SpaceâTime Coding Over Correlated Fading ... - Semantic Scholar
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Space–Time Coding Over Correlated Fading Channels With Antenna Selection Israfil Bahceci, Yucel Altunbasak, Senior Member, IEEE, and Tolga M. Duman, Senior Member, IEEE
Abstract—In a previous paper by Bahceci et al., antenna selection for multiple-antenna transmission systems under the assumption that the subchannels between antenna pairs fade independently was studied. In this paper, the performance of such systems when the subchannels experience correlated fading is considered. It is assumed that the channel-state information (CSI) is available only at the receiver, the antenna selection is performed only at the receiver, and the selection is based on the instantaneous received signal power. The effects of channel correlations on the diversity and coding gain when the receiver system is a subset of the antennas are quantified. Theoretical results indicate that the correlations in the channel do not degrade the diversity order, provided that the channel is full rank. However, it does result in some performance loss in the coding gain. Index Terms—Antenna selection, diversity, multiple-antenna communications, multiple-input–multiple-output (MIMO) systems, pairwise error probability (PEP), space–time coding, spatial fading correlation.
I. I NTRODUCTION
D
URING the last decade, multiple-input–multiple-output (MIMO) antenna technology has emerged as a key technology for enabling high-speed wireless communications. In [2] and [3], it is shown that multiple-antenna systems are able to achieve larger capacities and improved performance in comparison to their single-antenna counterparts. These benefits can be reaped by recently proposed space–time coding techniques for MIMO systems [4]. For practical applications, however, the cost and the complexity of implementing such systems are significant because of the large number of RF chains required. Such concerns have led many researchers to develop methods that can reduce the implementation cost while retaining the benefits of MIMO systems. Antenna-subset selection, which seeks the utilization of a subset of all available antennas at the transmitter and/or receiver, is such a technique [5]–[10]. Most of the work on multiple-antenna systems make the assumption that the subchannels between transmit/receive antenna pairs experience independent identically distributed (i.i.d.) fading [3], [4], [11]–[13]. A more realistic assumption, Manuscript received November 22, 2003; revised September 20, 2004; accepted December 27, 2004. The editor coordinating the review of this paper and approving it for publication is H. Jafarkhani. The work of I. Bahceci and Y. Altunbasak is supported by the National Science Foundation (NSF) Award CCR-0105654 and the work of T. M. Duman is supported by the NSF CAREER Award CCR-9984237. I. Bahceci was with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0250 USA. He is now with the Department of Electrical Engineering, University of Waterloo, Waterloo, ON N2L 3G1, Canada (e-mail: [email protected]). Y. Altunbasak is with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0250 USA (e-mail: [email protected]). T. M. Duman is with the Department of Electrical Engineering, Arizona State University, Tempe, AZ 85287-5706 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TWC.2005.858004
however, is that the fades are not independent, because of insufficient spacing between antenna elements, placement of scatterers, etc.[14]. The nonzero correlation between subchannels may significantly reduce the capacity as shown in [15]. The effect of such correlations on the system performance is studied in [16] and [17]. Recent work on correlated fading includes [18]–[21]. Hong et al. investigated the design and performance of spatial multiplexing for MIMO correlated fading channels in [18]. In [19], Ivrlac et al. studied the effects of fading correlations and transmitter-channel knowledge on the capacity and cutoff rate for MIMO systems, and in [20], Chiani et al. derived a closedform expression for the characteristic functions for MIMOsystem capacity for the correlated-fading case. Smith et al. also studied the capacity of MIMO systems, but they focused on semicorrelated flat fading [21]. The effects of subchannel correlation when antenna-subset selection is employed have interesting implications. For instance, in [5], Gore et al. considered the capacity of MIMO systems with antenna selection when the channel is rank deficient, and show that a larger capacity can be achieved by using a “good” subset of transmit antennas (i.e., by using those antennas that result in a full-rank channel). In [6], following the work in [5], Sandhu et al. propose an efficient method to find the optimal subset of antennas. Another line of work investigates antenna selection based on error probability [7], [8]. In [7] and [8], Gore et al. assume that the channel statistics change very slowly and that the selected antenna subset remain the same over the transmission period. To develop the criteria for selection, the authors derive bounds on average pairwise error probabilities (PEPs) for the full-complexity system over correlated fading that depend on the channel covariance matrix and select the subset of antennas that minimize those bounds. Note that in these studies, error probability for a system using antenna selection is not formulated at all. Only the error-probability expressions for the full-complexity system are considered. In [1], we studied the performance of received-power-based antenna selection for MIMO systems when the subchannels undergo (independent) quasi-static fading. We derive upper bounds on the PEP for the system with antenna selection and show that one can achieve, by using antenna selection, the same diversity order as that obtained by the full-complexity system while experiencing some loss in the coding gain. We also show that the diversity order decreases dramatically when the space–time code employed is rank deficient. In this paper, we consider the case when correlations exist among the subchannels. Assuming the presence of only transmit correlation, we investigate the performance of MIMO antenna selection. Our analysis quantifies the dependence of the selection performance on the correlation matrix of the channel. We show that one can achieve the same diversity order as
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the full-complexity system, even under heavy correlation, as long as the channel correlation matrix is nonsingular. With extensive simulations, we validate our theoretical analysis and demonstrate the effect of fading correlations when antenna selection is employed. In the next section, we describe the signal model and present the performance bound for the full-complexity MIMO system. We then derive the PEP for the system with antenna selection in Section III. The numerical results are presented in Section IV and the concluding remarks are summarized in Section V.
where λm are the nonzero eigenvalues of ∆∆H and r = rank ˆ denoting the codeword difference (∆∆H ) with ∆ = S − S matrix. From Inequality (2), we conclude that the diversity of the multiple-antenna system over correlated fading is the same as that obtained for the uncorrelated fading channel while there is some loss in coding gain, depending on the correlation structure. Note that these results are valid when the channel is full rank, i.e., the covariance matrix of the channel is nonsingular.
