On the Evolution of the Capacity-Achieving Input Support for the Amplitude-Constrained AWGN Channel

2026-09-09

AI Summary

研究了幅值受限AWGN信道中容量实现输入分布的支持集如何随幅值约束变化，并证明了非零支持点的局部稳定性及支持集基数变化仅发生在原点。

Abstract

We consider an additive white Gaussian noise channel subject to a peak-amplitude constraint and study how the support of the capacity-achieving input distribution (CAID) evolves as the amplitude constraint $A$ varies. Although the CAID is known to be unique, symmetric, discrete, and finitely supported, the structure of its support transitions has remained largely unresolved. We show that the origin is the only possible degenerate support point and the only possible inactive contact point of the KKT function. We then establish continuity properties of the optimal distribution and the KKT function and prove that every nonzero support point is locally stable: under small changes in $A$, it persists as a unique nearby atom whose location and probability mass vary continuously. Combining these results, we prove that, locally, the support cardinality at a nearby amplitude is either unchanged or larger by one and that any such increase can occur only at the origin. Consequently, all local changes in support cardinality are confined to the origin: locally, the only possible transition mechanisms are the appearance or disappearance of the point at the origin and the splitting or merging of the origin into a symmetric pair.

Problem

• Amplitude-Constrained AWGN Channel
• Capacity-Achieving Input Distribution (CAID)
• Support Evolution

Innovation

• capacity-achieving input distribution
• peak-amplitude constraint
• KKT function
• support cardinality