Space Time Coding over Correlated Fading Channels with Antenna Selection

Space Time Coding over Correlated Fading Channels with Antenna Selection İsrafil Bahçeci,Yücel Altunbaşak and Tolga M. Duman School of Electrical and Computer Engineering Department of Electrical Engineering Georgia Institute of Technology Arizona State University Atlanta, GA 30332-0250 Tempe, AZ 85287-5706 {bahceci,yucel}@ece.gatech.edu duman@asu.edu Abstract In ], antenna selection for multiple antenna transmission systems has been studied under the assumption that the subchannels between antenna pairs fade independently. In this paper, we consider the performance of such systems when the subchannels experience correlated fading. We assume that the channel state information 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. We quantify the effects of channel correlation on the diversity and coding gain when the receiver system uses all or a subset of the antennas. Theoretical results indicate that the correlations in the channel does not degrade the diversity order provided that the channel is full-rank. However, it does result in some performance loss in the coding gain. Furthermore, for non-full-rank channels, the diversity order of the system degrades significantly and is determined by the rank of the channel correlation matrix. Index terms: Antenna selection, diversity, multiple input-multiple output systems, multiple antenna communications, spatial fading correlation, space-time coding, pairwise error probability I. INTRODUCTION During the last decade, the MIMO antenna technology has emerged as a key technology to enable high-speed wireless communications. In 2,3], 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 is significant because of the large number of RF chains required. Such concerns have lead 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 smaller number of antennas at the transmitter and/or receiver, is such a technique 5 7]. Most of the work on multiple antenna systems make the assumption that the subchannels between transmit/receive antenna pairs experience independent and identically distributed fading 3,4,8]. A more realistic assumption, however, is that the fades are not independent, because of insufficient spacing between antenna elements, placement of scatterers, etc. 9]. The effect of such correlations on the system performance is studied in 0]. In 0], Bolcksei quantified the system s diversity order as a function of the ranks of the transmit and receive correlation matrices. The recent work on correlated fading includes 3]. In ], Ivrlac et al. study the effects of fading correlations and transmitter channel knowledge on the capacity and cutoff rate for MIMO systems, and in 2], Chiani et al. derive closed form expression for the characteristic functions for MIMO system capacity for correlated fading case. Smith et al. also studies the capacity of MIMO systems, but they focus on semicorrelated flat fading 3]. Hong et al. investigate the design and performance of spatial multiplexing for MIMO correlated fading channels in 4]. This work is supported by NSF Award CCR-005654 and NSF CAREER Award CCR-9984237 The effects of subchannel correlation when antenna subset selection is employed have interesting implications. For instance, in 5], Gore et al. consider 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. Another line of work investigates antenna selection based on error probability 6,5]. To develop the criteria for selection, the authors derive bounds on average pairwise error probabilities 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 is considered. In ], we study the performance of the received-power based antenna selection for MIMO systems when the subchannels undergo (independent quasi-static fading. We derive upper bounds on the 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 spacetime code employed is rank-deficient. In this paper, we consider the case when the subchannels are correlated, and investigate the performance of the multipleantenna systems with antenna selection. In particular, we derive explicit bounds on the under