II. S YSTEM M ODEL AND PEP
In this section, we present the PEP analysis for systems employing antenna selection. First, we consider the case where a single antenna is selected. Then, we generalize our analysis to the case where more than one antenna are selected.
We consider a system equipped with M transmit and N receive antennas. We denote by H the N × M channel transfer matrix h11 · · · h1M . .. .. H = .. . . . h N 1 · · · hN M The channel is modeled as a flat Rayleigh-fading (i.e., hnm ∼ CN (0, 1)) channel that remains constant over a block of t symbols and changes independently from one block to the next. The received signal matrix XN ×t is given by ρ X= HS + W (1) M where S is the M × t transmitted signal matrix (selected from a space–time codeword alphabet), and W is the N × t additive white Gaussian noise matrix. The average energy of the transmitted signal is normalized to unity over M antennas so that ρ is the expected signal-to-noise ratio at each receive antenna. We assume that channel-state information (CSI), i.e., H, is known at the receiver, but not at the transmitter. The assumption of i.i.d. fading is typically made to model the channel [4]. However, in real-time propagation, the presence of local scatterers around the transmitter and the receiver induces correlations among subchannels that can be modeled as H = 1/2 1/2 1/2 H/2 R(r) Hw R(t) , where R(r) = R(r) R(r) is the receive co1/2
H/2
variance matrix, R(t) = R(t) R(t) is the transmit covariance matrix, and Hw is a matrix with i.i.d. CN (0, 1) entries [14].1 In this paper, we consider the existence of only transmit correlation, e.g., the antenna separation at the receiver is sufficiently large so that fading associated with each receive antenna is 1/2 uncorrelated. Such channels can be modeled as H = Hw R(t) [14]. Hence, it is assumed that the N rows of H are i.i.d. complex Gaussian vectors with covariance matrix R(t) . The PEP for a full-rank channel,2 assuming maximumlikelihood decoding at the receiver, is given by [8] ˆ ≤ P (S → S) R(t) N (
1 r m=1
N
λm )
ρ −N r 4M
(2)
III. E RROR -P ROBABILITY A NALYSIS W ITH A NTENNA S ELECTION
A. PEP Analysis With Single-Antenna Selection If only one antenna is selected out of the N receive antennas, the selection rule is reduced to choosing the antenna element that observes the largest instantaneous SNR, i.e., i = argmax |hi1 |2 + · · · + |hiM |2 . i=1,...,N
In this case, the Chernoff bound on the PEP can be expressed as (see [1]) ˆ P (S → S)
N −1 ρ 2 ≤ N e− 4M r∆ FZ r 2 fR (r)dr
−1 H N −1 ρ 1 2 e−rR(t) r dr = N e− 4M r∆ FZ r 2 π M R(t) (3) where FZ (·) is the cumulative distribution function (cdf) of Z = rrH . Using the singular-value decomposition (SVD) ∆∆H = UΛUH in (3), and then letting β = rU, we obtain ˆ P (S → S)
N −1 ρ H ≤ N e− 4M βΛβ FZ β 2 −βUH R−1 Uβ H
(t) ×e dβ
ρ H H −1 N e−β 4M Λ+U R(t) U β FZ β 2 N −1 dβ. = M π R(t) (4)
To find the FZ (·), we need to evaluate the probability FZ (a) = P {Z ≤ a}
= P |hi1 |2 + · · · + |hiM |2 ≤ a
= fRi (r)dr C
1 (·)H
denote the Hermitian transpose. (full-rank) channels refer to channels with singular (nonsingular) covariance matrices within this manuscript. 2 Low-rank
1 π M R(t)
=
1 π M R(t)
e C
−1 H −rR(t) r
dr
(5) (6)
(7)
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where C is the region {hi1 , . . . , hiM , : |hi1 |2 + · · · + |hiM |2 ≤ a}. Using the SVD of R(t) = VMVH in (7), and the routine integration tools, we obtain FZ (a) =
1 |M|
−1
e−(µ1
u1 +···+µ−1 uM ) M
ˆ ≤ P (S → S) du1 · · · duM .
u1 +···+uM ≤a
(8) The evaluation of this integral results in FZ (a) = 1 −
M
−1
kj e−µj
For high SNR and full-rank space–time codes, i.e., ρ → ∞, and rank(∆) = M , we can further simplify the expression as
a
(9) ˆ ≤ P (S → S)
where kj = M
i=1,i =j
µj − µi
,
j = 1, . . . , M.
(10)
Substituting (9) in (4) and simplifying the resulting expression, we arrive at
ˆ ≤N P (S → S) e−(λ1 u1 +···+λM uM ) M j=1 µj (N −1) M u +···+uM − 1 µ j × 1 − kj e du1 · · · duM (11)
FZ (a) = 1 −
−1 −1 µ2 µ1 e−µ2 a − e−µ1 a , µ2 −µ1 µ1 −µ2
FZ (a) = 1 −
ˆ ≤ P (S → S)
µ1 (µ1 −µ2 )
2 1 − µ1 µ2 |Λ | Λ +
µ2 (µ2 −µ1 )
− Λ +
I µ1
.
