关于幅度受限AWGN信道容量可达输入支撑集的演化
On the Evolution of the Capacity-Achieving Input Support for the Amplitude-Constrained AWGN Channel
- Politecnico di Milano（米兰理工大学）
- Qualcomm Flarion Technology, Inc.（高通Flarion科技有限公司）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究幅度受限AWGN信道中容量可达输入分布支撑集随幅度约束的演化，证明原点为唯一退化点且支撑基数局部变化仅发生在原点，机制为原点出现/消失或分裂/合并。
AI中文摘要：
我们考虑受峰值幅度约束的加性高斯白噪声信道，并研究容量可达输入分布（CAID）的支撑集如何随幅度约束$A$的变化而演化。尽管已知CAID是唯一的、对称的、离散的且具有有限支撑集，但其支撑集转移的结构在很大程度上仍未解决。我们证明原点是唯一可能的退化支撑点，也是KKT函数唯一可能的不活跃接触点。然后，我们建立了最优分布和KKT函数的连续性性质，并证明每个非零支撑点都是局部稳定的：在$A$的微小变化下，它作为唯一的邻近原子持续存在，其位置和概率质量连续变化。综合这些结果，我们证明在局部，邻近幅度处的支撑基数要么不变，要么增加一，且任何此类增加只能发生在原点。因此，支撑基数的所有局部变化都局限于原点：局部上，唯一可能的转移机制是原点处点的出现或消失，以及原点分裂或合并为对称对。
英文摘要：
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.