correlated fading. We consider a channel model where the correlation exists either on the transmitter side or on the receiver side as in 9]. Our analysis quantifies the dependency of the performance on correlation matrix of the channel. We show that one can achieve the same diversity order as the full-complexity system, even under heavy correlation, as long as the channel correlation matrix is nonsingular. We also present some results for the channels whose covariance matrix is singular. We show that over low-rank channels the diversity and coding gains for the full-complexity MIMO systems with full-rank space-time codes is equivalent to those of a MIMO system with non-full-rank space-time codes over fullrank channel. 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 analysis for the full-complexity MIMO system. We then derive the 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. II. SYSTEM MODEL AND PAIRWISE ERROR PROBABILITY We consider a system equipped with M transmit and N receive antennas. Each receive antenna observes a noisy superposition of the M transmitted signals corrupted by Rayleigh flat Low-rank channels refer to channels with singular correlation matrices within this manuscript. IEEE Communications Society 832

fading. The signal at the n th receive antenna element at time t, x tn, is given by x tn = ρ/m h nm s tm + w tn,t=, 2,,l, ( m= where h nm is the complex-valued channel gain from the m th transmit antenna to the n th receive antenna, and w tn is the additive noise at the n th receive antenna. Both h nm and w tn CN(0,. The average energy of the transmitted signal at time t is normalized to unity over M antennas so that ρ is the expected signal-to-noise ratio at each receive antenna. The equation in ( can be written in vector form as ρ X = HS + W (2 M where X is the N l received signal vector, S is the M l transmitted signal vector, H is the N M channel transfer matrix, h h M H =....., h N h NM and W is the N l additive white Gaussian noise vector. We assume that channel state information (CSI, i.e., H, is known at the receiver, but not at the transmitter. In general, one can classify four different types of fading correlations: uncorrelated, semicorrelated, semicorrelated type 2, and fully-correlated fading ]. In this paper, we consider semicorrelated fading channels for which there exist either receive correlation or transmit correlation. It has been shown that urban radio is well approximated by semicorrelated fading channel models. Transmit Correlation: Let us assume that the receiver is surrounded by many scatterers while the transmitter is placed high enough that there are not many scatterers in its vicinity. Assume also that the antenna spacing at the receiver is sufficiently large so that fading associated with each receive antenna is (almost independent 9]. In such a set-up, each antenna at the receiver observes correlated fading gains from the transmitter antennas. Explicitly, each row (denoted by r i of the channel gain matrix is a circularly symmetric complex Gaussian distributed vector with covariance matrix R (t = E{r H i r i} and all the rows are independent and identically distributed. The for this system model can be obtained as 6] P (S Ŝ Nr R (t N ( r m= λ m N (3 where rank( =r<m, and H = UΛU H with = S Ŝ. From (3, we conclude that the diversity of the multiple antenna system over correlated fading is the same as that obtained for the uncorrelated fading channel. The Inequality (3 also reveal that 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. Receive Correlation: Now, we exchange the roles of the transmitter and the receiver, that is, the transmitter is now in a richly scattering environment while the receiver is not. In this case, the correlations exist only among the subchannels from a certain transmit antenna to all receive antennas. Mathematically, the columns of H, c i, are iid circularly symmetric complex Gaussian with an N N correlation matrix R (r = E{c i c H i }. For this case, we obtain the as 6] P (S Ŝ R (r M ( r m= λ m N Nr. (4 The Inequality (4 is the analogous of (3 for the case of receive