On rate-optimal MIMO signalling with mean and covariance feedback

We consider a single-user multiple-input multiple-output (MIMO) communication system in which the transmitter has access to both the channel covariance and the channel mean. For this scenario, we provide an explicit second-order approximation of the ergodic capacity of the channel, and we use this approximation to show that when the channel has a non-zero mean, the basis of the optimal input covariance matrix depends on the input signal power. (This basis is independent of the signal power in the zero-mean case.) The second-order approximation also provides insight into the way in which the low-signal-to-noise-ratio (SNR) optimal input covariance matrix is related to the optimal input covariance matrix at arbitrary SNRs. Furthermore, we show that the design of the input covariance matrix that optimizes the second-order approximation can be cast as a convex optimization problem for which the Karush-Kuhn-Tucker (KKT) conditions completely characterize the optimal solution. Using these conditions, we provide an efficient algorithm for obtaining second-order optimal input covariance matrices. The resulting covariances confirm our theoretical observation that, in general, the low-SNR optimal signal basis does not coincide with the optimal basis at higher SNRs. Finally, we show how our second-order design algorithm can be used to efficiently obtain input covariance matrices that provide ergodic rates that approach the ergodic capacity of the system.