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

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