Separability of parallel Gaussian MIMO broadcast channels with shaping constraints

It is well known that separate coding on each carrier is capacity-achieving in parallel Gaussian multiple-input multiple-output (MIMO) broadcast channels in the sense that the sum rate capacity under a sum power constraint is achievable with separate coding if the noise is independent on each carrier. In this paper, we generalize this statement by proving that the optimality of separate coding also holds under per-carrier shaping constraints, i.e., a sum covariance constraint on each carrier instead of a sum power constraint. Moreover, we show that not only the sum rate optimal point, but the whole capacity region can be achieved with separate coding. Finally, we prove that the worst-case noise in a MIMO broadcast channel with shaping constraints is independent across carriers.