Signaling for MISO Channels Under First- and Second-Moment Constraints

Consider a multiple-input single-output system, where the nonnegative, peak-limited inputs ${X_1}, \ldots ,{X_{{n_{\text{T}}}}} \in [0,\mathcal{A}]$ are subject to first- and second-moment sum-constraints on all antennas. The paper characterizes all probability distributions that can be induced for the "channel image," which is given by the inner product of the input vector with a given channel vector. Key to this result is the description of input vectors that achieve a given deterministic channel image with the smallest energy, where "energy" of an input vector refers to a weighted sum of its one- and two-norms. Minimum-energy input vectors have an interesting structure: depending on the desired channel image, some of the weakest antennas are silenced, and the remaining antennas are chosen according to a shifted and amplitude-constrained beamforming rule.