Ergodic Capacity, Capacity Distribution and Outage Capacity of MIMO Time-Varying and Frequency-Selective Rayleigh Fading Channels

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1 Ergodic Capacity, Capacity Distribution and Outage Capacity of MIMO Time-Varying and Frequency-Selective Rayleigh Fading Channels Chengshan Xiao and Yahong R. Zheng Department of Electrical & Computer Engineering University of Missouri, Columbia, MO 65211, USA Abstract The ergodic capacity, capacity distribution and outage capacity are investigated for MIMO wireless systems under time-varying and frequency-selective Rayleigh fading channels. Our theoretical results and Monte-Carlo simulations show that: (1) if the frequency-selective Rayleigh fading channel impulse response has no inter-tap correlations, then its ergodic capacity is the same as that of the frequency flat Rayleigh fading channel, otherwise, its ergodic capacity is less than that of the frequency flat Rayleigh fading channel; (2) time variations have no influence on ergodic capacity but have significant impact on the capacity distribution and outage capacity in the case of finite coding length; (3) the maximum diversity level of a MIMO channel is quantified as M N L N DPR, where M, N, and L are the number of receive antennas, the number of transmit antennas, and the channel length, respectively. The fourth-dimensional diversity is provided by time variations due to Doppler spread with the order being N DPR = 1 + 2F d, where is the coding length, F d the maximum Doppler, and the symbol period. 1 Introduction The ergodic capacity, capacity distribution and outage capacity are to be investigated for MIMO wireless systems under time-varying and frequency-selective Rayleigh fading channels. Currently, the majority of the MIMO channel capacity analysis assumes quasi-static channels, or block fading channels [1]-[12], where the channel remains a constant through a transmission block (codeword), and varies randomly from one block to another. This assumption implies that the MIMO channels have no Doppler spread, which is not the case in moderate and high user mobility wireless communication systems. In this paper, we consider the MIMO channels with both time variations and frequency selectivity. In other words, both the Doppler spread and the delay spread are included in the fading channels. We assume that the channel state information is unknown to the transmitter but perfectly known to the receiver. Theoretical formulas are derived for the ergodic capacity and the probability density function (PDF) of the instantaneous capacity upper bound. It is verified, through Monte-Carlo simulations on the PDFs of the true instantaneous capacity and its upper bound, that the shapes of the upper bound PDFs are similar to the true capacity PDFs except for their slightly higher mean values than the true capacity PDFs. 346

2 Moreover, three important findings are also presented. First, if the frequency-selective Rayleigh fading channel impulse response has no inter-tap correlations, then its ergodic capacity is the same as that of the frequency flat Rayleigh fading channel; otherwise, if the frequency selective Rayleigh fading channel has inter-tap correlation, then its ergodic capacity is less than that of the frequency flat Rayleigh fading channel. Second, a slow time-varying channel and a fast time-varying channel have the same ergodic capacity but different capacity distributions. The larger the time variations, the higher and narrower the PDF curves, and the larger outage capacity at acceptably low outage probabilities. Therefore time variations provide a dimension of diversity which is referred to as Doppler diversity in this paper. The Doppler diversity order is quantified as N DPR = 1 + 2F d, where is the coding length, F d the maximum Doppler, and the symbol period. Increasing the Doppler diversity order will significantly improve the outage capacity at low outage probabilities. This is of particular interest in practical wireless communication systems with high data rate and low error probability. Third, the maximum diversity level of a MIMO channel is also quantified as M N L (1 + 2F d ), where, L is the channel length. 