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

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