correlation. Once again, we note that the diversity of the multiple antenna system over correlated fading is the same as that obtained for the uncorrelated fading channel. The Inequalities in (3 and (4 indicate that the fading correlations can not improve the system performance. This can be seen by observing R (t or R (r that follows from the Hadamard s Inequality. The equality is satisfied only if the correlation matrices are identity, which is the case for iid fading. Hence, the system performance will be the best for iid fading, and it will get worse as correlations occur among the subchannels. III. ERROR PROBABILITY ANALYSIS WITH ANTENNA SELECTION In this section, we present the 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 antennas are selected. A. Analysis with Single Antenna Selection A. Transmit Correlation If only one antenna is selected out of the N receive antennas, the selection rule reduces to choosing the antenna element that observes the largest instantaneous SNR, i.e., i = argmax i=,,n h i 2 + + h im 2. In this case, the can be expressed as (see ] P (S Ŝ N e ρ r 2 F Z ( r 2 N e rr(t rh dr π M, (5 R (t where F Z ( is the cumulative distribution function (cdf of Z = rr H.UsingtheSVD H = UΛU H in (5, and then letting β = ru, we obtain P (S Ŝ N = H R e ρ βλβh F Z ( β 2 N e βu (t UβH dβ N π M R (t π M R (t e β Λ+UH R (t U β H F Z ( β 2 N dβ. To find the F Z (, we need to evaluate the probability F Z (a = P {Z a} = P { h i 2 + + h im 2 a} = π M e rr (t rh dr, (7 R (t C where C is the region { hi,,h im, : h i 2 + + h im 2 a }. Using the SVD of R (t = VMV H in (7, and the routine integration tools, we obtain F Z (a = e (µ u+ +µ M um du du M. M u + +u M a (8 (6 IEEE Communications Society 833

The evaluation of this integral results in F Z (a = = k e µ a where k, =,,M are constants that depend on µ,,µ M. Substituting (9 in (6 and simplifying the resulting expression, we arrive at P (S Ŝ N M = µ e (λ u + +λ M u M (N (0 k e u + +u M µ du du M, = where λ i,i=,,m are the eigenvalues of Λ = Λ + ζ with ζ = U H R (t U. For specific values of N, (0 can easily be evaluated. In fact, a closed form expression for any values of M,N can also be obtained. In terms of Λ and µ, the final result can be expressed as in ( on top of the next 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 : M =3and N =2 When M =3, assuming that µ i µ, for i, i, =, 2, 3, the cdf is given as in (2 on the next page. When the space-time code is full-rank, the bound for M =3and N =2 is given by (3 on the next page. At high SNR, this bound can be approximated by P (S Ŝ 2 (µ µ 2 µ 3 2 g(λ,λ 2,λ 3 (λ λ 2 λ 3 2 ρ (9 6, (4 where g( is a function that depends only on λ i, i =, 2, 3. Thus, the diversity order is equal to MN =6, which is equal to the diversity order of the full-complexity system. On the other hand, we can obtain Chernoff bounds, if rank( =<M,as P (S Ŝ ξ (ζ,r (t, (5 and if rank( =2<M,as P (S Ŝ ξ 2(ζ,R (t 2, (6 where ξ ( and ξ 2 ( 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. A.2 Receive Correlation For this case, we will need the ( P (S C Ŝ exp ρ r H r H f ˆR (rdr, (7 M where f ˆR(r is the pdf of the row of H with the maximum norm. Using a similar analysis as in ], we obtain this pdf as f ˆR(r =NP r ( r 2 < r 2,, r n 2 < r 2, r n+ 2 < r 2,, r N 2 < r 2 f Rn (r, (8 where f Rn (r is the marginal distribution of any row of H, i.e., f Rn (r = f H (r,, r n, r n+,, r N dr dr n dr n+ dr N, and P r ( is the probability given by P r( = f H (r,, r n, r n+,, r N dr dr n dr n+ dr N Ω where Ω is the region Ω= { r i : r i 2 < r 2,i=,,n,n+,,N }. Unlike the case of the transmit correlation where it is possible to obtain simpler expressions for the, further simplification does not seem to be warranted. However, we can perform numerical calculations to evaluate the given by (7, which we will do in Section IV. B. Selection of More Than One Antenna Since the receive correlation model does not allow a mathematically tractable analysis, we will study only the case of fading correlation at the transmitter side. For fading correlation at the receiver side, we will resort to numerical