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An Identity of Hankel Matrices Generated from the Moments of Gaussian Distribution

Sha Hu

TL;DR

The paper addresses exact identities between Hankel matrices generated from Gaussian moments and their use in optimizing a nonlinear distortion function for a maximum-likelihood receiver. It derives closed-form Cholesky-based decompositions and reveals a commuting identity via Hermite-polynomial structure, recasting the optimization as a spectral problem. The key contributions are (i) a commuting identity that reduces the transformed Hankel product to a scalar multiple of a diagonal matrix, (ii) explicit factorizations of the Hankel matrices in terms of diagonal scaling, and (iii) the result that the maximal gain is $G = N/\sigma^2$ with the optimal distortion function being the $N$-th Hermite polynomial $H_N(s)$. These findings provide exact, scalable tools for NL distortion optimization in Gaussian channels and highlight deep connections between Hankel structure and Hermite polynomials.

Abstract

In this letter, we proved a matrix identity of Hankel matrices that seems unrevealed before, generated from the moments of Gaussian distributions. In particular, we derived the Cholesky decompositions of the Hankel matrices in closed-forms, and showed some interesting connections between them. The results have potential applications in such as optimizing a nonlinear (NL) distortion function that maximizes the receiving gain in wireless communication systems.

An Identity of Hankel Matrices Generated from the Moments of Gaussian Distribution

TL;DR

The paper addresses exact identities between Hankel matrices generated from Gaussian moments and their use in optimizing a nonlinear distortion function for a maximum-likelihood receiver. It derives closed-form Cholesky-based decompositions and reveals a commuting identity via Hermite-polynomial structure, recasting the optimization as a spectral problem. The key contributions are (i) a commuting identity that reduces the transformed Hankel product to a scalar multiple of a diagonal matrix, (ii) explicit factorizations of the Hankel matrices in terms of diagonal scaling, and (iii) the result that the maximal gain is with the optimal distortion function being the -th Hermite polynomial . These findings provide exact, scalable tools for NL distortion optimization in Gaussian channels and highlight deep connections between Hankel structure and Hermite polynomials.

Abstract

In this letter, we proved a matrix identity of Hankel matrices that seems unrevealed before, generated from the moments of Gaussian distributions. In particular, we derived the Cholesky decompositions of the Hankel matrices in closed-forms, and showed some interesting connections between them. The results have potential applications in such as optimizing a nonlinear (NL) distortion function that maximizes the receiving gain in wireless communication systems.
Paper Structure (3 sections, 3 theorems, 41 equations)

This paper contains 3 sections, 3 theorems, 41 equations.

Table of Contents

  1. Origin of the Problem
  2. Main Results
  3. Some Remarks

Key Result

Theorem 1

The Hankel matrices $\boldsymbol{A}$ and $\boldsymbol{B}$, both of sizes $M\!\times\!M$ and generated from the even moments of a Gaussian distribution $\mathcal{N}(0, \sigma^2)$ as in (A) and (B), respectively, are equal to where the diagonal matrix $\boldsymbol{D}_\sigma$ is The Hankel matrices $\boldsymbol{A}_0$ and $\boldsymbol{B}_0$ are generated with the case $\sigma\!=\!1$ from (A) and (B)

Theorems & Definitions (10)

  • Example 1
  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • proof