[X a ] =
c 2008 IEEE 1536-1276/08$25.00
1 Γ(a)
0
∞
a
MX (−s) sa−1 ds.
(4)
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 4, APRIL 2008
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b1 p1
Input Data
BPSK Modulation
h1
Re{ r
H
Re{ r
H
v1
^
v2
^
b1
w1 }
n Input Data
b 2 p2 BPSK Modulation
h2
+
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w 2}
. . .
. . .
+
r
bM pM
vM
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hM
Re{r
H
^
bM
wM }
Fig. 1. Model of a Gaussian vector channel. The channel propagation vectors {hi } and the thermal noise vector n have a complex Gaussian distribution. This model corresponds to a multiuser SIMO system with arbitrary-power users, and the received signal at each user’s N -branch receiver being subject to cochannel interference, fading, and additive thermal noise.
The utility of (1) stems from the minimal constraints on the variables X and Y , and the constants a and b. The N -dimensional vector having a complex Gaussian distribution with mean m and covariance matrix R is written as fx (x) =
1 π N |R|
exp(−(x − m) R−1 (x − m)) H
(5)
and denoted x ∼ CN (m, R). We wish to examine the ratio of quadratic forms where X = xH Ax and Y = xH Bx with x ∼ CN (m, R). The bivariate MGF exp(s1 xH Ax + s2 xH Bx)fx (x) d x MX ,Y (s1 , s2 ) = x H
exp(−m R−1 m) |I − s1 RA − s2 RB| × exp mH R−1 (R−1 − s1 A − s2 B)−1 R−1 m =
can be derived via the integral property x
exp(−xH Ax − aH x − xH b) dx =
πN exp(aH A−1 b). |A|
We restrict attention to the evaluation of (3) for k = 1 and (1 ,0 ) b = 1/2, and derive MX ,Y (0 , −s) via the properties ∂|F(x)| ∂x ∂F−1 (x) ∂x
−1
= |F(x)|trace F = −F−1 (x)
∂F(x) (x) ∂x
∂F(x) −1 F (x). ∂x
(1 ,0 )
Subsequent substitution of MX ,Y (0 , −s) into (3) allows us to arrive at the integral shown in (6), where we have used a change of variable t = exp(−s) to obtain an integral with finite limits. In the more specific scenario of x ∼ CN (0, R)
equation (6) reduces to
H x Ax
E √
xH Bx m=0 1 trace(RA(I − ln(t)RB)−1 ) 1 = Γ(0.5) 0 |I − ln(t)RB| × t−3/2 dt.
(7)
It is noteworthy that (6) and (7) consist of a single integral although N variates are present. III. P ERFORMANCE A NALYSIS The conventional approach to performance analysis of MRC in the presence of cochannel interference has relied upon obtaining the distribution of the signal-to-interference plus noise ratio (SINR). This is followed by averaging a conditional BER expression for a particular modulation scheme over the distribution of the SINR. In this paper we shall take the alternate approach of performing the analysis directly on the decision statistic rather than on the SINR. We consider the output of the vector channel in Fig. 1 to be r = HPb + n
(8)
with the ith column matrix H denoted as hi , the √ N ×M √ √ of the matrix P = diag( P1 , P2 , . . . , PM ), and Pi denoting the transmit power of the ith user. Confining analysis to a single bit interval, we define the vector b = [b1 , b2 , . . . , bM ]T as the transmitted information at a given time instant, with bi ∈ {−1, 1} via binary phase-shift keying (BPSK) modulation and P [bi = 1] = P [bi = −1] = 0.5 ∀ i. We shall assume hi ∼ CN (mi , Ri ) ∀ i, and the additive receiver noise will be accounted for via the vector n ∼ CN (0, N). For the remainder of this presentation we shall restrict attention to the system of Fig. 1 with the columns of the channel matrix corresponding to the propagation vectors of the M users.
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xH Ax E √ xH Bx
= × = ×
exp(−mH R−1 m) ∞ exp mH R−1 (R−1 + sB)−1 R−1 m s−1/2 Γ(0.5) |I + sRB| 0 H −1 −1 −1 −1 −1 −1 m R (R + sB) A(R + sB) R m + trace(RA(I + sRB)−1 ) ds exp(−mH R−1 m) 1 exp mH R−1 (R−1 − ln(t)B)−1 R−1 m t−3/2 Γ(0.5) |I − ln(t)RB| 0 H −1 −1 −1 −1 m R (R − ln(t)B) A(R − ln(t)B)−1 R−1 m + trace(RA(I − ln(t)RB)−1 ) dt
Thus, the analysis will be applicable to a multiuser singleinput multiple-output (SIMO) system with the received vector r consisting of a superposition of the desired user’s signal, L = M − 1 interfering signals, and additive noise. By virtue of users transmitting signals independently, the N -dimensional propagation vectors present at a given receiver are assumed to be mutually independent. However, the individual vector elements, denoting the received signal present at different branches but originating from a common source, may share a certain amount of correlation. Mathematically, this translates to the joint probability density function (pdf) of the desired signal and interference as seen by user d from a set of M active users, being given by fhd ,h1 ,...,hL (hd , h1 , . . . , hL ) = fhd (hd )
L
fhm (hm ).
