The Capacity Achieving Distribution for the Amplitude Constrained Additive Gaussian Channel: An Upper Bound on the Number of Mass Points

Abstract

This paper studies an $n$-dimensional additive Gaussian noise channel with a peak-power-constrained input. It is well known that, in this case, when $n=1$ the capacity-achieving input distribution is discrete with finitely many mass points, and when $n>1$ the capacity-achieving input distribution is supported on finitely many concentric shells. However, due to the previous proof technique, neither the exact number of mass points/shells of the optimal input distribution nor a bound on it was available. This paper provides an alternative proof of the finiteness of the number mass points/shells of the capacity-achieving input distribution and produces the first firm bounds on the number of mass points and shells, paving an alternative way for approaching many such problems. Roughly, the paper consists of three parts. The first part considers the case of $n=1$. The first result, in this part, shows that the number of mass points in the capacity-achieving input distribution is within a factor of two from the downward shifted capacity-achieving output probability density function (pdf). The second result, by showing a bound on the number of zeros of the downward shifted capacity-achieving output pdf, provides a first firm upper on the number of mass points. Specifically, it is shown that the number of mass points is given by $O(\mathsf{A}^2)$ where $\mathsf{A}$ is the constraint on the input amplitude. The second part generalizes the results of the first part to the case of $n>1$. In particular, for every dimension $n>1$, it is shown that the number of shells is given by $O(\mathsf{A}^2)$ where $\mathsf{A}$ is the constraint on the input amplitude. Finally, the third part provides bounds on the number of points for the case of $n=1$ with an additional power constraint.