2 Channel Model and Preliminaries Consider an M N MIMO Rayleigh fading channel described by the following discrete-time model [13] L 1 y(k)= H(l, k)x(k l)+v(k), k =, 1,,, (1) l= where x(k) = [x 1 (k), x 2 (k),, x N (k)] t, v(k) = [v 1 (k), v 2 (k),, v M (k)] t and y(k) = [y 1 (k), y 2 (k),, y M (k)] t are the modulated signal input vector, the circularly symmetric complex additive white Gaussian noise vector with zero mean and variance σ 2, and the output signal vector at time instant k, respectively. The superscript ( ) t is the transpose. The channel matrix H(l, k) is the lth-tap delayed channel impulse responses at time instant k. Its (m, n)th element is the channel coefficient denoted h m,n (l, k). The channel coefficients are zero-mean Gaussian random variables. The correlation function between the channel coefficients h m,n (l, k) and h p,q (l, k) is given by [13] E [ h m,n (l 1, k 1 ) h p,q(l 2, k 2 ) ] = Ψ RX (m, p) Ψ TX (n, q) Ψ ISI (l 1, l 2 ) J [2πF d (k 1 k 2 ) ], (2) where J ( ) is the zero-order Bessel function of the first kind, F d is the maximum Doppler frequency and is the symbol period. The matrices Ψ RX, Ψ TX and Ψ ISI are the receive correlation coefficient matrix, the transmit correlation coefficient matrix, and the intersymbol interference (ISI) inter-tap correlation coefficient matrix, respectively. They satisfy tr (Ψ RX ) = M, tr (Ψ TX ) = N and tr (Ψ ISI ) = 1. This MIMO channel model is a general model describing trebly selective MIMO channels. It contains many existing channel models that are currently discussed in the literature. If L = 1, and F d =, then the channel model becomes the quasi-static, spatially correlated and frequency flat model discussed in [5]. If L = 1, F d =, Ψ TX = I N, and Ψ RX = I M, then the model becomes the quasi-static, i.i.d., and frequency flat model [1]. If M = 1 and N = 1, then our model becomes the doubly selective model for single-input single-output (SISO) systems [14]. In the sequel, we concentrate our study in the time-varying and frequencyselective MIMO channel where Ψ TX = I N and Ψ RX = I M with I M being the M M identity matrix. 347

3 Due to the intersymbol interference, the mutual information has to be calculated based on a block of output symbols {y(k + 1), y(k + 2),, y(k + )} at the receiver. The MIMO channel with ISI can be represented by or more compactly y(k+1). y(k+) = H(L 1, k+1)... H(, k+1) H(L 1, k+)... H(, k+) x(k L + 1) v(k + 1). +. x(k + ) v(k + ) (3) Y = H X +L 1 + V, (4) where the input vector X +L 1 is circularly symmetric complex Gaussian, and the noise vector V is the additive white complex Gaussian random vector whose entries are i.i.d. and circularly symmetric. When the channel matrix H is perfectly known to the receiver, the instantaneous mutual information (per input symbol) is derived as 1 ( I(Y, X +L 1 ) = + L 1 log 2 det I M + σ 2 HQH h), (5) where Q = E [ X +L 1 X+L 1] h. For a large L, the factor 1/( + L 1) in (5) can be approximated by 1/. When the transmitter does not know the channel, the equal power allocation scheme is generally employed. Therefore, Q = P I, where P is the total transmit power across N (+L 1)N the N antennas. In this case, the instantaneous capacity C (k) is defined by C (k) = 1 [ ( log 2 det I M + P )] Nσ 2 HHh, b/s/hz. (6) Since the channel matrix H is random in nature, the channel capacity C (k) is also a random variable, especially when is finite. We will investigate its statistic properties including the ergodic capacity (the mean), the capacity distribution (the PDF) and the outage capacity. 