simulations. B. Transmit Correlation We can easily extend the results of Section III-A. for single antenna selection to obtain the bounds on the for the case when L out of N antennas are selected. Using a similar line of argument as in ], Equation 30], we arrive at (9, which can be re-written as (20 where the region R l is defined as R l = {r,, r L : r l < r k, k =,,l,l+,,l}, and H is the L M matrix formed by deleting the rows of H corresponding to the antennas that are not selected. Analytical evaluation of this integral over R l 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 R l will not depend on l and the analysis results in the upper bound (2 on the next page. Note that this expression reduces to Inequality (0 when L =. Since further simplification of this expression is not analytically tractable, we have to resort to numerical examples to present the system performance. Example 2: When M = N =3and L =2, the bound in (2 can be written as in (22 on the next page. At high SNR, we can further simplify this bound using (4 to obtain P (S Ŝ 3 (µ µ 2 µ 3 2 Λ Finally, letting ρ,wearriveat g(λ,λ 2,λ 3 µ µ 2 µ 3 (λ λ 2 λ 3 2 6 ]. (23 P (S Ŝ 3g(λ,λ 2,λ 3 9. (24 (µ µ 2 µ 3 3 (λ λ 2 λ 3 3 Hence, the diversity order of the system using antenna selection is MN =9, which is equal to the diversity order achieved by the full-complexity system. For rank- and rank-2 space-time codes, i.e., rank( =or 2, the asymptotic performance is given by, respectively, 2 P (S Ŝ ξ (ζ,r (t, (25 IEEE Communications Society 834

N P (S Ŝ N M = µ C(N,l( l l=0 = l = k k l Λ +(µ + + µ l I M. ( µ 2 3 F Z (a = (µ 3 µ 2 (µ 3 µ e µ 3 a µ 2 2 (µ 2 µ 3 (µ 2 µ e µ 2 a µ 2 (µ µ 2 (µ µ 3 e µ a. (2 P (S Ŝ 2 µ µ 2 µ 3 Λ µ2 /(µ µ 2 (µ µ 3 Λ µ2 2 /(µ 2 µ (µ 2 µ 3 + I/µ Λ µ2 3 /(µ ] 3 µ (µ 3 µ 2 + I/µ 2 Λ. (3 + I/µ 3 L P (S Ŝ l= R l e ρ 2 H N! (N L!L!L = k r l 2 /µ N L π ML R (t L e (r R (t rh + +r LR (t rh L dr dr L. (9 and 4 P (S Ŝ ξ 2(ζ,R (t, (26 where ξ ( and ξ 2( can be obtained in a similar fashion as ξ ( and ξ 2 (. The expressions in (25 and (26 indicate that the diversity order with antenna selection is degraded significantly when the space-time code is rank-deficient. IV. EXAMPLES In the previous sections, we have theoretically analyzed the performance issues and derived several bounds on the. 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 non-full-rank channels, and compare the performance of the system against that of the same system over uncorrelated fading (results of ]. In Figure, we compare the bounds for full-complexity system and the one using antenna selection for the case of transmit correlation. In the figure, the performance of a fullrank code (solid lines and a rank-deficient code (dashed lines is presented. Figure illustrates the results for the case of double-transmit and the double-receive antenna system when r c =0.54 + 0.72. The full-rank space-time codeword pairs are selected from the 2 bits/sec/hz 8 state space-time trellis codes using 4 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 bounds when there is no correlation, i.e., iid fading. We observe that even for a high level of correlation, i.e., r c =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 theoretically evaluated Chernoff bound is also plotted, and it is about 2 db away from the exact. From the curves obtained for rank-deficient space-time codeword pairs, we observe that (i the performance of the full-complexity system under correlated fading and iid fading is very close to each other, while the performance of the system with antenna selection is superior when there is correlated fading, and (ii the diversity order is reduced when antenna selection is performed. The performance of a system over a fading channel with receive correlation is presented in Figure 2. The correlation matrix is R (t = ( 0.6 0.4 0.6 0.4 0.45 0.45. (27 We use Monte-Carlo integration of (7 to obtain the curves for the system using antenna selection. We present the performance for both a rank-deficient space-time code and a full-rank space-time codes. The conclusions on the diversity orders and 0 0 0 0 0 2 0 3 0 4 0 0 0 0 2 0 3 0 4 