I µ2
j=1
N −1
µj
l=0
C(N − 1, l)(−1)
l
M j1=1
−1 µ23 e−µ3 a (µ3 − µ2 )(µ3 − µ1 )
−
−1 µ22 e−µ2 a (µ2 − µ3 )(µ2 − µ1 )
−
−1 µ21 e−µ1 a . (µ1 − µ2 )(µ1 − µ3 )
µ1
µ22
−
(µ2 −µ1 )(µ2 −µ3 )
Λ +
I µ2
−
µ23 (µ3 −µ1 )(µ3 −µ2 )
Λ +
I µ3
. (16)
At high SNR, this bound can be approximated by ˆ ≤ P (S → S)
2 g(λ1 , λ2 , λ3 ) ρ −6 (µ1 µ2 µ3 )2 (λ1 λ2 λ3 )2 4M
(17)
where g(·) is a function that depends only on λi , i = 1, 2, 3. Thus, the diversity order is equal to M N = 6, which is equal to the diversity order of the full-complexity system. On the other hand, we can obtain Chernoff bounds, if rank(∆) = 1 < M , as ρ −1 ˆ ≤ ξ1 ζ, R(t) P (S → S) 4M
(13)
ˆ ≤N P (S → S) M
(15)
When the space–time code is full rank, the bound for M = 3 and N = 2 is given by µ21 2 1 (µ −µ )(µ1 −µ3 ) 1 2 ˆ ≤ P (S → S) − µ1 µ2 µ3 |Λ | Λ + I
for µ1 = µ2 .
Using the cdf, the Chernoff bound in (12) for M = N = 2 can be simplified to
ρ −1 2 . (µ1 µ2 )λ1 4M
This result implies that for rank-deficient space–time codes, antenna selection degrades the diversity order significantly, i.e., the diversity order with selection is 1, while it is N r = 2 with the full-complexity system. Example 2: When M = 3, assuming that µi = µj , for i = j, i, j = 1, 2, 3, we have
j=1
where λi , i = 1, . . . , M are the eigenvalues of Λ = −1 U. For specific values of N , (ρ/4M )Λ + ζ with ζ = UH R(t) (11) can easily be evaluated. In fact, a closed-form expression for any values of M and N can also be obtained. In terms of Λ and µj , the final result can be expressed as shown in (12) at the bottom of the page. Unfortunately, for the general case, the closed-form expression does not give much insight about the effect of the correlation on the system performance. Therefore, in what follows, we will present a few special cases. We will also provide some numerical results later in Section IV. Example 1: For M = 2, the cdf in (9) is given by
(14)
Hence, the diversity order remains the same as that of the fullcomplexity system, while there is some loss in the coding gain that depends on the determinant of the correlation matrix. If the underlying space–time code is rank deficient, i.e., rank(∆) = 1 < M , then we have
j=1
−1 µM j
ρ −4 1 2 . (µ1 µ2 )2 (λ1 λ2 )2 4M
···
M jl =1
−1 −1 kj1 · · · kjl Λ + µ−1 j1 + · · · + µjl IM
(18)
(12)
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and if rank(∆) = 2 < M , as ρ −2 ˆ ≤ ξ2 ζ, R(t) P (S → S) 4M
(19)
where ξ1 (·) and ξ2 (·) depend only on ζ and R(t) , but not ρ. These bounds indicate the degradation in the diversity order due to antenna selection when we have low-rank space–time codes. We note that if some of the eigenvalues of R(t) are identical, we can evaluate the integral in (8) as well. We also note that the above results are valid only if R(t) is nonsingular. B. Selection of More Than One Antenna We now obtain the bounds on the PEP for the case when L out of N antennas are selected. Using a similar line of argument as in [11, eq. (30)], we arrive at ˆ ≤ P (S → S)
L
ρ
˜
e− 4M H∆
l=1 R
l
× 1 −
2
M
N! (N − L)!L!L −rl 2 µj
kj
πM L
j=1
×e
−1 H −1 H − r1 R(t) r1 +···+rL R(t) rL
Analytical evaluation of this integral over Rl is a formidable task. Integrating over the whole space, although resulting in a looser bound, yields a mathematically tractable analysis. In this case, integration over Rl will not depend on l and the analysis results in the upper bound given in (22) at the bottom of the page. Since further simplification of this expression is not analytically tractable, we have to resort to numerical examples to present the system performance. However, it is possible to obtain simpler expressions for special cases, as shown in the next example. Example 3: When M = N = 3 and L = 2, the bound in (22) can be written as 3 ˆ ≤ 1 − P (S → S) 2 (µ1 µ2 µ3 ) |Λ | |Λ | −
1 R(t) L
dr1 · · · drL (20)
ˆ ≤ P (S → S)
L
e
−
j=1
rj
ρ 4M
l=1 R
l
×
−1 Λ+UH R(t) U rH j
M
I µ2
−
I µ1
µ23 (µ3 −µ1 )(µ3 −µ2 )
Λ +
I µ3
.
(23)
3 g(λ1 , λ2 , λ3 ) ρ −6 . (µ1 µ2 µ3 )2 |Λ | µ1 µ2 µ3 (λ1 λ2 λ3 )2 4M (24)
ˆ ≤ P (S → S)
L
Λ +
Λ +
Finally, letting ρ → ∞, we arrive at
which can be rewritten as ˆ ≤ P (S → S)
µ22 (µ2 −µ1 )(µ2 −µ3 )
µ21 (µ1 −µ2 )(µ1 −µ3 )
At high SNR, we can further simplify this bound using (17) to obtain
N −L
37
r 2 − µl j
N! 1 − kj e (N − L)!L!L j=1
N −L
ρ −9 3g(λ1 , λ2 , λ3 ) . (µ1 µ2 µ3 )3 (λ1 λ2 λ3 )3 4M
(25)
Hence, the diversity order of the system using antenna selection is M N = 9, which is equal to the diversity order achieved by the full-complexity system. For rank-1 and rank-2 space–time codes, i.e., rank(∆) = 1 or 2, the asymptotic performance is given by, respectively ρ −2 ˆ ≤ ξ ζ, R(t) P (S → S) 1 4M
(26)
Rl = {r1 , . . . , rL : rl < rk , k = 1, . . . , l − 1, l + 1, . . . , L}
ρ −4 ˆ ≤ ξ ζ, R(t) P (S → S) 2 4M
(27)
˜ is the L × M matrix formed by deleting the rows of H and H corresponding to the antennas that are not selected.
where ξ1 (·) and ξ2 (·) can be obtained in a similar fashion as ξ1 (·) and ξ2 (·). The expressions in (26) and (27) indicate
1 × L dr1 · · · drL M L R(t) π
(21) and
where the region Rl is defined as
ˆ ≤ P (S → S)
1 N! L (N − L)!L! R(t) |Λ |L−1 N −L M M −1 −1 × C(N − L, l)(−1)l ··· kj1 · · · kjl Λ µ−1 j1 + · · · + µjl IM l=0
j1 =1
jl =1
(22)
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Fig. 1. PEP versus SNR.
