Some Relations between Divergence Derivatives and Estimation in Gaussian channels

TL;DR

This paper investigates the behavior of MMSE and non-Gaussianity in Gaussian channels, revealing how estimation error diminishes with increasing signal components and analyzing the Taylor expansion of non-Gaussianity for small noise levels.

Contribution

It introduces a general continuous-time channel model and analyzes the asymptotic behavior of MMSE and non-Gaussianity as the number of signal components increases.

Findings

MMSE converges to signal energy as N increases.

Non-Gaussianity terms vanish in the Taylor expansion for small q.

Behavior of non-Gaussianity is characterized for large N and small q.

Abstract

The minimum mean square error of the estimation of a non Gaussian signal where observed from an additive white Gaussian noise channel's output, is analyzed. First, a quite general time-continuous channel model is assumed for which the behavior of the non-Gaussianess of the channel's output for small signal to noise ratio q, is proved. Then, It is assumed that the channel input's signal is composed of a (normalized) sum of N narrowband, mutually independent waves. It is shown that if N goes to infinity, then for any fixed q (no mater how big) both CMMSE and MMSE converge to the signal energy at a rate which is proportional to the inverse of N. Finally, a known result for the MMSE in the one-dimensional case, for small q, is used to show that all the first four terms in the Taylor expansion of the non-Gaussianess of the channel's output equal to zero.