M=2,N=2,r c =0.54+0.72, transmit correlation exact, full rank, full complexity,independent exact, full rank, selection, independent exact, full rank, full complexity, r c exact, full rank, selection, r c Chernoff, full rank, selection, r c exact, rank deficient, full complexity,independent exact, rank deficient, selection, independent exact, rank deficient, full complexity, r c exact, rank deficient, selection, r c Chernoff, rank deficient, selection, r c 0 5 0 5 20 25 Fig.. vs. SNR M=2, N=3, L= or 2, receive correlation Chernoff, full complexity, full rank Chernoff, selection, L=, full rank Chernoff, selection, L=2, full rank Chernoff, optimal selection, L=2, full rank Chernoff, full complexity, rank deficient Chernoff, selection, L=, rank deficient Chernoff, selection, L=2, rank deficient Chernoff, optimal selection, L=2, rank deficient 5 0 5 0 5 20 Fig. 2. vs. SNR the coding gains for this fading model is similar to the previous results. In addition, we note that (i performing optimal selection that maximizes the instantaneous channel capacity gives very similar performance as that obtained by SNR-based selection provided that the space-time code is full-rank, (ii and if the space-time code is rank deficient, one can obtain a lower by using capacity-based criterion for selection; however, the diversity order achieved with optimal selection is the same as that achieved by SNR-based selection. Figure 3 shows the Chernoff bounds on the for a sys- IEEE Communications Society 835

L P (S Ŝ l= e ( L= r ρ Λ+UH R (t U R l r H N! (N L!L!L N L P (S Ŝ N! (N L!L! R (t L Λ L C(N L, l( l l=0 = N L k e r l 2 /µ = l = k k l Λ +(µ π ML R (t L dr dr L. (20 + + µ l I M. (2 P (S Ŝ 3 (µ µ 2 µ 3 2 Λ Λ µ2 /(µ µ 2 (µ µ 3 Λ µ2 2 /(µ 2 µ (µ 2 µ 3 + I/µ Λ µ2 3 /(µ ] 3 µ (µ 3 µ 2 + I/µ 2 Λ. (22 + I/µ 3 0 0 2 0 3 0 4 M=2, N= 3 or 4, L=2, rank 2 channel, receiver correlation N=3, full complexity, full rank N=3, L=2, selection N=3, L=2, optimal selection N=4, full complexity N=4, L=2, selection N=4, L=2, optimal selection N=3, full complexity, rank deficient N=3, L=2, selection N=3, L=2, optimal selection N=4, full complexity N=4, L=2, selection N=4, L=2, optimal selection 5 0 5 0 5 20 Fig. 3. when channel-rank = 2 tem with M =2transmit antennas and N =3and N =4 receive antennas. L =2antennas are selected for systems using antenna selection. We plot the s also for the optimal selection that maximizes the instantaneous channel capacity. To show the effect of rank-deficiency in the space-time code, we provide the s for a rank- space-time code as well. From the plots, the following observations are made: (i For a low-rank channel with a correlation matrix of rank-r, the diversity order remains as Mr, where r = rank(r r, as we increase the number of receive antennas; (ii For the full-rank space-time code, the performance of optimal selection is either very close to that of the SNR-based selection or it is slightly worse, i.e, SNR-based selection performs about 0.7 db better than optimal selection for N =4; (iii For rank-deficient space-time codes over low-rank channels, optimal selection achieve the same diversity order as that achieved by the full-complexity system, but SNR-based selection experiences some loss in the diversity gain. V. CONCLUSIONS We analyzed the performance of MIMO systems with antenna selection under correlated fading channels. We considered a semicorrelated fading channel model. 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 a low level correlation, the loss in the coding gain is not significant for both full-complexity systems and systems with antenna selection. For low-rank channels, however, there is considerable loss in the diversity order; in this case, the diversity order of the full-complexity system is determined by the rank of the channel covariance matrix. When we employ antenna selection for low-rank channels, we can still achieve the same diversity order as that achieved by the full-complexity system, provided that the underlying space-time code is full-rank. REFERENCES ] I. Bahceci, T. M. Duman, and Y. 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