On The Minimum Mean-Square Estimation Error of the Normalized Sum of Independent Narrowband Waves in the Gaussian Channel

TL;DR

This paper analyzes the minimum mean-square error in estimating a signal composed of many independent narrowband waves transmitted over a Gaussian channel, showing that errors decrease proportionally to 1/N as the number of waves increases.

Contribution

It demonstrates that both causal and non-causal MMSEs converge to the signal energy at a rate of 1/N when the number of narrowband waves tends to infinity.

Findings

CMMSE and MMSE approach the signal energy as N increases.

The convergence rate is proportional to 1/N regardless of the signal-to-noise ratio.

Results hold for any fixed signal energy to noise energy ratio.

Abstract

The minimum mean-square error of the estimation of a signal where observed from the additive white Gaussian noise (WGN) channel's output, is analyzed. 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 signal energy to noise energy ratio (no mater how big) both the causal minimum mean-square error CMMSE and the non-causal minimum mean-square error MMSE converge to the signal energy at a rate which is proportional to 1/N.