We prove convergence of the projected gradient algorithm with inexact projections when applied to linear inverse problems with constraint sets that are unions of subspaces. Such an algorithm is useful for joint angle and delay estimation in MIMO radar, where classical estimators for angle estimation can be integrated into compressive sensing methods for range estimation.

This paper considers the so-called partial decode-and-forward (DF) strategy for the Gaussian multiple-input multiple-output (MIMO) relay channel. Unlike for the DF strategy or point-to-point (P2P) transmission from source to destination, for which Gaussian channel inputs are known to maximize the achievable rates, the input distribution that attains the maximum achievable partial DF rate for the Gaussian MIMO relay channel has remained unknown so far. For some special cases, e.g., for relay channels where the partial DF strategy reduces to the DF or P2P transmission, it could be deduced that Gaussian inputs maximize the rate that can be achieved with the partial DF strategy. For the general case, however, the problem has remained open until now. In this paper, we solve this problem by proving that the maximum achievable partial DF rate for the Gaussian MIMO relay channel is always attained by Gaussian channel inputs. Our proof relies on the channel enhancement technique, which was originally introduced by Weingarten et al. to derive the (private message) capacity region of the Gaussian MIMO broadcast channel. By combining this technique with a primal decomposition approach, we first establish that jointly Gaussian source and relay inputs maximize the achievable partial DF rate for the aligned Gaussian MIMO relay channel. Subsequently, we use a limiting argument to extend this result from the aligned to the general Gaussian MIMO relay channel.

Proper (i.e., circularly symmetric) Gaussian signals are known to be capacity-achieving in Gaussian multiple-input multiple-output (MIMO) broadcast channels with proper noise in the sense that the sum rate capacity under a sum power constraint is achievable with proper Gaussian signaling. In this paper, we generalize this statement by proving that the optimality of proper Gaussian signals also holds under a shaping constraint, i.e., a sum covariance constraint instead of a power constraint. Moreover, we show that not only the sum rate optimal point, but the whole capacity region can be achieved with proper Gaussian signals. Finally, we prove that the worst-case noise in a MIMO broadcast channel with shaping constraints is proper.

In this paper, we consider achievable rates for the Gaussian multiple-input multiple-output (MIMO) relay channel that can be obtained with the relay using the partial decode-and-forward scheme. The partial decode-and-forward strategy allows to optimize the amount of information the relay has to decode and can hence be seen as a generalization of the decode-and-forward strategy, where the relay must decode the entire source message. Since we cannot determine the maximal achievable partial decode-and-forward rate, we propose a suboptimal approach that is based on zero-forcing the interference the relay would suffer from the part of the source signal that it is not required to decode. For this purpose, a zero-forcing receive filter is introduced at the relay. We then show that, if the receive filter is fixed, standard convex optimization techniques can be used to evaluate the best rate our suboptimal partial decode-and-forward scheme can achieve. Simulation results demonstrate that the coding scheme we propose significantly outperforms the decode-and-forward scheme and/or approximates the cut-set bound for different network scenarios.