Optimal Signaling Schemes and Capacities of Non-Coherent Correlated MISO Channels Under Per-Antenna Power Constraints

This paper investigates the optimal signaling schemes and capacities of non-coherent correlated multiple-input single-output (MISO) channels in fast Rayleigh fading. We consider both channels under per-antenna power constraints as well as channels under joint per-antenna and sum power constraints. For per-antenna power constraint channels, we first establish the convex and compact properties of the feasible sets, and demonstrate the existence of optimal input distribution and the uniqueness of optimal effective magnitude input distribution. By exploiting the solutions of a quadratic optimization problem, we show that the Kuhn-Tucker condition on the optimal inputs can be simplified to a single dimension. As a result, we can apply the Identity Theorem to show the discrete and finite nature of the optimal effective magnitude distribution, with a mass point located at the origin. By using this distribution, we then construct a finite and discrete optimal input vector distribution. The use of this input allows us to determine the capacity gain of MISO over SISO via the phase solutions of a constrained quadratic optimization problem on a sphere, which can be obtained using a proposed penalized optimization algorithm. We also extend the results to MISO channels subject to the joint per-antenna and sum power constraints. Under this consideration, it is shown that not all per-antenna constraints are active. While the finiteness and discreteness of the optimal effective magnitude and the optimal input vector distributions still hold, the optimal phases and the optimal power allocation among the transmit antennas need to be determined simultaneously via a quadratic optimization problem under inequality constraints. These solutions can finally be used to obtain the MISO capacity gain.