3 Ergodic Capacity This section presents an explicit formula for the ergodic (average) capacity of MIMO channels under time-varying and frequency-selective Rayleigh fading senario. An asymptotic approximation for the MIMO channel ergodic capacity is obtained for large numbers of transmit and receive antennas. Simplified exact solutions are also derived for SISO, SIMO, MISO Rayleigh fading channels. All proofs of our new results are omitted for brevity. Theorem 1: For a time-varying and frequency-selective MIMO Rayleigh fading channel, if Ψ TX = I N and Ψ RX = I M, then the ergodic capacity is given by C av MIMO = ( log γ ) m 1 N γ λ i! [ ISI L n m 2 i (λ)] λ n m e λ dλ (7) (i + n m)! i= where m = min{m, N}, n = max{m, N}, L j i is the associated Laguerre polynomial [1] of order i, and γ = P/σ 2 is the SNR, γ ISI is the SNR degradation factor due to the channel ISI inter-tap correlations. It is determined by γ ISI = (2 Cγ 1)/γ (8) 348

4 with C γ = 1 2π L 1 log 2π 2 [1 + γ f(ω)] dω, f(ω) = a i cos(iω), a i = i=1 L 1 i l= Ψ ISI (l, l + i). (9) The formula of the ergodic capacity is somewhat complicated for the general MIMO Rayleigh fading channel. However, when m is small, such as that in SISO, SIMO, MISO, MI2O and 2IMO systems, much simpler formulas can be easily derived from (7). For a large m, the ergodic capacity can also be approximated with high accuracy. These results are presented in the following corollaries. Corollary 1: When the numbers of transmit and receive antennas N and M are large, the ergodic capacity of the MIMO time-varying and frequency-selective Rayleigh fading channel is approximately C av MIMO Capprox = N b (λ a)(b λ) log MIMO 2π 2 (1+γ γ ISI λ) dλ, (1) a λ where a = ( M/N 1 ) 2 and b = ( M/N + 1 ) 2. Remark: Eqn. (1) implies that if the number of antennas increases with a fixed ratio N, then the ergodic capacity increases linearly with N (or M). M Corollary 2: For a SISO time-varying and frequency-selective Rayleigh fading channel, the ergodic capacity is given by C av SISO = log 2 (1 + γ γ ISI λ) e λ dλ, (11) where γ ISI is given by (8). Remark: If the ISI taps have no inter-tap correlation, then the SNR degradation factor γ ISI = 1, which means that there is no SNR degradation. In such a case, the ergodic capacity of the frequency-selective Rayleigh fading channel is the same as that of the frequency flat Rayleigh fading channel 1. However, if the ISI taps have inter-tap correlations, which is generally the case even if the physical channel is WSSUS [12], [13], then the ergodic capacity of the frequency-selective Rayleigh fading will be always smaller than that of the frequency flat Rayleigh fading. This degradation is quantized by the SNR degradation factor γ ISI 1. Its value is dependent on the level of SNR. Meanwhile, it is discovered that the time variation (due to Doppler spread) has no impact on the ergodic capacity, meaning that the fast timevarying and slow time-varying fading channels have the same ergodic capacity. Corollary 3: For time-varying and frequency-selective MISO and SIMO Rayleigh fading channels, if Ψ TX = I N and Ψ RX = I M, then the ergodic capacities are given by where γ ISI is given by (8). C av 1 ( log MISO (N 1)! 2 1+ γ ) N γ λ ISI λ N 1 e λ dλ (12) C av 1 log SIMO (M 1)! 2 (1+γ γ ISI λ) λ M 1 e λ dλ, (13) 1 It is noted that this special case has been discussed by Ozarow, Shamai and Wyner in [15] for quasi-static Rayleigh fading channel. Our result is suitable for both quasi-static and symbol-wise time-varying Rayleigh fading channels. 349

