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, not even a bound on the exact number of mass points/shells was available. This paper provides an alternative proof of the finiteness of the number mass points/shells of the capacity-achieving input distribution while producing the first firm bounds on the number of mass points and shells, paving an alternative way for approaching many such problems.The first main result of this paper is an order tight implicit bound which shows that the number of mass points in the capacity-achieving input distribution is within a factor of two from the number of zeros of the downward shifted capacityachieving output probability density function. Next, this implicit bound is utilized to provide a first firm upper on the support size of optimal input distribution, an O(A 2 ) upper bound where A denotes the constraint on the input amplitude. The second main result of this paper generalizes the first one to the case when n > 1, showing that, for each and every dimension n ≥ 1, the number of shells that the optimal input distribution contains is O(A 2 ). Finally, the third main result of this paper reconsiders the case n = 1 with an additional average power constraint, demonstrating a similar O(A 2 ) bound.

The Secrecy Capacity of Cost-Constrained Wiretap Channels

“…Another possible direction in the continuous channel setup is imposing a peak power constraint on the input. For point-to-point channels, it is known that the peak-constrained capacity achieving distribution is discrete with a finite support [35]- [38]. Exploring the properties of optimal distributions when an eavesdropper is present seems like a natural extension.…”

Properties of the Support of the Capacity-Achieving Distribution of the Amplitude-Constrained Poisson Noise Channel

“…2) Oscillation Theorem: To find an upper bound on the number of points in the support of P X , we will follow the proof technique developed in [28] for the Gaussian noise channel. The key step that we borrow from [28] is the use of the variation-diminishing property, which is captured by the oscillation theorem of Karlin [29]. To state the oscillation theorem we need the following definition.…”

On the Capacity of Amplitude Modulated Soliton Communication over Long Haul Fibers

“…In [ 34 , 35 ], it is shown for the AWGN channel with amplitude constraints that the capacity-achieving distribution is discrete with a finite number of mass points for such channels. An upper bound is proposed in [ 36 ] for the number of mass points. However, these works focus on the MI-based capacity formulation.…”