An Improved Lower Bound on Cardinality of Support of the Amplitude-Constrained AWGN Channel

2025/12/27 by Haiyang Wang, Wang, Haiyang, Luca Barletta +3 · 1 citation
Engineering · #Advanced MIMO Systems Optimization #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Molecular Communication and Nanonetworks #Statistics Theory (math.ST) #Wireless Communication Security Techniques

paper · doi:10.48550/arxiv.2512.22691

openalex publication_date 2025/12/27 · openalex created_date 2025/12/31 · openalex updated_date 2026/07/28

Abstract

We study the amplitude-constrained additive white Gaussian noise channel. It is well known that the capacity-achieving input distribution for this channel is discrete and supported on finitely many points. The best known bounds show that the support size of the capacity-achieving distribution is lower-bounded by a term of order A and upper-bounded by a term of order A², where A denotes the amplitude constraint. It was conjectured in [1] that the linear scaling is optimal. In this work, we establish a new lower bound of order A√(log A), improving the known bound and ruling out the conjectured linear scaling. To obtain this result, we quantify the fact that the capacity-achieving output distribution is close to the uniform distribution in the interior of the amplitude constraint. Next, we introduce a wrapping operation that maps the problem to a compact domain and develop a theory of best approximation of the uniform distribution by finite Gaussian mixtures. These approximation bounds are then combined with stability properties of capacity-achieving distributions to yield the final support-size lower bound.

Citations

• Super-Linear Growth of the Capacity-Achieving Input Support for the Amplitude-Constrained AWGN Channel
• On 2 × 2 MIMO Gaussian Channels with a Small Discrete-Time Peak-Power Constraint
• Binomial Channel: On the Capacity-Achieving Distribution and Bounds on the Capacity
• On the best approximation by finite Gaussian mixtures
• Properties of the Support of the Capacity-Achieving Distribution of the Amplitude-Constrained Poisson Noise Channel
• The Capacity Achieving Distribution for the Amplitude Constrained\n Additive Gaussian Channel: An Upper Bound on the Number of Mass Points
• On the Capacity of the Peak Power Constrained Vector Gaussian Channel:\n An Estimation Theoretic Perspective
• Capacity Bounds for Discrete-Time, Amplitude-Constrained, Additive White\n Gaussian Noise Channels
• On the Capacity of Vector Gaussian Channels With Bounded Inputs
• Mutual Information and Minimum Mean-square Error in Gaussian Channels

Cited by

• Phase Transitions of the Additive Uniform Noise Channel with Peak Amplitude and Cost Constraint
• The Binomial Channel: On Capacity, Optimal Inputs, and Beta-Binomial Approximation

Related

• Super-Linear Growth of the Capacity-Achieving Input Support for the Amplitude-Constrained AWGN Channel
• Capacity Bounds for Discrete-Time, Amplitude-Constrained, Additive White\n Gaussian Noise Channels
• The Capacity of the Amplitude-Constrained Vector Gaussian Channel
• The Capacity Achieving Distribution for the Amplitude Constrained\n Additive Gaussian Channel: An Upper Bound on the Number of Mass Points
• Finite-Support Capacity-Approaching Distributions for AWGN Channels