Super-Linear Growth of the Capacity-Achieving Input Support for the Amplitude-Constrained AWGN Channel

TL;DR

The paper tackles how the capacity-achieving input support size $K_A$ scales with amplitude $A$ for the amplitude-constrained AWGN channel. It shows the capacity-achieving output becomes nearly uniform on the bulk as $A$ grows and develops a novel theory for best approximation by finite Gaussian mixtures in $\chi^2$ divergence. By combining these insights, it proves a first non-trivial lower bound $K_A \gg A$, disproving the conjecture $K_A \propto A$ and establishing super-linear growth. This advances fundamental understanding of discrete capacity-achieving inputs under constraints and informs both analytical and numerical approaches to related channel models.

Abstract

We study the growth of the support size of the capacity-achieving input distribution for the amplitude-constrained additive white Gaussian noise (AWGN) channel. While it is known since Smith (1971) that the optimal input is discrete with finitely many mass points, tight bounds on the number of support points $K_A$ as the amplitude constraint $A$ increases remain open. Not much is known until recently, when Dytso et al. (2019) proved that $K_A$ grows at least linearly and at most quadratically in $A$. Here, we provide a novel method, building on Ma et al. (2024); Zhang (1994), to derive the first non-trivial lower bound showing that KA grows super-linearly in A.

Theorems & Definitions (6)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 4
• Lemma 5
• Lemma 6