m=1
In effect, the N -dimensional vector hi = denotes the [ρi,1 ejφi,1 , ρi,2 ejφi,2 , . . . , ρi,N ejφi,N ]T complex random channel gains experienced by the transmitted signal of the ith user among N diversity branches. After conditioning upon the vector hd , the distribution of√the channel output in (8) is obtained as r ∼ CN (bd Pd hd , i=d Pi Ri + N). It is common for the output of a vector channel to be processed via a linear operation to yield the decision statistic vd = {wdH r}
(9)
and the recovered bit according to ˆbd = sgn{vd } . Within a detection-theoretic framework, the complex vector wd is a function of the matrix H for minimum meansquare error (MMSE) and decorrelator detectors [11], or alternatively a function of the vector hd for simpler (and inferior) detectors. In this presentation we shall restrict attention to the latter class of detectors when specifying on hd we obtain vd ∼ wd . Having √ already conditioned N ( {bd Pd wdH hd }, 12 wdH ( i=d Pi Ri + N)wd ) with the corresponding BER given by E[vd ] P [e|hd ] = Q V ar[vd ] ⎛ ⎞ H {wd hd } ⎠ (.10) = Q ⎝ 1 H w ( P R + N)w i i d i=d 2Pd d In the derivation of the above expression it was recognized that, by symmetry, the probability of error conditioned on bd =
(6)
1 will be equal to that when bd = −1. The unconditional BER is ⎞⎤ ⎡ ⎛ H {wd hd } ⎠⎦ Pe = E ⎣Q ⎝ 1 H w ( P R + N)w d i=d i i 2Pd d ⎛ ⎡ ⎤⎞ H {w h } d d ⎦⎠(11) ≥ Q ⎝E ⎣ 1 H( w P R + N)w d i=d i i 2Pd d with the expectation taken with respect to hd . The lower bound follows from Jensen’s inequality and the convexity of the Q-function over [0, +∞). Within the diversity combining literature the MRC specification requires that wd = hd . With MRC, the lower bound in (11) can be written as ⎤⎞ ⎛ ⎡ H h h d d ⎦⎠(12) Pe ≥ Q ⎝ E ⎣ 1 H h ( P R + N)h i i d i = d d 2Pd with the argument inside the Q-function expressed in the form of (6) with B = 2P1 d P R + N , A = I, i=d i i R = Rd , and m = md . It should be apparent that the substitution of (7) into (12) would correspond to the more specific scenario in which the desired user’s signal experiences arbitrarily correlated Rayleigh fading. The tightness of the above bound will be investigated in the following section, however, two benefits seem evident at this point. Firstly, the general applicability of the lower bound to various statistical fading conditions, and secondly, the fact that the bound will consist of a single integral (inside the Q-function) with finite limits irrespective of the diversity order (N ) or the number of users (M ). For reference, we provide the expression for the exact BER with MRC combining ⎞⎤ ⎡ ⎛ H hd hd ⎠⎦ Pe = E ⎣Q ⎝ 1 H h ( P R + N)h i i d i=d 2Pd d π/2 2 1 h ) −Pd (hH d d = exp 2 π 0 hH hd i=d Pi Ri + N)hd sin θ d ( × fhd (hd )dhd dθ
(13)
by considering an alternate form of the Q-function which has finite limits [12]. We note that the exact BER expression consists of an N -fold integral, whereas the lower bound consisted of a single integral with finite limits irrespective of P R the diversity order. A further simplification of i=d i i + N = ( i=d Pi Ωi + η)I exists if N = ηI, and the interferers have independent identically distributed (i.i.d.) fading statistics at each diversity branch, that is if Ri = Ωi I ∀ i = d with
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 4, APRIL 2008
Ωi E[|hij |2 ] = E[ρ2ij ] ∀ j, i = d. The conditional BER of (10) will reduce to ⎞ ⎛ π/2 H 1 Pd hd hd ⎠ dθ, exp ⎝− P [e|hd ] = 2 π 0 P Ω + η sin θ i i i=d (14) with the unconditional probability derived as −1 exp(−mH d Rd md ) Pe = π N +1 π/2 −1 −1 a −1 −1 exp mH Rd md d Rd (Rd + sin2 θ I)
dθ ×
I + a2 Rd
0 sin θ with a = Pd /( i=d Pi Ωi + η). Furthermore, if md = 0 and Rd = Ωd I with Ωd E[|hdj |2 ] ∀ j, then the exact BER can be expressed as N sin2 θ 1 π/2 Pe = dθ π 0 sin2 θ + c k N −1 1 − β2 1 2k = (15) 1−β k 2 4 k=0 c/(1 + c) and c = Pd Ωd /( i=d Pi Ωi + η). with β = Although the above expression provides an exact BER in closed-form, the applicability of the result is limited to the scenario of i.i.d. Rayleigh fading for the received signal of the desired user, and i.i.d. fading among the components of the interferers’ propagation vectors.
IV. S IMULATION R ESULTS AND D ISCUSSION The analytical results derived in the previous section warrant quantitative investigation. The objective of this section is twofold. First, we wish to investigate the tightness of the lower bound on the performance of MRC over a correlated Gaussian vector channel. Secondly, with the scenario of mutually independent fading among the diversity branches as a baseline, we wish to examine the impact of correlated fading on system performance. Prior to the quantitative investigation, we examine the application of the correlated Gaussian vector channel model of Fig. 1 to a multiuser frequency diversity system. Subsequently, in the performance analysis, we shall restrict attention to the users experiencing Rayleigh faded channels. Frequency diversity in the form of orthogonal frequency division multiplexing (OFDM) has been adopted for high data-rate wireless multimedia applications as well as being a candidate for upcoming cellular radio standards. The most promising and most heavily investigated multiple access scheme incorporating OFDM has been multicarrier CDMA (MC-CDMA) [1]. Within a MC-CDMA system, encoding (decoding) of N subcarriers is performed prior to transmission (detection) in order to provide a level of orthogonality among the M active users. Considering ci as a length N ”codeword” with ci [n] ∈ {−1, 1} we define the matrix Ci diag(ci ). Adopting the notation of [13] for the more general scenario of uplink transmission, the channel output is represented by r = SPb + n
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with the ith column of the N × M matrix S defined as si Ci hi . With BPSK modulation the desired user’s transmitted bit will be recovered via ˆbd = sgn{ {tH d r}} with td Cd wd , where wd = hd for MRC. Conditioning on the vector hd followed√by straightforward Manalysis yields νd = 1 H ∼ N ( Pd sH {tH d r} d sd , 2 sd ( i=1, i=d Pi Ci Ri Ci + N)sd ), and thus P [e|hd ]
=
Q
=
Q
M
1 sH ( 2Pd d
sH d sd
i=1, i=d
M
1 hH ( 2Pd d
Pi Ci Ri Ci + N)sd
hH d hd
i=1, i=d Pi Ci Ri Ci + N )hd
.
