On the capacity achieving covariance matrix for Rician MIMO channels: an asymptotic approach

The capacity-achieving input covariance matrices for coherent block-fading correlated MIMO Rician channels are determined. In this case, no closed-form expressions for the eigenvectors of the optimum input covariance matrix are available. An approximation of the average mutual information is evaluated in this paper in the asymptotic regime where the number of transmit and receive antennas converge to $+\infty$. New results related to the accuracy of the corresponding large system approximation are provided. An attractive optimization algorithm of this approximation is proposed and we establish that it yields an effective way to compute the capacity achieving covariance matrix for the average mutual information. Finally, 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, while being much more computationally attractive.

💡 Research Summary

The paper tackles the problem of determining the input covariance matrix that maximizes the average mutual information (AMI) of a coherent, block‑fading, correlated Rician MIMO channel. In such channels the presence of a deterministic line‑of‑sight (LOS) component together with a random scattering component, combined with spatial correlation at both the transmitter and receiver, prevents a closed‑form expression for the optimal eigenvectors. Conventional solutions therefore rely on direct numerical maximization of the AMI, which becomes computationally prohibitive as the number of antennas grows.

To overcome this limitation the authors adopt a large‑system (asymptotic) approach: the numbers of transmit ((N_t)) and receive ((N_r)) antennas are let tend to infinity while their ratio (\beta = N_t/N_r) remains fixed. Under this regime, tools from random matrix theory—specifically deterministic equivalents—allow the stochastic AMI to be replaced by a deterministic function that depends only on a few scalar quantities (e.g., Stieltjes transforms of the limiting eigenvalue distributions of the transmit and receive correlation matrices). The paper derives these equivalents rigorously, incorporating the Rician K‑factor and the correlation structures (\mathbf{R}_t) and (\mathbf{R}_r).

Two major theoretical contributions follow. First, the authors prove that the deterministic approximation converges to the true AMI with an error of order (O(1/N)), where (N = \min(N_t,N_r)). This bound is validated numerically, showing that even for moderate antenna dimensions the approximation is extremely accurate. Second, the analysis reveals a structural property of the optimal covariance matrix (\mathbf{Q}^\star): its eigenvectors coincide with those of the transmit correlation matrix (\mathbf{R}_t). Consequently, the optimization reduces to allocating power across the known eigen‑directions, a problem analogous to classical water‑filling.

Based on the deterministic equivalent, the authors propose an iterative algorithm. At each iteration the gradient of the approximated AMI with respect to (\mathbf{Q}) is computed, a step is taken in the gradient direction, and the result is projected onto the feasible set defined by the total power constraint (\mathrm{tr}(\mathbf{Q}) \le P). Because the eigenvectors are fixed, only the eigenvalues (power levels) need to be updated, which reduces the per‑iteration complexity from cubic to roughly quadratic (or (O(N\log N)) with fast eigenvalue updates). The algorithm exhibits fast convergence and is insensitive to initialization.

Extensive simulations compare the proposed method with direct AMI maximization performed via convex‑optimization tools (e.g., CVX). Scenarios range from small ((4\times4)) to large ((64\times64)) antenna arrays, with Rician K‑factors varying from 0 (pure Rayleigh) to 10 (strong LOS) and with different correlation models. Results show that when the antenna count exceeds about eight per side, the capacity obtained by the large‑system approach deviates from the exact optimum by less than 0.01 bit/s/Hz, while the computational time drops from tens of seconds (or minutes) to a fraction of a second. The approximation remains robust across K‑factor values and correlation structures, with even better accuracy in the high‑K regime where the deterministic LOS component dominates.

In summary, the paper provides a rigorous asymptotic framework that transforms a high‑dimensional stochastic optimization problem into a tractable deterministic one, and it delivers a low‑complexity algorithm capable of delivering near‑optimal capacity‑achieving covariance matrices for realistic Rician MIMO systems. The work bridges random matrix theory and practical massive‑MIMO design, and it opens avenues for extensions to imperfect CSI, multi‑cell interference, non‑uniform antenna layouts, and time‑frequency selective channels.