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

Haiyang Wang, Luca Barletta, Alex Dytso

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 $A^2$, 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 $A\sqrt{\log A}$, 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.

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

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 and upper-bounded by a term of order , where 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 , 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.
Paper Structure (10 sections, 8 theorems, 49 equations)

This paper contains 10 sections, 8 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Outline and Contributions
  3. Notation
  4. Main Result
  5. Proofs and Techniques
  6. A Best Approximation Theory With Gaussian Mixtures
  7. Some Properties of Capacity-Achieving Distribution
  8. Proof of the Main Theorem
  9. Conclusion
  10. Proof of Proposition \ref{['prop:cap_properties']}

Key Result

Theorem 1

Fix some $A>0$ and let $P_{X^*}$ be the capacity-achieving input distribution in eq:Capacity_def. Then, for some explicit constant $c>0$, where $\log^{+}(x) := \max\{\log(x), 0\}$.

Theorems & Definitions (15)

  • Theorem 1
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • proof
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • ...and 5 more