MIMO Capacity in Correlated Interference-Limited Channels

Transcription

1 MIMO Capacity in Correlated Interference-Limited Channels Jin Sam Kwak LG Electronics Anyang , Korea Jeffrey G. Andrews The University of Texas at Austin AustiX 78712, USA Angel Lozano Bell Labs Alcatel-Lucent Holmdel NJ 07733, USA Abstract This paper analyzes the capacity of MIMO channels in the presence of both antenna correlation and co-channel interference. We investigate the optimization of the input covariance, characterize the optimality of beamforming, and study the behavior of the input covariance in the low and highpower regimes. For the special case of separable correlations, we also derive analytical expressions for the key statistical properties of the spectral efficiency achievable with an arbitrary input covariance. Altogether, our analysis enables assessing the oint impact of correlation and interference on the capacity of multiantenna architectures in a cellular system. I. INTRODUCTION For noise-limited MIMO channels with instantaneous channel state information CSI at the receiver and statistical CSI at the transmitter, the capacity-achieving input covariance and the corresponding ergodic capacity have been studied for i.i.d. Rayleigh-faded channels in [1], [2] and for spatially correlated channels in [3]-[11]. One of the main features of cellular systems is the strong level of co-channel interference from neighboring cells. In contrast with thermal noise, interference is spatially colored and subect to fading. The impact of interference on the capacity of MIMO channels has been investigated in [12]-[19], mostly within the context of i.i.d. Rayleig fading. One-sided correlated channels were considered in [18]. Only asymptotically in the number of antennas has the capacity-achieving input covariance with both correlation and interference been studied [19]. In this paper, we present new analytical solutions that greatly facilitate the computation of this covariance. In addition, we also derive the MGF of the mutual information from which the ergodic spectral efficiency and other statistical properties thereof can be then readily obtained. This complements and expands the analytical framework available to date to assess the oint impact of correlation and interference on the performance limits of MIMO channels. Notation: λ i X denotes the ith eigenvalue of X, VX denotes a Vandermonde matrix whose entries are given by VX} i, λ i X 1 with the eigenvalues of X properly ordered, and X k represents the submatrix of X obtained by removing the kth row and kth column thereof. II. MIMO SIGNAL AND CHANNEL MODEL Consider a cellular system where a MIMO transmitter with antennas intends to communicate with a desired user equipped with n R receive antennas. There are L I interferers whose codebooks are unknown, each equipped with,k transmit antennas, k 1, 2,..., L I. The n R -dimensional received signal vector is r L I P D H D s D + PI,k H I,k s I,k +n 1 where s D is the 1 input signal vector with unit power, E [ s D 2] 1, s I,k is the,k 1 signal vector transmitted by the kth interferer with E [ s I,k 2] 1, and n is Gaussian noise with covariance σ 2 I nr. To model the n R channel matrix for the desired user, H D, and the n R,k channel matrices for the interferers, H I,k, we invoke the broad unitaryindependent-unitary UIU channel model [10] whereby H D U R,0 HD U T,0 2 H I,k U R,k HI,k U T,k 3 such that U R,k and U T,k, k 0, 1,..., L I, are deterministic unitary matrices while H D and H I,k are complex random matrices whose entries are zero-mean and independent with arbitrary marginal distributions. The variances of the entries can be assembled into gain matrices G D } i, E [ H D } i, 2] 4 G I,k } i, E [ H I,k } i, 2]. 5 The transmit powers P D and P I,k are normalized such that E[trH D H D }] n R and E[trH I,k H I,k }],kn R, respectively. The appeal of the UIU channel structure lies in that it encompasses most zero-mean fading channels of interest, including the popular separable correlation model for spatial diversity, the virtual representation propounded in [9], and the independent nonidentically distributed IND model needed for polarization or pattern diversity topologies. In the more restrictive separable correlation model, which we will resort to at times, H D Σ 1/2 R,0 H D Σ 1/2 T,0 6 H I,k Σ 1/2 R,k H I,k Σ 1/2 T,k 7 where Σ R,k and Σ T,k are deterministic receive and transmit correlation matrices, whose eigenvector matrices equal respectively U R,k and U T,k in the UIU representation, while H D

