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

This paper studies an n-dimensional additive Gaussian noise channel with a peak-power-constrained input. It is well known that, in this case, the capacity-achieving input distribution is supported on finitely many concentric shells. However, due to the previous proof technique, neither the exact number of 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 shells of the capacity-achieving input distribution and produces the first firm upper bound on the number of shells, paving an alternative way for approaching many such problems. In particular, for every dimension n, it is shown that the number of shells is given by O(A <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) where A is the constraint on the input amplitude. Moreover, this paper also provides bounds on the number of points for the case of n = 1 with an additional power constraint.