Optimality of proper signaling in Gaussian MIMO broadcast channels with shaping constraints

Abstract

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.

Block-Skew-Circulant Matrices in Complex-Valued Signal Processing

“…However, until recently, it had not been shown that the optimality of proper signals also holds under a shaping constraint, i.e., a constraint on the sum transmit covariance matrix instead of on the sum power. 7 In our recent works [34], [35], we used the minimax duality with linear conic constraints from [56], [57] in combination with a power shaping matrix and an impropriety matrix to show this more general result. As discussed in Section III.D, these matrices fit into the framework proposed in this paper.…”

“…In order to briefly sketch the key idea of this approach, we reproduce the proof for the special case of a sum rate maximization in a system with two users. The more general weighted sum rate maximization for an arbitrary number of users is studied in [35].…”

“…In the above discussion, a so-called composite real representation of vectors and matrices was used. Apart from being closely related to the practical implementation and to our intuitive understanding of complex numbers, this representation has the following important advantage, which was exploited, e.g., in [11], [16], [17], [32]- [35] (and for the special case of scalar complex random variables in [3], [24], [28]). Whenever we know a method to solve a given problem involving real vectors, we can readily apply the method to composite real representations of general (proper or improper) complex vectors.…”

Minimax Duality for MIMO Interference Networks

“…Recent results use our minimax duality to proof the optimality of proper signaling for the MIMO broadcast channel under a shaping constraint [26] and for the MIMO relay channel with partial decode-and-forward [34]. A key ingredient for the proof is the observation that in a composite real representation a covariance matrix can be partitioned in a power shaping component and an impropriety component, such that the components are elements of linear subspaces that are orthogonal complements.…”

“…The use of the linear conic constraints is not motivated by a specific application, instead we will see that they are general enough to model a large class of constraints, including sum-power constraints, per antenna power constraints, or shaping constraints. Shaping constraints for the optimization of transmit covariances in MIMO systems are for example considered in [23][24][25][26][27].…”