5 4 Capacity Distribution and Outage Capacity It is mathematically intractable [3] to the derivation of the probability density function of the instantaneous capacity C (k) of a time-varying and frequency-selective MIMO channel. However, based on the structure of H given by (3), we can prove that C (k) M log 2 [1 + γ R(k)] = C ub (k), (14) where C ub (k) denotes the instantaneous upper bound of C (k), and R(k) = 1 MN M N L 1 i=1 m=1 n=1 l= h m,n (l, k + i) 2. (15) Since the channel coefficients h m,n (l, k) are zero mean Gaussian random variables whose cross-correlations are given by (2), we can prove that R(k) has the same PDF as the following random variable R = M N L 1 k=1 m=1 n=1 l= h w σ TX,n σ RX,m σ DPR,i σ ISI,l m,n (l, k) 2, (16) where h w m,n(l, k) are circularly symmetric complex zero mean unit variance white Gaussian random variables which are white in the four dimensions (m, n, l, and k). And σ TX,n, σ RX,m, and σ ISI,l are, respectively, the nth singular value of Ψ TX /N, the mth singular value of Ψ RX /M, and the lth singular value of Ψ ISI. The parameter σ DPR,k is the kth singular value of the matrix 1 Ψ whose (i, j)th element is given by DPR Ψ DPR (i, j) = J [2πF d (i j)]. (17) It can be shown that the matrix Ψ DPR has about N DPR non-negligible positive singular values compared to its largest singular value, where N DPR = 1 + 2F d. The introduction of the intermediate random variable (16) leads to a tractable analysis of the PDF of the instantaneous capacity upper bound as follows. Theorem 2: For a time-varying and frequency-selective MIMO Rayleigh fading channel, when Ψ TX = I N and Ψ RX = I M, the PDF of the instantaneous capacity upper bound C ub (k) is given by (C) = ln 2 2C/M pc ub M γ p R 2C/M 1, γ C, (18) with p R (R) given by p R (R) = p MNL (R/σ DPR,1 ) σ DPR,1 p MNL (R/σ DPR,N DPR ) σ DPR,NDPR, for Ψ ISI = I L /L, R p MN (R/α 1 ) α 1 p MN (R/α L N DPR ) α L NDPR, for σ ISI,l σ ISI,j, l, j, R, (19) where is the convolution with respect to R, and α i = σ ISI,j σ DPR,k for j = 1,, L and k = 1,, N DPR ; and p MNL (R) = (MNL)(MNL) (MNL 1)! RMNL 1 e (MNL)R, p MN (R) = (MN)(MN) (MN 1)! RMN 1 e (MN)R. (2) 35

6 Remark: For any given M, N and L, if N DPR = 1 + 2F d approaches infinity, then the PDF of C ub ub approaches an impulse delta function. Therefore, C becomes a constant. In such a case, each individual random realization of H will lead to the same deterministic C ub Ȧlthough the PDF formulas are quite complicated, it is possible to calculate (18) and (19) numerically. The results will be presented in Section 5. For the special cases of SISO channels, the PDF of C ub (k) can be simplified to closed form formulas. These are presented in the following two corollaries. Corollary 4: For a time-varying, frequency flat SISO Rayleigh fading channel, the PDF of C ub (k) is given by where (C) = pc ub ( C ln 2 2C 2 γ exp + 1 γ N DPR ln 2 2C γ β DPR,n n=1 σ DPR,n β DPR,n = ), for F d 1, C ( C ) exp, for N γ σ DPR 2, C, DPR,n N DPR k=1,k n (21) σ DPR,n σ DPR,n σ DPR,k. (22) Corollary 5: For a time-varying, frequency-selective SISO Rayleigh fading channel, the PDF of C ub (k) is given by (C) = pc ub where ( ln 2 2C γ L L (L 1)! ln 2 2C γ L N DPR i=1 β ISI,l = ) L 1exp ( 2 C 1 γ β i α i exp L k=1,k l ( 2C 1 γ α i L 2C 1 γ ) ), for F d 1, σ ISI,l = 1 L, l, C, for N DPR 2, σ ISI,l σ ISI,j, l, j, C, (23) L N σ ISI,l DPR α i, β i =. (24) σ ISI,l σ ISI,k k=1,k i α i α k With the PDF of C ub (k) at hand, we can study the statistical behavior of the instantaneous capacity C (k), and the outage capacity following the procedures in [12]. Details are omitted for brevity. 5 Simulation Results To verify the theoretical analysis presented in Section 3 and Section 4, the ergodic capacity, the instantaneous capacity C (k) and the instantaneous capacity upper bound C ub (k) were simulated extensively using the discrete-time MIMO channel model described in Section 2. All our simulation results are in excellent agreement with the theoretical results. To keep the paper within the length limit, we only present some of the simulation results. 351