2 and H I,k are i.i.d. fading matrices. The entries of the gain matrices in the UIU model relate to Σ R,k and Σ T,k via G D } i, λ i Σ R,0 λ Σ T,0 8 G I,k } i, λ i Σ R,k λ Σ T,k. 9 Since the same local scattering at the receiver is encountered by desired signal and by each of the interferers, we consider identical unitary matrices for all of them at the receiver, i.e., U R U R,k for k 0, 1,..., L I. In the separable case, this corresponds to Σ R Σ R,k for k 0, 1,..., L I. If we decompose the input covariance matrix as E[s D s D ] U DPU D 10 where U D is unitary and P diagp 1,, p nt }, then, borrowing the approach in [11] we can show that, in order to achieve capacity, the signalling must take place along the eigenvectors U D U T,0. Likewise, we consider that each interferer signals along the eigenvectors U T,k corresponding to its own channel and we denote by P I,k the eigenvalue matrix of its signalling covariance. Thus, E[s I,k s I,k ] U T,kP I,k U T,k. 11 For analytical tractability, we consider an interferencelimited environment where the thermal noise n can be neglected. This is appropriate when the interference-to-noise ratio is high, i.e., when L I P I P I,k σ and L I n I,k n R. 13 The resulting mutual information with a Gaussian input and Gaussian interfering signals can be written as Iγ, P log 2 det I nr + γ H D P H D H I P I H I 1 14 where γ P D /P I is the average SIR per receive antenna, P I diag, p I,2,, p I,nI }/P I with trp I 1, and H I [ H I,1 HI,LI ]. The capacity is C γ max E [I γ, P]. 15 P:trP 1 Altogether, the interference has n I degrees of freedom and the channel coupling it with the receiver is IND with gain matrix G I [G I,1 G I,LI ]. Note that, for n I /n R, the interference behaves as white noise. III. CAPACITY-ACHIEVING POWER ALLOCATION Having already optimized the signalling eigenvectors, the task that remains in terms of input optimization is the computation of the power allocation P. In order to leverage the noise-limited optimization in [11], let us consider an MMSE linear estimator at the receiver and define the output SIR of the signal that originated at the th transmit antenna, with power p and direction of the corresponding eigenvector of U T,0, as 1 SIR p γp h H Ξ H h 16 where Ξ diagγp, P I }, h is the th column of H D, and H is the matrix obtained by removing the th column from H [ H D HI ]. The corresponding MMSE in interferencelimited conditions is MMSE lim P I /σ 2 MMSE SIR p. 17 Expressed in terms of MMSE, the necessary and sufficient conditions for P diagp 1,, p nt to be capacityachieving are the same ones derived in [11] for noise limited MIMO channels: 1 E [ MMSE ] p nt i1 1 E 18 [MMSE i ] for all p > 0 and E [SIR 1] i1 1 E [MMSE i ] 19 for all p 0. To find the power allocation that satisfies these conditions, we can directly utilize the iterative algorithm given in [11] or any other iterative approach geared towards the solution of nonlinear equations. The burden, however, is that each iteration of such algorithms will require that the expected value over the fading of the MMSE for each transmit signal be computed. It would clearly be advantageous to have closed forms for such MMSE that avoided the need for lengthy Montecarlo expectations within each iteration. Only for the noise-limited Rayleigh-faded MISO channel has such closedform been given [8]. In the following, we provide more general closed forms for MIMO channels with separable correlations. 1 Theorem 1: Consider a Rayleigh-faded MIMO channel with separable correlations and with a power allocation matrix P having q nonzero powers p whose corresponding eigenvalues of Σ T,0 are σ for 1,, q. Defining a q +n I q +n I diagonal matrix Z q such that γσ Z q }, p 1,, q Λ T,I P I } q, q q + 1,, q + n I 20 where Λ T,I diagλ T,1,, Λ T,LI } with Λ T,k being the diagonal eigenvalue matrix of Σ T,k, then E [ MMSE ] 1 γσ p det[ 1 Z q, γσ p ] 21 det[vz q ] det [ 2 Z q ] E [SIR 1] γσ 22 det [V Z q ] where, for a given l l diagonal matrix Z diagz 1,, z l }, 1 Z, x and 2 Z are l l matrices whose entries are 1 Z, x} i, 2 Z} i, V Z} i,, i l n R For sake of brevity, the proofs of the results are not included.