Fig. 2. PEP versus SNR.
that the diversity order with antenna selection is degraded significantly when the space–time code is rank deficient. IV. E XAMPLES In the previous sections, we have theoretically analyzed the performance and derived several bounds on the PEP. We now evaluate those bounds for several codeword pairs that
are selected from the codes developed in [4]. We present the effect of fading correlation (including the case of both full-rank nonfull-rank space–time codes), and compare the performance of the system against that of the same system over uncorrelated fading (results of [1]). In Figs. 1 and 2, we compare the PEP bounds for the fullcomplexity system and the one using antenna selection for the case of transmit correlation. In both figures, the performance
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of a full-rank code (solid lines) and a rank-deficient code (dashed lines) is presented. Fig. 1 illustrates the results for the case of double transmit and the double-receive antenna system when rc = 0.54 + 0.72j. The full-rank space–time codeword pairs are selected from the 2 bit/s/Hz eight-state space–time trellis codes using 4-phase-shift keying (PSK) modulation (with M = 2). This code provides a diversity advantage of 6 [4], i.e., full spatial diversity. The two codewords considered differ in three consecutive symbols. For comparison purposes, we also present the PEP bounds when there is no correlation, i.e., i.i.d. fading. We observe that even for a high level of correlation, i.e., |rc | = 0.9, although there is some loss in the coding gain, the diversity orders of both the full-complexity system and the system using antenna selection are the same. The Chernoff bound evaluated using (13) is also plotted, and it is about 2-dB away from the exact PEP. From the PEP curves obtained for rank-deficient space–time codeword pairs, we observe that: 1) the performance of the full-complexity system under correlated fading and i.i.d. fading is very close to each other, while the performance of the system with antenna selection is superior when there is correlated fading; and 2) the diversity order is reduced when antenna selection is performed. The PEP for the case of M = 3 transmit and N = 2 receive antennas is presented in Fig. 2. We assume that the channel correlation matrix is given by
R(t)
1 = 0.6 0.4
0.6 1 0.45
0.4 0.45 . 1
(28)
For this correlation structure, we observe that the loss in the coding gain due to antenna selection is about 4 dB for the case of both i.i.d. fading and correlated fading. We note that the Chernoff bound (16) for the system with antenna selection is about 2-dB away from the exact PEP. The bound, on the other hand, is not as strict as this when the space–time code is rank deficient, i.e., it is about 6-dB away from the exact PEP. V. C ONCLUSION We analyzed the performance of MIMO systems with antenna selection under correlated fading channels. We considered a semicorrelated fading-channel model assuming the presence of only transmit correlation. We derived closed-form expressions for the Chernoff bounds on the pairwise error probability (PEP). The analysis for the system employing antenna selection has shown that the correlation between subchannels degrade the coding gain of the system but does not effect the diversity advantage as long as the channel is full rank. For low-rank space–time codes, however, there may be considerable loss in the diversity order when antenna selection is performed.
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R EFERENCES [1] I. Bahceci, T. M. Duman, and Y. Altunbasak, “Antenna selection for multiple antenna transmission systems: Performance analysis and code construction,” IEEE Trans. Inf. Theory, vol. 49, no. 10, pp. 2669–2881, Oct. 2003. [2] E. Telatar, “Capacity of multi-antenna Gaussian channels,” Eur. Trans. Telecommun., vol. 10, no. 6, pp. 585–595, Nov./Dec. 1999. [3] G. J. Foschini, Jr. and M. J. Gans, “On limits of wireless communication in a fading environment when using multiple antennas,” Wireless Pers. Commun., vol. 6, no. 2, pp. 311–335, Mar. 1998. [4] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space–time codes for high data rate wireless communication: Performance criterion and code construction,” IEEE Trans. Inf. Theory, vol. 44, no. 2, pp. 744–765, Mar. 1998. [5] D. A. Gore, R. U. Nabar, and A. Paulraj, “Selecting an optimal set of transmit antennas for a low rank matrix channel,” in Proc. Int. Conf. Acoustics, Speech Signal Processing (ICASSP), Istanbul, Turkey, 2000, pp. 2785–2788. [6] S. Sandhu, R. Nabar, D. A. Gore, and A. Paulraj, “Near optimal antenna selection of transmit antennas for a MIMO channel based on Shannon capacity,” in Proc. 34th Asilomar Conf., Pacific Grove, CA, Nov. 1999, pp. 567–571. [7] D. A. Gore and A. Paulraj, “Statistical MIMO antenna sub-set selection with space–time coding,” in Proc. Int. Conf. Communications (ICC), New York, 2002, pp. 641–645. [8] D. A. Gore, R. Heath, and A. Paulraj, “Statistical antenna selection for spatial multiplexing systems,” in Proc. IEEE Int. Conf. Communications (ICC), New York, 2002, vol. 1, pp. 450–454. [9] A. Ghrayeb and T. M. Duman, “Performance analysis of MIMO systems with antenna selection over quasi-static fading channels,” IEEE Trans. Veh. Technol., vol. 52, no. 2, pp. 281–288, Mar. 2003. [10] I. Bahceci, T. M. Duman, and Y. Altunbasak, “Antenna selection for multiple antenna transmission systems: Performance analysis and code construction,” in Proc. IEEE Int. Symp. Information Theory (ISIT), Yokohoma, Japan, Jun. 2003, p. 91. [11] J. C. Guey, M. P. Fitz, M. R. Bell, and W. Y. Kuo, “Signal design for transmitter diversity wireless communication systems over Rayleigh fading channels,” IEEE Trans. Commun., vol. 47, no. 4, pp. 527–537, Apr. 1999. [12] A. R. Hammons, Jr. and H. El Gamal, “On the theory of space–time codes for PSK modulation,” IEEE Trans. Inf. Theory, vol. 46, no. 2, pp. 524– 542, Mar. 2000. [13] A. Stefanov and T. M. Duman, “Turbo coded modulation for systems with transmit and receive antenna diversity over block fading channels: System model, decoding approaches and practical considerations,” IEEE J. Sel. Areas Commun., vol. 19, no. 5, pp. 958–968, May 2001. [14] D.