7 5.1 Ergodic Capacity Figure 1 depicts the ergodic capacity for the SISO, 2 2, and 4 4 systems under three fading channels. It is shown that every simulated curve is in excellent agreement with the corresponding theoretical curve. Comparing the ergodic capacities of the three different systems, we can see that the ergodic capacity increases as the number of antennas. The inter-tap uncorrelated frequency-selective channel has the same ergodic capacity as that of the frequency flat fading channel. However, when the frequency-selective Rayleigh fading channel has inter-tap correlations, its ergodic capacity is smaller than that of the channel with no inter-tap correlations. Ergodic Capacity (b/s/hz) Flat Fading, Simulation Flat Fading, Theory Two tap uncorrelated fading, Simulation Two tap uncorrelated fading, Theory Two tap correlated fading, Simulation Two tap correlated fading, Theory M = N = 4 M = N = 2 Ergodic Capacity (b/s/hz) F d =.1 2F d = 1. 2F d = 5. AGWN channel Frequency flat Rayleigh fading channel AWGN channel 5 M = N = 1 2 Two tap correlated Rayleigh fading channel Figure 1: Ergodic Capacity vs SNR for the SISO, 2 2 and 4 4 systems under: (1) frequency flat fading channel, (2) frequency-selective two-tap uncorrelated fading channel with Ψ ISI = [.5 ;.5], (3) frequency-selective two-tap correlated fading channel with Ψ ISI = [.5.475;.475.5]. All fading is Rayleigh distributed with 2F d = Figure 2: Time variation (or Doppler spread) effect on ergodic capacity of SISO channels. Three maximum Dopplers are selected such that 2F d T S =.1, 2F d T S = 1. and 2F d T S = 5.. The fading channels are the same as those in Fig. 1. Observation: the time variations do not have impact on the ergodic capacity. Figure 2 shows the time variation (or Doppler spread) effect on the ergodic capacity with the example of the SISO system. The curves for the frequency-selective inter-tap uncorrelated fading channel are omitted because they are the same as those of the frequency flat fading channel, as shown in Fig. 1. All curves were obtained by simulations. It is clearly shown that the time variations have almost no effect on the ergodic capacities, which agrees with our theoretical analysis. The figure also indicates that the ergodic capacities of the fading channels are smaller than that of the AWGN channel. 5.2 Capacity Distribution Figure 3 shows the capacity distributions for the instantaneous capacity C (k) and its upper bound C ub (k) using the example of the 2 2 system. The instantaneous capacities C (k) were obtained by simulations only. The upper bound capacities C ub (k) were computed by the theoretical formulas given in Section 4 and verified by simulations. The PDFs of the frequency flat fading channel and the two-tap uncorrelated frequency-selective fading channel are shown in Fig. 3a and Fig. 3b, respectively. Both slow and fast time-varying channels were considered. Upon careful examination of the figures, we can see that all PDF curves are (approximately) bell shaped with the center at their corresponding ergodic capacity (the mean value). 352

8 2 1.5 SNR = db C (k), 2F d =.1 C ub (k), 2Fd =.1 C (k), 2F d = 5. C ub (k), 2Fd = SNR = db C (k), 2F d =.1 C ub (k), 2Fd =.1 C (k), 2F d = 5. C ub (k), 2Fd = PDF 1 SNR = 1 db SNR = 2 db SNR = 3 db PDF 1 SNR = 1 db SNR = 2 db SNR = 3 db Capacity (bps/hz) Capacity (bps/hz) (a) Frequency Flat Rayleigh fading channel (b) Frequency-selective Rayleigh fading channel Figure 3: PDFs of the instantaneous capacity C (k) and its upper bound C ub (k) with various SNRs and Dopplers for the 2 2 MIMO system. The higher the SNR, the larger the width of the bell. The instantaneous capacities C (k) have slightly lower mean values than their corresponding upper bounds C ub (k). The lower the SNR, the tighter the bounds. Comparing the PDFs obtained by slow and fast time-varying channels, we found that the centers of the