3 1 Z, x} l nr +1, zl nr+1 x z x 2 log x 1 z z 2 Z} l nr +1, z l n R 1 logz. Example 1: Consider a uniform linear transmit array with antenna spacing d wavelengths and correlations matrix Σ T,0 } i, e 0.05d2 i 2 23 corresponding to a broadside truncated Gaussian power azimuth spectrum with 2 root-mean-square spread. Let the receive antennas be uncorrelated. Fig. 1 shows the ergodic capacity versus antenna spacing obtained via Montecarlo simulation with random realizations in the presence of equal-power interferers and Rayleigh fading when n R 2, n I 6 and γ 5 db. The power allocation was optimized using the iterative algorithm in [11] in conunction with Theorem 1. Also shown is the ergodic spectral efficiency with an isotropic input. Not surprisingly, optimizing the power allocation is increasingly impactful as the antenna spacing diminishes and correlation grows. Let us now consider the low- and high-power behaviors of the capacity-achieving power allocation. Corollary 1: In the low-power regime, i.e., for γ 0, the power should be evenly distributed among the th eigenvectors satisfying arg max tr diagg D, }E [ ] 1 HI P I H I 24 where g D, is the -th column of G D. If is unique, then beamforming is the optimum signalling strategy at low SIR. In noise-limited scenarios, power should be allocated only to the eigenvectors corresponding to the columns of G D with largest sum of entries [11]. With interference, however, the directions are affected by the inverse of the interference covariance matrix. Interestingly, if the correlations are separable then 24 reduces to arg max λ Σ T,0 implying that the power should be allocated to the maximal eigenvalue eigenspace irrespective of the interference and receiver-side correlations. Invoking Theorem 1, we can further characterize the optimality of beamforming in interference-limited MIMO as follows. Corollary 2: With separable correlations and with σ 1 and σ 2 where σ 1 > σ 2 being the two largest eigenvalues of Σ T,0, the region of beamforming optimality is R σ1, σ2 σ } 2 ψ nr γσ1, P I 25 where σ 1 ψ nr γσ 1, P I det P I γσ 1 I ni det 1 P I, γσ 1 det 2 ˆP I with ˆP I diagγσ 1, P I }. Example 2: For P I diag }, 1 n I I 1 n I 1, 26 the region of beamforming optimality is illustrated in Fig. 2 by depicting ψ nr γσ 1, P I. The region broadens as the number of interferers increases or the interfering power is equally distributed or the number of receive antennas decreases. Furthermore, for given σ 1 and σ 2, as the SIR γ decreases the chance of beamforming being optimum increases. Finally, we turn our attention to the high-power regime. Corollary 3: Let q be the number of nonzero powers within P. When n R, E[MMSE ] 0 as γ. When > n R and P is capacity-achieving, q n R for γ and each nonzero power is no larger than 1/n R with lim γ i1 1 E [MMSE i ] n R. 27 This corollary reveals that, as in the noise-limited case [18], P 1/ I nt for γ when n R. When > n R, however, such is not the case and we have to resort to the solution of 18 and 19. Corollary 3 can help establish good initial conditions therefore when the power is high. IV. CAPACITY WITH SEPARABLE CORRELATIONS Having characterized the behavior of the capacity-achieving power allocation, we turn our attention back to the capacity itself. Henceforth, we restrict ourselves to Rayleigh-faded channels with separable correlations. Let us start by considering an isotropic input, i.e., P 1/ I nt, and the worst-case scenario for a given total interference power: equal-power interferers, i.e., P I 1/n I I ni. Based on some results from [20], we investigate the MGF of the mutual information in 14, which in turn enables us to evaluate the ergodic spectral efficiency in 15. With an isotropic input, 14 becomes I iso γ log 2 det I nt + n I γh H DH I H H I 1 H D. 28 Note that, since we are considering signal and interference to be subect to the same receive correlation, the effects thereof vanish in interference-limited conditions given that Σ 1/2 R,0 Σ 1 R,I Σ1/2 R,0 I n R. Theorem 2: Denote by σ 1,..., σ nt the eigenvalues of the transmit correlation matrix Σ T,0 and let m min, n R, n max, n R and l max n I, n I + n R. The MGF of I iso γ is M iso s K m, n, l, Σ T,0 det F Σ T,0, s 29 where F, s is an matrix function of s whose entries are Y σ i, s 1,..., m F Σ T,0, s} i, m l 1 σ i m + 1,...,. 30