In the above expression we have used the fact that sH i si = h as well as assigned C = C C = C C and N = hH i i d d i i i Cd NCd . Hence, the lower bound on the unconditioned BER will be given as
Pe
≥ Q E
M
1 hH ( 2Pd d
hH d hd
i=1, i=d Pi Ci Ri Ci + N )hd
(16)
with the argument inside the Q-function expressed as the integral A = I, R = Rd , and B = Min (7) with 1 ( P C R i=1, i=d i i i Ci + N ). Of course, in the more 2Pd specific scenario of N = ηI and i.i.d. fading among the received subcarriers of each of the M users, the exact BER can be represented in closed-form via (15). In order to quantify the impact of correlated fading on the desired user’s BER, we shall consider a specific empirical channel model when specifying the users’ covariance matrices. Incorporating the notation and definitions of [14] for an isotropic scattering environment, we specify the parameters u1ij =
Jo (2πfm τij ) 2 1 + kij
u2ij = −kij u1ij
with Jo (.) denoting the zero-order Bessel function. In correspondence with the notion of equal frequency spacing between each pair of adjacent subcarriers (i.e., OFDM-based systems) we shall assume kij = |i − j|k, ∀ i, j; where k = 2π(Δf )στ indicates an equal frequency separation of Δf between adjacent channels. Thus, the elements of the desired user’s covariance matrix are determined as ⎧ i>j ⎨ Ωd u1ij + jΩd u2ij i<j Ωd u1ij − jΩd u2ij Rd (i, j) = (17) ⎩ i = j. Ωd For convenience, in the simulations we shall assume the fading statistics of the interferers to be identical to that of the desired user via the specification Ri = Rd ∀ i. It is desired to determine the tightness of the derived lower bound on the performance of MRC over a correlated Gaussian vector channel. Also, with the scenario of mutually independent fading among the diversity branches as a baseline, we wish to examine the impact of correlated fading on system performance. The MC-CDMA system will consist of N = 32 subcarriers with each user randomly selecting a codeword. Furthermore, a frequency separation of Δf = 312.5 kHz among adjacent subcarriers (IEEE 802.11a), channel delay
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Fig. 2. Illustration of the tightness of the lower bound within the context of a frequency diversity system with N = 32 subcarriers used for transmission.
spread of στ = 0.1μs, and τij = 0 indicating no time offset between reception of the subcarriers will be assumed. The mean signal-to-noise ratio (SNR) per bit of the desired user will be defined as SN R =
10
N Pd Ωd 2η
with SN R = 5, 10, and 20 dB used in the simulations and referred to as low, moderate, and high SNR, respectively. We shall presume all users to be of equal transmit power via the assignment Pi = Pd ∀ i. The tightness of our derived lower bound on the BER is illustrated in Fig. 2. The exact BER was obtained via a 106 sample Monte Carlo simulation of (13) and subsequent averaging over the number of iterations. Generally, an improvement in computation time and processing will be realized in the Monte Carlo simulation of such equations as opposed to fullscale system-level simulation intended to arrive at the same numerical performance results. The finite-limit integral in the lower bound was computed via Simpson’s rule. It is observed that the lower bound is within an order of magnitude of the exact BER irrespective of the SNR value or the number of interferers. The lower bound is tighter for a larger number of interferers with the deviation from the exact BER being less than a factor of six for L ≥ 20. The MRC performance degradation due to correlated fading is shown in Fig. 3 via simulation of (13) for correlated fading and computation of (15) in the case of mutually independent fading among the components of all the users’ channel vectors. In the high SNR regime, a performance loss of an order of magnitude is apparent for L < 6 with the degradation diminishing for higher levels of interference. Similarly, in the moderate SNR region the degradation appears to be at least an order of magnitude with L < 4 and diminishing steadily afterwards. A maximal degradation of a factor of two is observed in the low SNR regime. This may be explained by noting that an increase in interference or noise will lead to a higher error floor. Such an error floor will exist regardless of the properties of the users’ channel covariance matrices. Thus, there will be a
−3
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SNR=5dB, correlated fading SNR=5dB, independent fading SNR=10dB, correlated fading SNR=10dB, independent fading SNR=20dB, correlated fading SNR=20dB, independent fading
−5
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5
10
15
20
25
30
Interferers (L)
Fig. 3. Comparison of the BER of a MC-CDMA system with correlated and uncorrelated fading among the components of the users’ propagation vectors. The results were obtained via computation of the analytical expressions for the exact BER with correlated and uncorrelated fading.
