On the Capacity of the Peak Power Constrained Vector Gaussian Channel: An Estimation Theoretic Perspective
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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 the necessary and sufficient conditions on the constraint R such that the input distribution supported on a single sphere is optimal. The maximum R̅_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 R̅_n scales as √(n) and the exact limit of R̅_n/√(n) is found.
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