-S. Shiu, G. J. Foschini, M. J. Gans, and J. M. Kahn, “Fading correlation and its effect on the capacity of multielement antenna systems,” IEEE Trans. Commun., vol. 48, no. 3, pp. 502–513, Mar. 2000. [15] C. Chuah, J. M. Kahn, and D. Tse, “Capacity of multi-antenna array systems in indoor wireless environment,” in Global Telecommunications (GLOBECOM), Sydney, Australia, Nov. 1998, vol. 4, pp. 1894–1899. [16] H. Bolcskei and A. J. Paulraj, “Performance of space–time codes in the presence of spatial fading correlation,” in Proc. 34th Asilomar Conf. Signals, Systems Computers, Pacific Grove, CA, Nov. 2000, vol. 1, pp. 687–693. [17] M. Uysal and C. N. Georghiades, “Effect of spatial fading correlation on performance of space–time codes,” Electron. Lett., vol. 37, no. 3, pp. 181– 183, Feb. 2001. [18] Z. Hong, K. Liu, R. W. Heath, and A. M. Sayeed, “Spatial multiplexing in correlated fading via the virtual channel representation,” IEEE J. Sel. Areas Commun., vol. 21, no. 5, pp. 856–866, Jun. 2003. [19] M. T. Ivrlac, W. Utschick, and J. A. Nossek, “Fading correlations in wireless MIMO communication systems,” IEEE J. Sel. Areas Commun., vol. 21, no. 5, pp. 819–828, Jun. 2003. [20] M. Chiani, M. Z. Win, and A. Zanella, “On the capacity of spatially correlated MIMO Rayleigh-fading channels,” IEEE Trans. Inf. Theory, vol. 49, no. 10, pp. 2363–2371, Oct. 2003. [21] P. J. Smith, S. Roy, and M. Shafi, “Capacity of MIMO systems with semicorrelated flat fading,” IEEE Trans. Inf. Theory, vol. 49, no. 10, pp. 2781–2788, Oct. 2003.
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Space–Time Coding Over Correlated Fading Channels With Antenna Selection Israfil Bahceci, Yucel Altunbasak, Senior Member, IEEE, and Tolga M. Duman, Senior Member, IEEE
Abstract—In a previous paper by Bahceci et al., antenna selection for multiple-antenna transmission systems under the assumption that the subchannels between antenna pairs fade independently was studied. In this paper, the performance of such systems when the subchannels experience correlated fading is considered. It is assumed that the channel-state information (CSI) is available only at the receiver, the antenna selection is performed only at the receiver, and the selection is based on the instantaneous received signal power. The effects of channel correlations on the diversity and coding gain when the receiver system is a subset of the antennas are quantified. Theoretical results indicate that the correlations in the channel do not degrade the diversity order, provided that the channel is full rank. However, it does result in some performance loss in the coding gain. Index Terms—Antenna selection, diversity, multiple-antenna communications, multiple-input–multiple-output (MIMO) systems, pairwise error probability (PEP), space–time coding, spatial fading correlation.
I. I NTRODUCTION
D
URING the last decade, multiple-input–multiple-output (MIMO) antenna technology has emerged as a key technology for enabling high-speed wireless communications. In [2] and [3], it is shown that multiple-antenna systems are able to achieve larger capacities and improved performance in comparison to their single-antenna counterparts. These benefits can be reaped by recently proposed space–time coding techniques for MIMO systems [4]. For practical applications, however, the cost and the complexity of implementing such systems are significant because of the large number of RF chains required. Such concerns have led many researchers to develop methods that can reduce the implementation cost while retaining the benefits of MIMO systems. Antenna-subset selection, which seeks the utilization of a subset of all available antennas at the transmitter and/or receiver, is such a technique [5]–[10]. Most of the work on multiple-antenna systems make the assumption that the subchannels between transmit/receive antenna pairs experience independent identically distributed (i.i.d.) fading [3], [4], [11]–[13]. A more realistic assumption, Manuscript received November 22, 2003; revised September 20, 2004; accepted December 27, 2004. The editor coordinating the review of this paper and approving it for publication is H. Jafarkhani. The work of I. Bahceci and Y. Altunbasak is supported by the National Science Foundation (NSF) Award CCR-0105654 and the work of T. M. Duman is supported by the NSF CAREER Award CCR-9984237. I. Bahceci was with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0250 USA. He is now with the Department of Electrical Engineering, University of Waterloo, Waterloo, ON N2L 3G1, Canada (e-mail: [email protected]). Y. Altunbasak is with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0250 USA (e-mail: [email protected]). T. M. Duman is with the Department of Electrical Engineering, Arizona State University, Tempe, AZ 85287-5706 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TWC.2005.858004
however, is that the fades are not independent, because of insufficient spacing between antenna elements, placement of scatterers, etc.[14]. The nonzero correlation between subchannels may significantly reduce the capacity as shown in [15]. The effect of such correlations on the system performance is studied in [16] and [17]. Recent work on correlated fading includes [18]–[21]. Hong et al. investigated the design and performance of spatial multiplexing for MIMO correlated fading channels in [18]. In [19], Ivrlac et al. studied the effects of fading correlations and transmitter-channel knowledge on the capacity and cutoff rate for MIMO systems, and in [20], Chiani et al. derived a closedform expression for the characteristic functions for MIMOsystem capacity for the correlated-fading case. Smith et al. also studied the capacity of MIMO systems, but they focused on semicorrelated flat fading [21]. The effects of subchannel correlation when antenna-subset selection is employed have interesting implications. For instance, in [5], Gore et al. considered the capacity of MIMO systems with antenna selection when the channel is rank deficient, and show that a larger capacity can be achieved by using a “good” subset of transmit antennas (i.e., by using those antennas that result in a full-rank channel). In [6], following the work in [5], Sandhu et al. propose an efficient method to find