Capacity of the Vector Gaussian Channel in the Small Amplitude Regime

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 $\overline{R}_{n}$, such that using only a single sphere is optimal, is shown to be a solution of an integral equation. Moreover, it is shown that $\overline{R}_{n}$ scales as $\sqrt{n}$ and the exact limit of $\overline{R}_{n}\overline{\sqrt{n}}$ is found.