On the Capacity Achieving Covariance Matrix for Frequency Selective MIMO Channels Using the Asymptotic Approach

Abstract

Abstract-In this contribution, an algorithm for evaluating the capacity-achieving input covariance matrices for frequency selective Rayleigh MIMO channels is proposed. In contrast with the flat fading Rayleigh case, no closed-form expressions for the eigenvectors of the optimum input covariance matrix are available. Classically, both the eigenvectors and eigenvalues are computed numerically and the corresponding optimization algorithms remain computationally very demanding. In this paper, it is proposed to optimize (w.r.t. the input covariance matrix) a large system approximation of the average mutual information derived by Moustakas and Simon. The validity of this asymptotic approximation is clarified thanks to Gaussian large random matrices methods. It is shown that the approximation is a strictly concave function of the input covariance matrix and that the average mutual information evaluated at the argmax of the approximation is equal to the capacity of the channel up to a O (1/t) term, where t is the number of transmit antennas. An algorithm based on an iterative waterfilling scheme is proposed to maximize the average mutual information approximation, and its convergence studied. Numerical simulation results show that, even for a moderate number of transmit and receive antennas, the new approach provides the same results as direct maximization approaches of the average mutual information.Index Terms-Ergodic capacity, frequency selective MIMO channels, large random matrices

Optimization of MIMO Systems Capacity Using Large Random Matrix Methods

“…In [16] the authors generalized this analysis in the context of the multiple access jointly correlated Rician MIMO channels. Interestingly, the mutual information provided by the optimization of the approximation appears numerically to be quite close from the true optimum mutual information, even for realistic values of r and t. These observations were confirmed by the theoretical results of [11,12], which showed that the relative error is a O 1 t 2 term. In this paper, we give a comprehensive introduction to the optimization of large system approximations of average mutual information of MIMO channels.…”

“…In this paper, we give a comprehensive introduction to the optimization of large system approximations of average mutual information of MIMO channels. We note that the main results of this paper were already presented in [11,12]. However, the focus of [11,12] is on the detailed derivation of the large system approximations, while the present paper concentrates on their optimization.…”

“…We note that the main results of this paper were already presented in [11,12]. However, the focus of [11,12] is on the detailed derivation of the large system approximations, while the present paper concentrates on their optimization. In particular, we emphasize on methodological issues that are often partially-if at all-addressed.…”

“…This optimization problem is easier because the approximation can be very often nearly expressed in closed form so that its maximization does not need Monte-Carlo simulations. Large system approximations of average mutual information were derived in the past using the replica method (see e.g., [6] for bi-correlated MIMO Rayleigh channels, [7] for frequency selective Rayleigh channels, [8] for bi-correlated MIMO Ricean channels, [9] for jointly correlated Rician channel) or rigorous random matrix methods ( [10] for bi-correlated MIMO Rayleigh channels, [11] for bi-correlated MIMO Ricean channels, [12] for frequency selective Rayleigh channels, [13] for multiple access Rayleigh MIMO channels). Optimization algorithms of the large system approximation were first proposed in [14] in the context of bi-correlated MIMO Rayleigh channels.…”

“…Optimization algorithms of the large system approximation were first proposed in [14] in the context of bi-correlated MIMO Rayleigh channels. In [11,12] the authors addressed in more details the Rician bi-correlated channels and the frequency selective Rayleigh channels respectively, while [15] considered the Rician channel with interference. In [9] the authors considered the case of jointly correlated Rician channel.…”