the optimal subset of antennas. Another line of work investigates antenna selection based on error probability [7], [8]. In [7] and [8], Gore et al. assume that the channel statistics change very slowly and that the selected antenna subset remain the same over the transmission period. To develop the criteria for selection, the authors derive bounds on average pairwise error probabilities (PEPs) for the full-complexity system over correlated fading that depend on the channel covariance matrix and select the subset of antennas that minimize those bounds. Note that in these studies, error probability for a system using antenna selection is not formulated at all. Only the error-probability expressions for the full-complexity system are considered. In [1], we studied the performance of received-power-based antenna selection for MIMO systems when the subchannels undergo (independent) quasi-static fading. We derive upper bounds on the PEP for the system with antenna selection and show that one can achieve, by using antenna selection, the same diversity order as that obtained by the full-complexity system while experiencing some loss in the coding gain. We also show that the diversity order decreases dramatically when the space–time code employed is rank deficient. In this paper, we consider the case when correlations exist among the subchannels. Assuming the presence of only transmit correlation, we investigate the performance of MIMO antenna selection. Our analysis quantifies the dependence of the selection performance on the correlation matrix of the channel. We show that one can achieve the same diversity order as
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the full-complexity system, even under heavy correlation, as long as the channel correlation matrix is nonsingular. With extensive simulations, we validate our theoretical analysis and demonstrate the effect of fading correlations when antenna selection is employed. In the next section, we describe the signal model and present the performance bound for the full-complexity MIMO system. We then derive the PEP for the system with antenna selection in Section III. The numerical results are presented in Section IV and the concluding remarks are summarized in Section V.
where λm are the nonzero eigenvalues of ∆∆H and r = rank ˆ denoting the codeword difference (∆∆H ) with ∆ = S − S matrix. From Inequality (2), we conclude that the diversity of the multiple-antenna system over correlated fading is the same as that obtained for the uncorrelated fading channel while there is some loss in coding gain, depending on the correlation structure. Note that these results are valid when the channel is full rank, i.e., the covariance matrix of the channel is nonsingular.
II. S YSTEM M ODEL AND PEP
In this section, we present the PEP analysis for systems employing antenna selection. First, we consider the case where a single antenna is selected. Then, we generalize our analysis to the case where more than one antenna are selected.
We consider a system equipped with M transmit and N receive antennas. We denote by H the N × M channel transfer matrix h11 · · · h1M . .. .. H = .. . . . h N 1 · · · hN M The channel is modeled as a flat Rayleigh-fading (i.e., hnm ∼ CN (0, 1)) channel that remains constant over a block of t symbols and changes independently from one block to the next. The received signal matrix XN ×t is given by ρ X= HS + W (1) M where S is the M × t transmitted signal matrix (selected from a space–time codeword alphabet), and W is the N × t additive white Gaussian noise matrix. The average energy of the transmitted signal is normalized to unity over M antennas so that ρ is the expected signal-to-noise ratio at each receive antenna. We assume that channel-state information (CSI), i.e., H, is known at the receiver, but not at the transmitter. The assumption of i.i.d. fading is typically made to model the channel [4]. However, in real-time propagation, the presence of local scatterers around the transmitter and the receiver induces correlations among subchannels that can be modeled as H = 1/2 1/2 1/2 H/2 R(r) Hw R(t) , where R(r) = R(r) R(r) is the receive co1/2
H/2
variance matrix, R(t) = R(t) R(t) is the transmit covariance matrix, and Hw is a matrix with i.i.d. CN (0, 1) entries [14].1 In this paper, we consider the existence of only transmit correlation, e.g., the antenna separation at the receiver is sufficiently large so that fading associated with each receive antenna is 1/2 uncorrelated. Such channels can be modeled as H = Hw R(t) [14]. Hence, it is assumed that the N rows of H are i.i.d. complex Gaussian vectors with covariance matrix R(t) . The PEP for a full-rank channel,2 assuming maximumlikelihood decoding at the receiver, is given by [8] ˆ ≤ P (S → S) R(t) N (
1 r m=1
N
λm )
ρ −N r 4M
(2)
III. E RROR -P ROBABILITY A NALYSIS W ITH A NTENNA S ELECTION
A. PEP Analysis With Single-Antenna Selection If only one antenna is selected out of the N receive antennas, the selection rule is reduced to choosing the antenna element that observes the largest instantaneous SNR, i.e., i = argmax |hi1 |2 + · · · + |hiM |2 . i=1,...,N
In this case, the Chernoff bound on the PEP can be expressed as (see [1]) ˆ P (S → S)
N −1 ρ 2 ≤ N e− 4M r∆ FZ r 2 fR (r)dr
−1 H N −1 ρ 1 2 e−rR(t) r dr = N e− 4M r∆ FZ r 2 π M R(t) (3) where FZ (·) is the cumulative distribution function (cdf) of Z = rrH . Using the singular-value decomposition (SVD) ∆∆H = UΛUH in (3), and then letting β = rU, we obtain ˆ P (S → S)
N −1 ρ H ≤ N e− 4M βΛβ FZ β 2 −βUH R−1 Uβ H
(t) ×e dβ
ρ H H −1 N e−β 4M Λ+U R(t) U β FZ β 2 N −1 dβ. = M π R(t) (4)
To find the FZ (·), we need to evaluate the probability FZ (a) = P {Z ≤ a}
= P |hi1 |2 + · · · + |hiM |2 ≤ a
= fRi (r)dr C
1 (·)H
denote the Hermitian transpose. (full-rank) channels refer to channels with singular (nonsingular) covariance matrices within this manuscript. 2 Low-rank
1 π M R(t)
=
1 π M R(t)
e C
−1 H −rR(t) r
dr
(5) (6)
(7)
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where C is the region {hi1 , . . . , hiM , : |hi1 |2 + · · · + |hiM |2 ≤ a}. Using the SVD of R(t) = VMVH in (7), and the routine integration tools, we obtain FZ (a) =
1 |M|
−1
e−(µ1
u1 +···+µ−1 uM ) M
ˆ ≤ P (S → S) du1 · · · duM .
