Capacity of the vector Gaussian channel in the small amplitude regime

This paper studies the capacity of an ndimensional 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 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 R^¯n scales as √n and the exact limit of ^√R¯n/ √n is found.