decreasing advantage in having full diversity (that is, complete independence among branches) simply because the benefits will be less apparent due to the high levels of interference or noise. It is also observed that, for both correlated and independent fading, the SNR heavily influences the BER in a lightly loaded system, but is of diminishing value for increasing levels of interference. V. C ONCLUSION In this paper the bit error rate of MRC was examined over a correlated Gaussian vector channel. A lower bound was derived with the variable-power users’ channel vectors following a complex Gaussian distribution with arbitrary means and covariance matrices. Exact BER expressions were examined in less general cases, such as with the interferers’ signals experiencing i.i.d. fading, and all users experiencing i.i.d. Rayleigh fading. Numerical results illustrating the impact of the number of cochannel interferers, different SNR conditions, and correlated fading on a user’s BER were presented within the context of a frequency diversity system. In general, the BER degradation caused by statistical dependence among diversity branches is most pronounced for increasing SNR values along with decreasing levels of interference. ACKNOWLEDGMENT The authors wish to thank Robert Berk for bringing reference [9] to their attention. We also thank Xiao-Li Meng for an insightful discussion. R EFERENCES [1] S. Hara and R. Prasad, “Overview of multicarrier CDMA,” IEEE Commun. Mag., vol. 35, no. 12, pp. 126-133, Dec. 1997. [2] R. L. Pickholtz, D. L. Schilling, and L. B. Milstein, “Theory of spreadspectrum communications - a tutorial,” IEEE Trans. Commun., vol. COM-30, pp. 855-884, May 1982. [3] S. Gezici, H. Kobayashi, H. V. Poor, and A. F. Molisch, “Performance evaluation of impulse radio UWB systems with pulse-based polarity randomization,” IEEE Trans. Signal Processing, vol. 53, pp. 2537-2548, July 2005.
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[4] M. K. Simon and M. S. Alouini, Digital Communication Over Fading Channels. New York: Wiley, 2000. [5] M. Z. Win, G. Chrisikos, and J. H. Winters, “MRC performance for M-ary modulation in arbitrary correlated Nakagami fading channels,” IEEE Commun. Lett., vol. 4, pp. 301-303, Oct. 2000. [6] Q. T. Zhang, “Maximal-ratio combining over Nakagami fading channels with an arbitrary branch covariance matrix,” IEEE Trans. Veh. Technol., vol. 48, pp. 1141-1150, July 1999. [7] V. Veeravalli, “On performance analysis for signaling on correlated fading channels,” IEEE Trans. Commun., vol. 49, pp. 1879-1883, Nov. 2001. [8] X. Zhang and N. C. Beaulieu, “Outage probability of MRC with unequal-power cochannel interferers in correlated Rayleigh fading,” IEEE Commun. Lett., vol. 10, pp. 7-9, Jan. 2006.
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[9] X. Meng, “From unit root to Stein’s estimator to Fisher’s k statistics: if you have a moment, I can tell you more,” Statistical Science, vol. 20, pp.141-162, 2005. [10] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products. San Diego, CA: Academic Press, 2000. [11] R. R. Muller and S. Verdu, “Design and analysis of low-complexity interference mitigation on vector channels,” IEEE J. Select. Areas Commun., vol. 19, pp. 1429-1441, Aug. 2001. [12] M. K. Simon and D. Divsalar, “Some new twists to problems involving the Gaussian integral,” IEEE Trans. Commun., vol. 46, pp. 200-210, Feb. 1998. [13] A. M. Tulino, L. Li, and S. Verdu, “Spectral efficiency of multicarrier CDMA,” IEEE Trans. Inform. Theory, vol. 51, pp. 479-505, Feb. 2005. [14] W. C. Jakes, Ed., Microwave Mobile Communications. New York: IEEE Press, 1974.
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Bit Error Rate Analysis of Maximal-Ratio Combining over Correlated Gaussian Vector Channels Siamak Sorooshyari and David G. Daut, Senior Member, IEEE
Abstract— Analytical results are derived for the performance of maximal-ratio combining (MRC) over correlated Gaussian vector channels. Generality is maintained by assuming arbitrarypower users and no specific form for the covariance matrices of the received faded signals. The scenarios of Rayleigh fading and statistical independence among diversity branches are considered as special cases of the general analysis. The analysis is presented within the context of a system with receive diversity and cochannel interference. The results obtained are applicable to antipodal signaling over a multiuser single-input multiple-output (SIMO) channel. Index Terms— Bit error rate, cochannel interference, maximal ratio combining, diversity, fading channels.
the channel matrix. Aside from providing a framework for performance analysis in the realistic scenario of correlated fading, such analysis provides quantitative insight into the degradation brought on by various degrees of statistical dependence among diversity branches. The remainder of this paper is organized as follows. Section II presents the statistical foundations used in the derivation of system performance. In Section III a lower bound is derived for the performance of MRC over arbitrarily correlated Gaussian vector channels. Several exact BER expressions are also presented for more specific fading conditions. Section IV presents simulation results illustrating the tightness of the lower bound, and quantifying the performance degradation incurred due to correlated fading.
I. I NTRODUCTION
T
HE prevalence of diversity has been instrumental in the advancement of reliable wireless communication characterized by a low bit error rate (BER). Vector channels typically arise in a diversity system where cochannel interferers are present. The vector output of such systems is mathematically described as the product of a channel matrix and an information-bearing vector. Prior to detection, the output vector of the channel may be processed via maximal-ratio combining (MRC) to form a decision statistic. The benefits of MRC arise from providing diversity gain in a dispersive channel at a reasonable complexity. The efficient implementation of MRC has been realized for multicarrier communication systems [1], spread spectrum systems with RAKE receivers [2], and ultra-wide bandwidth (UWB) systems [3]. While the performance of MRC over correlated fading channels has been extensively studied [4] [5] [6] [7]; few results exist for the BER analysis of MRC over vector channels subject to correlated fading. In fact, the only existing work neglects thermal noise in the analysis [8]. The importance of this research stems from its application to various wireless systems employing spatial, frequency, or temporal diversity. Correlated fading in diversity systems is often an unavoidable consequence of system characteristics and the physics of the radio propagation channel. The analysis presented in this paper applies to variable-power users and correlated fading with arbitrary covariance matrices for the vectors comprising
Manuscript received October 16, 2006; revised March 27, 2007; accepted April 17, 2007. The associate editor coordinating the review of this paper and approving it for publication was A. Conti. This work was presented in part at the Conference on Information Sciences and Systems (CISS 2006), Princeton, New Jersey, March 2006. S. Sorooshyari is with Alcatel-Lucent, Whippany, NJ 07981 USA (e-mail: [email protected]). D. G. Daut is with the Department of Electrical and Computer Engineering, Rutgers University, Piscataway, NJ 08854 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TWC.2008.060835.
