On the Capacity of the Peak Power Constrained Vector Gaussian Channel: An Estimation Theoretic Perspective

This paper studies the capacity of an $n$-dimensional vector Gaussian noise channel subject to the constraint that an input must lie in the ball of radius $R$ centered at the origin. It is known that in this setting the optimizing input distribution is supported on a finite number of concentric spheres. However, the number, the positions and the probabilities of the spheres are generally unknown. This paper characterizes necessary and sufficient conditions on the constraint $R$ such that the input distribution supported on a single sphere is optimal. The maximum $\bar{R}_n$, such that using only a single sphere is optimal, is shown to be a solution of an integral equation. Moreover, it is shown that $\bar{R}_n$ scales as $\sqrt{n}$ and the exact limit of $\frac{\bar{R}_n}{\sqrt{n}}$ is found.