curves are the same for different time variations at a given SNR. This confirms the conclusion drawn in Fig. 2 that the time variations have no effect on the ergodic capacity. However, the height and width of the curves change dramatically with the time variations. The PDF curves for the larger time variation are much higher and narrower. This greatly affects their outage capacities as it is directly related with the PDF curve through an integration operation [12]. This will be shown in Fig. 4. Comparing Fig. 3(a) and Fig. 3(b), we found that the PDFs of the frequency selective fading channel are higher and narrower than the corresponding ones of the frequency flat fading channel although they have the same ergodic capacities. The larger the number of fading taps L, the narrower the PDF shape. The narrow PDFs will lead to larger outage capacities at low outage rate, which is similar to the cases of large time variations. It is worthwhile to note that, although they do not have influence on the ergodic capacity, the frequency selective fading and the time variations do improve the outage capacity at low outage rates. This is of particular interest for high data rate and low error probability communications in practical wireless communication systems. 5.3 Outage capacity Figure 4 shows three-dimensional surfaces of the outage capacity vs SNR vs outage probability for both SISO and 2 2 frequency flat Rayleigh fading channels. Both block fading (2F d = ) and time-varying fading (2F d = 5) are included. These surfaces represent the performance limit of the corresponding Rayleigh fading channels, assuming optimal coding and large enough coding length. Only the region underneath the surface is achievable for practical wireless communication systems. The surface of 2 2 block fading channel, as shown in Fig. 4b, was previously reported in [12, pp ]. It was regarded as the fundamental limit of the corresponding 2 2 fading channel. However, the surfaces of the time-varying fading channels, shown in Fig. 4 (c) and (d), are much higher than those of the block fading channels, especially at low outage probabilities. For example, at the outage 353

9 probability of 1 3 and SNR = 2 db, the SISO time-varying channel can achieve the outage capacity of 3.64 bps/hz while that of the SISO block fading channel is only.15 bps/hz. Similarly, the outage capacity of the 2 2 time-varying channel is 9.7 bps/hz while that of the 2 2 block fading channel is only 5.46 bps/hz Outage Capacity (bps/hz) Outage Capacity (bps/hz) Outage Probablity 1 4 Outage Probablity (a) SISO block fading channel (2F d = ) (b) 2 2 block fading channel (2F d = ) 12 2 Outage Capacity (bps/hz) Outage Capacity (bps/hz) Outage Probablity Outage Probablity (c) SISO time-varying fading (2F d = 5) (d) 2 2 time-varying fading (2F d = 5) Figure 4: The 3-D surface of the outage capacity vs SNR vs outage probability for frequency flat Rayleigh fading channels. With optimal coding, the SNR can be traded for outage probability at a given signaling rate, or traded for signaling rate at a given outage probability. The region underneath the surface is the achievable region for practical wireless communication systems. It is clear from Fig. 4 that time variations have significant impacts on the outage capacity, especially when the outage probability is small. In fact, time variations provide a dimension of diversity in addition to the existing three major types of diversities (transmit diversity, receive diversity, and frequency selective (path) diversity). We call it the Doppler diversity. The Doppler diversity order is quantified as N DPR = 1 + 2F d. Increasing the Doppler diversity order will significantly improve the outage capacity at acceptably low outage rate. When N DPR approaches infinity, the random instantaneous capacity will become a deterministic constant which is equal to the ergodic capacity at the same SNR. 354