II. S TATISTICAL P RELIMINARIES The calculation of moments of ratios frequently arises in the performance analysis of communication systems. The utility of the moment generating function (MGF) for the calculation of moments of random variables is well known. Within the context of performance analysis of communication systems, the MGF approach has been well documented in [4] with an emphasis on faded diversity channels. In [9], Meng has elegantly addressed the calculation of noninteger moments of ratios. Incorporating the notation of Meng, we designate x as the smallest integer exceeding x, define x x − x, and (n,m) allow MX ,Y (s1 , s2 ) to represent the nth and mth partial derivatives of the bivariate MGF MX ,Y (s1 , s2 ) with respect to the variables s1 and s2 , respectively. A primary finding of [9] states that for 0 ≤ a, 0 < b, and P [X ≥ 0, Y > 0] = 1 a ∞ ∞ X 1 (a,0 ) E MX ,Y (−s1 , −s2 ) = Yb Γ(a)Γ(b) 0 0 a−1 b−1 s2
× s1
ds1 ds2
(1)
where Γ(.) denotes the standard Gamma function [10]. The proof of this result is given in Appendix A of [9], and is dependent upon noting that E[g(X, Y )] =
∞ 0
∞ 0
h(s1 , s2 )MX ,Y (−s1 , −s2 ) ds1 ds2
(2)
∞∞ with h(s1 , s2 ) 0 0 g(x, y)es1 x+s2 y dxdy. Two limiting cases of (1) are given for a nonnegative integer k, 0 ≤ a, 0 < b, and P [X > 0] = 1 as k ∞ X 1 (k ,0 ) MX ,Y (0 , −s) sb−1 ds (3) E = Yb Γ(b) 0 and E[X a ] =
c 2008 IEEE 1536-1276/08$25.00
1 Γ(a)
0
∞
a
MX (−s) sa−1 ds.
(4)
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b1 p1
Input Data
BPSK Modulation
h1
Re{ r
H
Re{ r
H
v1
^
v2
^
b1
w1 }
n Input Data
b 2 p2 BPSK Modulation
h2
+
Input Data
b2
w 2}
. . .
. . .
+
r
bM pM
vM
BPSK Modulation
hM
Re{r
H
^
bM
wM }
Fig. 1. Model of a Gaussian vector channel. The channel propagation vectors {hi } and the thermal noise vector n have a complex Gaussian distribution. This model corresponds to a multiuser SIMO system with arbitrary-power users, and the received signal at each user’s N -branch receiver being subject to cochannel interference, fading, and additive thermal noise.
The utility of (1) stems from the minimal constraints on the variables X and Y , and the constants a and b. The N -dimensional vector having a complex Gaussian distribution with mean m and covariance matrix R is written as fx (x) =
1 π N |R|
exp(−(x − m) R−1 (x − m)) H
(5)
and denoted x ∼ CN (m, R). We wish to examine the ratio of quadratic forms where X = xH Ax and Y = xH Bx with x ∼ CN (m, R). The bivariate MGF exp(s1 xH Ax + s2 xH Bx)fx (x) d x MX ,Y (s1 , s2 ) = x H
exp(−m R−1 m) |I − s1 RA − s2 RB| × exp mH R−1 (R−1 − s1 A − s2 B)−1 R−1 m =
can be derived via the integral property x
exp(−xH Ax − aH x − xH b) dx =
πN exp(aH A−1 b). |A|
We restrict attention to the evaluation of (3) for k = 1 and (1 ,0 ) b = 1/2, and derive MX ,Y (0 , −s) via the properties ∂|F(x)| ∂x ∂F−1 (x) ∂x
−1
= |F(x)|trace F = −F−1 (x)
∂F(x) (x) ∂x
∂F(x) −1 F (x). ∂x
(1 ,0 )
Subsequent substitution of MX ,Y (0 , −s) into (3) allows us to arrive at the integral shown in (6), where we have used a change of variable t = exp(−s) to obtain an integral with finite limits. In the more specific scenario of x ∼ CN (0, R)
equation (6) reduces to
H x Ax
E √
xH Bx m=0 1 trace(RA(I − ln(t)RB)−1 ) 1 = Γ(0.5) 0 |I − ln(t)RB| × t−3/2 dt.
(7)
It is noteworthy that (6) and (7) consist of a single integral although N variates are present. III. P ERFORMANCE A NALYSIS The conventional approach to performance analysis of MRC in the presence of cochannel interference has relied upon obtaining the distribution of the signal-to-interference plus noise ratio (SINR). This is followed by averaging a conditional BER expression for a particular modulation scheme over the distribution of the SINR. In this paper we shall take the alternate approach of performing the analysis directly on the decision statistic rather than on the SINR. We consider the output of the vector channel in Fig. 1 to be r = HPb + n
(8)
with the ith column matrix H denoted as hi , the √ N ×M √ √ of the matrix P = diag( P1 , P2 , . . . , PM ), and Pi denoting the transmit power of the ith user. Confining analysis to a single bit interval, we define the vector b = [b1 , b2 , . . . , bM ]T as the transmitted information at a given time instant, with bi ∈ {−1, 1} via binary phase-shift keying (BPSK) modulation and P [bi = 1] = P [bi = −1] = 0.5 ∀ i. We shall assume hi ∼ CN (mi , Ri ) ∀ i, and the additive receiver noise will be accounted for via the vector n ∼ CN (0, N). For the remainder of this presentation we shall restrict attention to the system of Fig. 1 with the columns of the channel matrix corresponding to the propagation vectors of the M users.