u1 +···+uM ≤a
(8) The evaluation of this integral results in FZ (a) = 1 −
M
−1
kj e−µj
For high SNR and full-rank space–time codes, i.e., ρ → ∞, and rank(∆) = M , we can further simplify the expression as
a
(9) ˆ ≤ P (S → S)
where kj = M
i=1,i =j
µj − µi
,
j = 1, . . . , M.
(10)
Substituting (9) in (4) and simplifying the resulting expression, we arrive at
ˆ ≤N P (S → S) e−(λ1 u1 +···+λM uM ) M j=1 µj (N −1) M u +···+uM − 1 µ j × 1 − kj e du1 · · · duM (11)
FZ (a) = 1 −
−1 −1 µ2 µ1 e−µ2 a − e−µ1 a , µ2 −µ1 µ1 −µ2
FZ (a) = 1 −
ˆ ≤ P (S → S)
µ1 (µ1 −µ2 )
2 1 − µ1 µ2 |Λ | Λ +
µ2 (µ2 −µ1 )
− Λ +
I µ1
.
I µ2
j=1
N −1
µj
l=0
C(N − 1, l)(−1)
l
M j1=1
−1 µ23 e−µ3 a (µ3 − µ2 )(µ3 − µ1 )
−
−1 µ22 e−µ2 a (µ2 − µ3 )(µ2 − µ1 )
−
−1 µ21 e−µ1 a . (µ1 − µ2 )(µ1 − µ3 )
µ1
µ22
−
(µ2 −µ1 )(µ2 −µ3 )
Λ +
I µ2
−
µ23 (µ3 −µ1 )(µ3 −µ2 )
Λ +
I µ3
. (16)
At high SNR, this bound can be approximated by ˆ ≤ P (S → S)
2 g(λ1 , λ2 , λ3 ) ρ −6 (µ1 µ2 µ3 )2 (λ1 λ2 λ3 )2 4M
(17)
where g(·) is a function that depends only on λi , i = 1, 2, 3. Thus, the diversity order is equal to M N = 6, which is equal to the diversity order of the full-complexity system. On the other hand, we can obtain Chernoff bounds, if rank(∆) = 1 < M , as ρ −1 ˆ ≤ ξ1 ζ, R(t) P (S → S) 4M
(13)
ˆ ≤N P (S → S) M
(15)
When the space–time code is full rank, the bound for M = 3 and N = 2 is given by µ21 2 1 (µ −µ )(µ1 −µ3 ) 1 2 ˆ ≤ P (S → S) − µ1 µ2 µ3 |Λ | Λ + I
for µ1 = µ2 .
Using the cdf, the Chernoff bound in (12) for M = N = 2 can be simplified to
ρ −1 2 . (µ1 µ2 )λ1 4M
This result implies that for rank-deficient space–time codes, antenna selection degrades the diversity order significantly, i.e., the diversity order with selection is 1, while it is N r = 2 with the full-complexity system. Example 2: When M = 3, assuming that µi = µj , for i = j, i, j = 1, 2, 3, we have
j=1
where λi , i = 1, . . . , M are the eigenvalues of Λ = −1 U. For specific values of N , (ρ/4M )Λ + ζ with ζ = UH R(t) (11) can easily be evaluated. In fact, a closed-form expression for any values of M and N can also be obtained. In terms of Λ and µj , the final result can be expressed as shown in (12) at the bottom of the page. Unfortunately, for the general case, the closed-form expression does not give much insight about the effect of the correlation on the system performance. Therefore, in what follows, we will present a few special cases. We will also provide some numerical results later in Section IV. Example 1: For M = 2, the cdf in (9) is given by
(14)
Hence, the diversity order remains the same as that of the fullcomplexity system, while there is some loss in the coding gain that depends on the determinant of the correlation matrix. If the underlying space–time code is rank deficient, i.e., rank(∆) = 1 < M , then we have
j=1
−1 µM j
ρ −4 1 2 . (µ1 µ2 )2 (λ1 λ2 )2 4M
···
M jl =1
−1 −1 kj1 · · · kjl Λ + µ−1 j1 + · · · + µjl IM
(18)
(12)
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and if rank(∆) = 2 < M , as ρ −2 ˆ ≤ ξ2 ζ, R(t) P (S → S) 4M
(19)
where ξ1 (·) and ξ2 (·) depend only on ζ and R(t) , but not ρ. These bounds indicate the degradation in the diversity order due to antenna selection when we have low-rank space–time codes. We note that if some of the eigenvalues of R(t) are identical, we can evaluate the integral in (8) as well. We also note that the above results are valid only if R(t) is nonsingular. B. Selection of More Than One Antenna We now obtain the bounds on the PEP for the case when L out of N antennas are selected. Using a similar line of argument as in [11, eq. (30)], we arrive at ˆ ≤ P (S → S)
L
ρ
˜
e− 4M H∆
l=1 R
l
× 1 −
2
M
N! (N − L)!L!L −rl 2 µj
kj
πM L
j=1
×e
−1 H −1 H − r1 R(t) r1 +···+rL R(t) rL
Analytical evaluation of this integral over Rl is a formidable task. Integrating over the whole space, although resulting in a looser bound, yields a mathematically tractable analysis. In this case, integration over Rl will not depend on l and the analysis results in the upper bound given in (22) at the bottom of the page. Since further simplification of this expression is not analytically tractable, we have to resort to numerical examples to present the system performance. However, it is possible to obtain simpler expressions for special cases, as shown in the next example. Example 3: When M = N = 3 and L = 2, the bound in (22) can be written as 3 ˆ ≤ 1 − P (S → S) 2 (µ1 µ2 µ3 ) |Λ | |Λ | −
1 R(t) L
dr1 · · · drL (20)
ˆ ≤ P (S → S)
L
e
−
j=1
rj
ρ 4M
l=1 R
l
×
−1 Λ+UH R(t) U rH j
M
I µ2
−
I µ1
µ23 (µ3 −µ1 )(µ3 −µ2 )
Λ +
I µ3
.