10 6 Conclusion Theoretical formulas have been derived for the ergodic capacity and the probability distribution of the instantaneous capacity upper bound for MIMO wireless systems under timevarying and frequency-selective Rayleigh fading channels. It has been shown that: (1) if the frequency selective Rayleigh fading channel has no inter-tap correlation, then its ergodic capacity is the same as that of the frequency flat Rayleigh fading channel, otherwise, its ergodic capacity is less than that of the frequency flat Rayleigh fading channel; (2) different time variations lead to the same ergodic capacity but to different capacity distributions and different outage capacities; (3) the maximum diversity level of a MIMO channel has been quantified as M N L (1 + 2F d ). Acknowledgment: the first author, C. Xiao, is grateful to Professor N.C. Beaulieu for his helpful discussion at the early stage of the work. References [1] I.E. Telatar, Capacity of multi-antenna Gaussian channels, Eur. Trans. Telecom., vol.1, pp , Nov Also in AT&T Bell Lab. Tech. Memo, June [2] G.J. Foschini and M.J. Gans, On limits of wireless communications in a fading environment when using multiple antennas, Wireless Personal Communications, vol.6, pp , [3] A.M. Sengupta and P.P. Mitra, Capacity of multivariate channels with multiplicative noise: I. Random matrix techniques and large-n expansions for full transfer matrices, Phy. Arch., no.181, 2. [4] L. Zheng and D.N.C. Tse, Communication on the Grassmann manifold: a geometric approach to the noncoherent multiple-antenna channel, IEEE Trans. Inform. Theory, vol.48, pp , Feb. 22. [5] C.N. Chuah, D.N.C. Tse, J.M. ahn, and R.A. Valenzuela, Capacity scaling in MIMO wireless systems under correlated fading, IEEE Trans. Info. Theory, vol.48, pp , March 22. [6] D. Chizhik, G.J. Foschini, M.J. Gans, and R.A. Valenzuela, eyholes, correlations, and capacities of multielement transmit and receive antennas, IEEE Trans. Wireless Com., vol.1, pp , April 22. [7] S. Verdu, Spectral efficiency in the wideband regime, IEEE Trans. Inform. Theory, vol.48, pp , June 22. [8] A. Sayeed and V. Veeravalli, The essential degrees of freedom in space-time fading channels, Proc. IEEE PIMRC, pp , Sept. 22. [9] A. Lozano and A.M. Tulino, Capacity of multiple-transmit multiple-receive antenna architectures, IEEE Trans. Info. Theory, vol.48, pp , Dec. 22. [1] A. Goldsmith, S.A. Jafar, N. Jindal, and S. Vishwanath, Capacity limits of MIMO channels, IEEE J. Selected. Areas Commun., vol.21, pp , June 23. [11] S. Wei, D. Goeckel, and R. Janaswamy, On the asymptotic capacity of MIMO systems with antenna arrays of fixed length, IEEE Trans. Wireless Commun., accepted for publication, 23. [12] A. Paulraj, R. Nabar, and D. Gore, Introduction to Space-Time Wireless Communications, Cambridge University Press, 23. [13] C. Xiao, J. Wu, S.-Y. Leong, Y.R. Zheng, and.b. Letaief, A discrete-time model for trebly selective MIMO Rayleigh fading channels, IEEE Trans. Wireless Commun., accepted for publication, 23. [14] X. Ma, G.B. Giannakis, and S. Ohno, Optimal training for block transmission over doubly selective wireless fading channels, IEEE Trans. Signal Proc., pp , May 23. [15] L.H. Ozarow, S. Shamai, and A.D. Wyner, Information theoretic considerations for cellular mobile radio, IEEE Trans. Veh. Technol., vol.43, pp , May [16] S. Barbarossa and A. Scaglione, On the capacity of linear time-varying channels, in Proc. IEEE ICASSP 99, pp , [17] M.-S. Alouini and A.J. Goldsmith, Capacity of Rayleigh fading channels under different adaptive transmission and diversity-combining techniques, IEEE Trans. Veh. Technol., pp , July [18] M. Medard, The effect upon channel capacity in wireless communications of perfect and imperfect knowledge of the channel, IEEE Trans. Inform. Theory, vol.46, pp , May