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xH Ax E √ xH Bx
= × = ×
exp(−mH R−1 m) ∞ exp mH R−1 (R−1 + sB)−1 R−1 m s−1/2 Γ(0.5) |I + sRB| 0 H −1 −1 −1 −1 −1 −1 m R (R + sB) A(R + sB) R m + trace(RA(I + sRB)−1 ) ds exp(−mH R−1 m) 1 exp mH R−1 (R−1 − ln(t)B)−1 R−1 m t−3/2 Γ(0.5) |I − ln(t)RB| 0 H −1 −1 −1 −1 m R (R − ln(t)B) A(R − ln(t)B)−1 R−1 m + trace(RA(I − ln(t)RB)−1 ) dt
Thus, the analysis will be applicable to a multiuser singleinput multiple-output (SIMO) system with the received vector r consisting of a superposition of the desired user’s signal, L = M − 1 interfering signals, and additive noise. By virtue of users transmitting signals independently, the N -dimensional propagation vectors present at a given receiver are assumed to be mutually independent. However, the individual vector elements, denoting the received signal present at different branches but originating from a common source, may share a certain amount of correlation. Mathematically, this translates to the joint probability density function (pdf) of the desired signal and interference as seen by user d from a set of M active users, being given by fhd ,h1 ,...,hL (hd , h1 , . . . , hL ) = fhd (hd )
L
fhm (hm ).
m=1
In effect, the N -dimensional vector hi = denotes the [ρi,1 ejφi,1 , ρi,2 ejφi,2 , . . . , ρi,N ejφi,N ]T complex random channel gains experienced by the transmitted signal of the ith user among N diversity branches. After conditioning upon the vector hd , the distribution of√the channel output in (8) is obtained as r ∼ CN (bd Pd hd , i=d Pi Ri + N). It is common for the output of a vector channel to be processed via a linear operation to yield the decision statistic vd = {wdH r}
(9)
and the recovered bit according to ˆbd = sgn{vd } . Within a detection-theoretic framework, the complex vector wd is a function of the matrix H for minimum meansquare error (MMSE) and decorrelator detectors [11], or alternatively a function of the vector hd for simpler (and inferior) detectors. In this presentation we shall restrict attention to the latter class of detectors when specifying on hd we obtain vd ∼ wd . Having √ already conditioned N ( {bd Pd wdH hd }, 12 wdH ( i=d Pi Ri + N)wd ) with the corresponding BER given by E[vd ] P [e|hd ] = Q V ar[vd ] ⎛ ⎞ H {wd hd } ⎠ (.10) = Q ⎝ 1 H w ( P R + N)w i i d i=d 2Pd d In the derivation of the above expression it was recognized that, by symmetry, the probability of error conditioned on bd =
(6)
1 will be equal to that when bd = −1. The unconditional BER is ⎞⎤ ⎡ ⎛ H {wd hd } ⎠⎦ Pe = E ⎣Q ⎝ 1 H w ( P R + N)w d i=d i i 2Pd d ⎛ ⎡ ⎤⎞ H {w h } d d ⎦⎠(11) ≥ Q ⎝E ⎣ 1 H( w P R + N)w d i=d i i 2Pd d with the expectation taken with respect to hd . The lower bound follows from Jensen’s inequality and the convexity of the Q-function over [0, +∞). Within the diversity combining literature the MRC specification requires that wd = hd . With MRC, the lower bound in (11) can be written as ⎤⎞ ⎛ ⎡ H h h d d ⎦⎠(12) Pe ≥ Q ⎝ E ⎣ 1 H h ( P R + N)h i i d i = d d 2Pd with the argument inside the Q-function expressed in the form of (6) with B = 2P1 d P R + N , A = I, i=d i i R = Rd , and m = md . It should be apparent that the substitution of (7) into (12) would correspond to the more specific scenario in which the desired user’s signal experiences arbitrarily correlated Rayleigh fading. The tightness of the above bound will be investigated in the following section, however, two benefits seem evident at this point. Firstly, the general applicability of the lower bound to various statistical fading conditions, and secondly, the fact that the bound will consist of a single integral (inside the Q-function) with finite limits irrespective of the diversity order (N ) or the number of users (M ). For reference, we provide the expression for the exact BER with MRC combining ⎞⎤ ⎡ ⎛ H hd hd ⎠⎦ Pe = E ⎣Q ⎝ 1 H h ( P R + N)h i i d i=d 2Pd d π/2 2 1 h ) −Pd (hH d d = exp 2 π 0 hH hd i=d Pi Ri + N)hd sin θ d ( × fhd (hd )dhd dθ
(13)
by considering an alternate form of the Q-function which has finite limits [12]. We note that the exact BER expression consists of an N -fold integral, whereas the lower bound consisted of a single integral with finite limits irrespective of P R the diversity order. A further simplification of i=d i i + N = ( i=d Pi Ωi + η)I exists if N = ηI, and the interferers have independent identically distributed (i.i.d.) fading statistics at each diversity branch, that is if Ri = Ωi I ∀ i = d with
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 4, APRIL 2008
Ωi E[|hij |2 ] = E[ρ2ij ] ∀ j, i = d. The conditional BER of (10) will reduce to ⎞ ⎛ π/2 H 1 Pd hd hd ⎠ dθ, exp ⎝− P [e|hd ] = 2 π 0 P Ω + η sin θ i i i=d (14) with the unconditional probability derived as −1 exp(−mH d Rd md ) Pe = π N +1 π/2 −1 −1 a −1 −1 exp mH Rd md d Rd (Rd + sin2 θ I)
dθ ×
I + a2 Rd
0 sin θ with a = Pd /( i=d Pi Ωi + η). Furthermore, if md = 0 and Rd = Ωd I with Ωd E[|hdj |2 ] ∀ j, then the exact BER can be expressed as N sin2 θ 1 π/2 Pe = dθ π 0 sin2 θ + c k N −1 1 − β2 1 2k = (15) 1−β k 2 4 k=0 c/(1 + c) and c = Pd Ωd /( i=d Pi Ωi + η). with β = Although the above expression provides an exact BER in closed-form, the applicability of the result is limited to the scenario of i.i.d. Rayleigh fading for the received signal of the desired user, and i.i.d. fading among the components of the interferers’ propagation vectors.