(23)
3 g(λ1 , λ2 , λ3 ) ρ −6 . (µ1 µ2 µ3 )2 |Λ | µ1 µ2 µ3 (λ1 λ2 λ3 )2 4M (24)
ˆ ≤ P (S → S)
L
Λ +
Λ +
Finally, letting ρ → ∞, we arrive at
which can be rewritten as ˆ ≤ P (S → S)
µ22 (µ2 −µ1 )(µ2 −µ3 )
µ21 (µ1 −µ2 )(µ1 −µ3 )
At high SNR, we can further simplify this bound using (17) to obtain
N −L
37
r 2 − µl j
N! 1 − kj e (N − L)!L!L j=1
N −L
ρ −9 3g(λ1 , λ2 , λ3 ) . (µ1 µ2 µ3 )3 (λ1 λ2 λ3 )3 4M
(25)
Hence, the diversity order of the system using antenna selection is M N = 9, which is equal to the diversity order achieved by the full-complexity system. For rank-1 and rank-2 space–time codes, i.e., rank(∆) = 1 or 2, the asymptotic performance is given by, respectively ρ −2 ˆ ≤ ξ ζ, R(t) P (S → S) 1 4M
(26)
Rl = {r1 , . . . , rL : rl < rk , k = 1, . . . , l − 1, l + 1, . . . , L}
ρ −4 ˆ ≤ ξ ζ, R(t) P (S → S) 2 4M
(27)
˜ is the L × M matrix formed by deleting the rows of H and H corresponding to the antennas that are not selected.
where ξ1 (·) and ξ2 (·) can be obtained in a similar fashion as ξ1 (·) and ξ2 (·). The expressions in (26) and (27) indicate
1 × L dr1 · · · drL M L R(t) π
(21) and
where the region Rl is defined as
ˆ ≤ P (S → S)
1 N! L (N − L)!L! R(t) |Λ |L−1 N −L M M −1 −1 × C(N − L, l)(−1)l ··· kj1 · · · kjl Λ µ−1 j1 + · · · + µjl IM l=0
j1 =1
jl =1
(22)
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Fig. 1. PEP versus SNR.
Fig. 2. PEP versus SNR.
that the diversity order with antenna selection is degraded significantly when the space–time code is rank deficient. IV. E XAMPLES In the previous sections, we have theoretically analyzed the performance and derived several bounds on the PEP. We now evaluate those bounds for several codeword pairs that
are selected from the codes developed in [4]. We present the effect of fading correlation (including the case of both full-rank nonfull-rank space–time codes), and compare the performance of the system against that of the same system over uncorrelated fading (results of [1]). In Figs. 1 and 2, we compare the PEP bounds for the fullcomplexity system and the one using antenna selection for the case of transmit correlation. In both figures, the performance
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of a full-rank code (solid lines) and a rank-deficient code (dashed lines) is presented. Fig. 1 illustrates the results for the case of double transmit and the double-receive antenna system when rc = 0.54 + 0.72j. The full-rank space–time codeword pairs are selected from the 2 bit/s/Hz eight-state space–time trellis codes using 4-phase-shift keying (PSK) modulation (with M = 2). This code provides a diversity advantage of 6 [4], i.e., full spatial diversity. The two codewords considered differ in three consecutive symbols. For comparison purposes, we also present the PEP bounds when there is no correlation, i.e., i.i.d. fading. We observe that even for a high level of correlation, i.e., |rc | = 0.9, although there is some loss in the coding gain, the diversity orders of both the full-complexity system and the system using antenna selection are the same. The Chernoff bound evaluated using (13) is also plotted, and it is about 2-dB away from the exact PEP. From the PEP curves obtained for rank-deficient space–time codeword pairs, we observe that: 1) the performance of the full-complexity system under correlated fading and i.i.d. fading is very close to each other, while the performance of the system with antenna selection is superior when there is correlated fading; and 2) the diversity order is reduced when antenna selection is performed. The PEP for the case of M = 3 transmit and N = 2 receive antennas is presented in Fig. 2. We assume that the channel correlation matrix is given by
R(t)
1 = 0.6 0.4
0.6 1 0.45
0.4 0.45 . 1
(28)
For this correlation structure, we observe that the loss in the coding gain due to antenna selection is about 4 dB for the case of both i.i.d. fading and correlated fading. We note that the Chernoff bound (16) for the system with antenna selection is about 2-dB away from the exact PEP. The bound, on the other hand, is not as strict as this when the space–time code is rank deficient, i.e., it is about 6-dB away from the exact PEP. V. C ONCLUSION We analyzed the performance of MIMO systems with antenna selection under correlated fading channels. We considered a semicorrelated fading-channel model assuming the presence of only transmit correlation. We derived closed-form expressions for the Chernoff bounds on the pairwise error probability (PEP). The analysis for the system employing antenna selection has shown that the correlation between subchannels degrade the coding gain of the system but does not effect the diversity advantage as long as the channel is full rank. For low-rank space–time codes, however, there may be considerable loss in the diversity order when antenna selection is performed.
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