4 whereas K m, n, l, Σ T,0 1 mn 1 det Σ T,0 nt n+l l n + m! l n! In 30, Y α, β is given by m det V Σ T,0 l + m nt l n + k Y α, β α b B a +, b β log 2 e 2 F 1 b, b β log 2 e; b + a β log 2 e; 1 ni αγ where a n, b m + l n + 1 and 2F 1 p, q; r; z Γr ΓqΓr q 1 is a Gauss hypergeometric function. In turn, Ba, b ΓaΓb Γa + b u q 1 1 u r q 1 1 zu p du is a beta function with Γ the Gamma function [21]. 33 Using Theorem 2, we can readily obtain solutions for some statistical properties of I iso γ. In particular, we obtain the ergodic spectral efficiency with an isotropic input as C iso γ K m, n, l, Σ T,0 det Ψ Σ T,0, k 34 where Ψ, is a matrix function whose entries are given by 35 at the bottom of the page with q q β q k α p k 1 Y p, q, α, β k log 2 p k 1 2 k0 2 F 1 p k 1, p k 1, p k, 1 αβ. 36 The above solution for interference-limited channels with transmit correlation only because of the mutually canceling identical receive correlations of signal and interference complements the one given in [18] for interference-limited channels with correlation at the receiver only possibly different for signal and interference in that case. In addition to the ergodic spectral efficiency, Theorem 2 enables obtaining other statistical moments that are useful in terms of outage characterizations. We can in fact establish an interesting reciprocity relationship that connects 34 with its counterpart in [18]. Corollary 4: The interference-limited, n R spectral efficiency with an isotropic input and SIR γ in the presence of n I interferers and given spatial correlation only at the transmitter respectively receiver is equivalent to the interference-limited n R, spectral efficiency in the presence of n I + n R interferers and that same spatial correlation at the receiver respectively transmitter and with SIR µγ where µ n R n I. 37 n I + n R Hence, if there are n I n R interferers beyond the number of degrees of freedom at the receiver, a, n R MIMO link with n R has a SIR gain of µ over a n R, MIMO link with the same correlation at the other end. When n R, a MIMO link with transmit correlation is equivalent to one with the same correlation at the receiver. Let us now consider an optimized power allocation, P q for q, obtained according to Section III. The statistical properties of the mutual information Iγ, P q can be obtained by replacing σ with σ p iheorem 2, where σ denotes the eigenvalue of the transmit correlation matrix Σ T,0 corresponding to each p > 0. Then, the ergodic capacity C γ equals C iso in 34 except with the roles of and Σ T,0 played by q and Λ q diagσ 1p 1,, σ qp q}, respectively. If the n I interferers are not equal power, as we had assumed at the onset of this section, then C γ computed with the same total interference power and equal-power interferers lower-bounds the actual capacity. Example 3: Consider the same correlation model in Example 1 with 4 and d 1. Fig. 3 shows the analytical solution for the capacity as function of the SIR with various n R and over Rayleigh fading channels when n I 5 with P I diag, 1 n I 1 I n I 1 }. 38 For validation purposes, Montecarlo simulation results are also shown realizations. We observe that the analytical solution, exact for equal-power interferers, tightly bounds the exact Montecarlo solution for unequal-power interferers over the entire range of SIR. V. CONCLUSION We have investigated the input optimization and the resulting capacity in interference-limited MIMO over correlated fading channels. First, we showed that MMSE-based conditions characterizing the optimum power allocation in noiselimited scenarios extent to interference-limited scenarios and, therefore, the solutions derived thereafter are applicable. We then presented closed forms for the expected value of the MMSE function over the fading, which greatly facilitate the computation of the optimum power allocation. The behavior of the power allocation in the low- and high-power regimes and σ b i B a +, a + b, 1,..., m and i k Y a + b, a + 1, n I γ/n R, σ i, 1,..., m and i k ΨΣ T,0, k} i, σ i m l 1, m + 1,..., and i k 0, m + 1,..., and i k. 35

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