IV. S IMULATION R ESULTS AND D ISCUSSION The analytical results derived in the previous section warrant quantitative investigation. The objective of this section is twofold. First, we wish to investigate the tightness of the lower bound on the performance of MRC over a correlated Gaussian vector channel. Secondly, with the scenario of mutually independent fading among the diversity branches as a baseline, we wish to examine the impact of correlated fading on system performance. Prior to the quantitative investigation, we examine the application of the correlated Gaussian vector channel model of Fig. 1 to a multiuser frequency diversity system. Subsequently, in the performance analysis, we shall restrict attention to the users experiencing Rayleigh faded channels. Frequency diversity in the form of orthogonal frequency division multiplexing (OFDM) has been adopted for high data-rate wireless multimedia applications as well as being a candidate for upcoming cellular radio standards. The most promising and most heavily investigated multiple access scheme incorporating OFDM has been multicarrier CDMA (MC-CDMA) [1]. Within a MC-CDMA system, encoding (decoding) of N subcarriers is performed prior to transmission (detection) in order to provide a level of orthogonality among the M active users. Considering ci as a length N ”codeword” with ci [n] ∈ {−1, 1} we define the matrix Ci diag(ci ). Adopting the notation of [13] for the more general scenario of uplink transmission, the channel output is represented by r = SPb + n
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with the ith column of the N × M matrix S defined as si Ci hi . With BPSK modulation the desired user’s transmitted bit will be recovered via ˆbd = sgn{ {tH d r}} with td Cd wd , where wd = hd for MRC. Conditioning on the vector hd followed√by straightforward Manalysis yields νd = 1 H ∼ N ( Pd sH {tH d r} d sd , 2 sd ( i=1, i=d Pi Ci Ri Ci + N)sd ), and thus P [e|hd ]
=
Q
=
Q
M
1 sH ( 2Pd d
sH d sd
i=1, i=d
M
1 hH ( 2Pd d
Pi Ci Ri Ci + N)sd
hH d hd
i=1, i=d Pi Ci Ri Ci + N )hd
.
In the above expression we have used the fact that sH i si = h as well as assigned C = C C = C C and N = hH i i d d i i i Cd NCd . Hence, the lower bound on the unconditioned BER will be given as
Pe
≥ Q E
M
1 hH ( 2Pd d
hH d hd
i=1, i=d Pi Ci Ri Ci + N )hd
(16)
with the argument inside the Q-function expressed as the integral A = I, R = Rd , and B = Min (7) with 1 ( P C R i=1, i=d i i i Ci + N ). Of course, in the more 2Pd specific scenario of N = ηI and i.i.d. fading among the received subcarriers of each of the M users, the exact BER can be represented in closed-form via (15). In order to quantify the impact of correlated fading on the desired user’s BER, we shall consider a specific empirical channel model when specifying the users’ covariance matrices. Incorporating the notation and definitions of [14] for an isotropic scattering environment, we specify the parameters u1ij =
Jo (2πfm τij ) 2 1 + kij
u2ij = −kij u1ij
with Jo (.) denoting the zero-order Bessel function. In correspondence with the notion of equal frequency spacing between each pair of adjacent subcarriers (i.e., OFDM-based systems) we shall assume kij = |i − j|k, ∀ i, j; where k = 2π(Δf )στ indicates an equal frequency separation of Δf between adjacent channels. Thus, the elements of the desired user’s covariance matrix are determined as ⎧ i>j ⎨ Ωd u1ij + jΩd u2ij i<j Ωd u1ij − jΩd u2ij Rd (i, j) = (17) ⎩ i = j. Ωd For convenience, in the simulations we shall assume the fading statistics of the interferers to be identical to that of the desired user via the specification Ri = Rd ∀ i. It is desired to determine the tightness of the derived lower bound on the performance of MRC over a correlated Gaussian vector channel. Also, with the scenario of mutually independent fading among the diversity branches as a baseline, we wish to examine the impact of correlated fading on system performance. The MC-CDMA system will consist of N = 32 subcarriers with each user randomly selecting a codeword. Furthermore, a frequency separation of Δf = 312.5 kHz among adjacent subcarriers (IEEE 802.11a), channel delay
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10
−1
10
10
10
−2
−2
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5
10
15 Interferers (L)
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25
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10
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Fig. 2. Illustration of the tightness of the lower bound within the context of a frequency diversity system with N = 32 subcarriers used for transmission.
spread of στ = 0.1μs, and τij = 0 indicating no time offset between reception of the subcarriers will be assumed. The mean signal-to-noise ratio (SNR) per bit of the desired user will be defined as SN R =
10
N Pd Ωd 2η
with SN R = 5, 10, and 20 dB used in the simulations and referred to as low, moderate, and high SNR, respectively. We shall presume all users to be of equal transmit power via the assignment Pi = Pd ∀ i. The tightness of our derived lower bound on the BER is illustrated in Fig. 2. The exact BER was obtained via a 106 sample Monte Carlo simulation of (13) and subsequent averaging over the number of iterations. Generally, an improvement in computation time and processing will be realized in the Monte Carlo simulation of such equations as opposed to fullscale system-level simulation intended to arrive at the same numerical performance results. The finite-limit integral in the lower bound was computed via Simpson’s rule. It is observed that the lower bound is within an order of magnitude of the exact BER irrespective of the SNR value or the number of interferers. The lower bound is tighter for a larger number of interferers with the deviation from the exact BER being less than a factor of six for L ≥ 20. The MRC performance degradation due to correlated fading is shown in Fig. 3 via simulation of (13) for correlated fading and computation of (15) in the case of mutually independent fading among the components of all the users’ channel vectors. In the high SNR regime, a performance loss of an order of magnitude is apparent for L < 6 with the degradation diminishing for higher levels of interference. Similarly, in the moderate SNR region the degradation appears to be at least an order of magnitude with L < 4 and diminishing steadily afterwards. A maximal degradation of a factor of two is observed in the low SNR regime. This may be explained by noting that an increase in interference or noise will lead to a higher error floor. Such an error floor will exist regardless of the properties of the users’ channel covariance matrices. Thus, there will be a
−3
−4
SNR=5dB, correlated fading SNR=5dB, independent fading SNR=10dB, correlated fading SNR=10dB, independent fading SNR=20dB, correlated fading SNR=20dB, independent fading
−5
−6
5
10
15
20
25
30
Interferers (L)
Fig. 3. Comparison of the BER of a MC-CDMA system with correlated and uncorrelated fading among the components of the users’ propagation vectors. The results were obtained via computation of the analytical expressions for the exact BER with correlated and uncorrelated fading.
decreasing advantage in having full diversity (that is, complete independence among branches) simply because the benefits will be less apparent due to the high levels of interference or noise. It is also observed that, for both correlated and independent fading, the SNR heavily influences the BER in a lightly loaded system, but is of diminishing value for increasing levels of interference. V. C ONCLUSION In this paper the bit error rate of MRC was examined over a correlated Gaussian vector channel. A lower bound was derived with the variable-power users’ channel vectors following a complex Gaussian distribution with arbitrary means and covariance matrices. Exact BER expressions were examined in less general cases, such as with the interferers’ signals experiencing i.i.d. fading, and all users experiencing i.i.d. Rayleigh fading. Numerical results illustrating the impact of the number of cochannel interferers, different SNR conditions, and correlated fading on a user’s BER were presented within the context of a frequency diversity system. In general, the BER degradation caused by statistical dependence among diversity branches is most pronounced for increasing SNR values along with decreasing levels of interference. ACKNOWLEDGMENT The authors wish to thank Robert Berk for bringing reference [9] to their attention. We also thank Xiao-Li Meng for an insightful discussion. R EFERENCES [1] S. Hara and R. Prasad, “Overview of multicarrier CDMA,” IEEE Commun. Mag., vol. 35, no. 12, pp. 126-133, Dec. 1997. [2] R. L. Pickholtz, D. L. Schilling, and L. B. Milstein, “Theory of spreadspectrum communications - a tutorial,” IEEE Trans. Commun., vol. COM-30, pp. 855-884, May 1982. [3] S. Gezici, H. Kobayashi, H. V. Poor, and A. F. Molisch, “Performance evaluation of impulse radio UWB systems with pulse-based polarity randomization,” IEEE Trans. Signal Processing, vol. 53, pp. 2537-2548, July 2005.
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[4] M. K. Simon and M. S. Alouini, Digital Communication Over Fading Channels. New York: Wiley, 2000. [5] M. Z. Win, G. Chrisikos, and J. H. Winters, “MRC performance for M-ary modulation in arbitrary correlated Nakagami fading channels,” IEEE Commun. Lett., vol. 4, pp. 301-303, Oct. 2000. [6] Q. T. Zhang, “Maximal-ratio combining over Nakagami fading channels with an arbitrary branch covariance matrix,” IEEE Trans. Veh. Technol., vol. 48, pp. 1141-1150, July 1999. [7] V. Veeravalli, “On performance analysis for signaling on correlated fading channels,” IEEE Trans. Commun., vol. 49, pp. 1879-1883, Nov. 2001. [8] X. Zhang and N. C. Beaulieu, “Outage probability of MRC with unequal-power cochannel interferers in correlated Rayleigh fading,” IEEE Commun. Lett., vol. 10, pp. 7-9, Jan. 2006.
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[9] X. Meng, “From unit root to Stein’s estimator to Fisher’s k statistics: if you have a moment, I can tell you more,” Statistical Science, vol. 20, pp.141-162, 2005. [10] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products. San Diego, CA: Academic Press, 2000. [11] R. R. Muller and S. Verdu, “Design and analysis of low-complexity interference mitigation on vector channels,” IEEE J. Select. Areas Commun., vol. 19, pp. 1429-1441, Aug. 2001. [12] M. K. Simon and D. Divsalar, “Some new twists to problems involving the Gaussian integral,” IEEE Trans. Commun., vol. 46, pp. 200-210, Feb. 1998. [13] A. M. Tulino, L. Li, and S. Verdu, “Spectral efficiency of multicarrier CDMA,” IEEE Trans. Inform. Theory, vol. 51, pp. 479-505, Feb. 2005. [14] W. C. Jakes, Ed., Microwave Mobile Communications. New York: IEEE Press, 1974.
