On the Equivalence of Maximum SNR and MMSE Estimation: Applications to Additive Non-Gaussian Channels and Quantized Observations

TL;DR

This paper establishes an equivalence between MMSE and maximum SNR estimators, extends it to suboptimal estimators with basis expansion models, and derives bounds for quantized observations, applicable to non-Gaussian channels.

Contribution

It formalizes the SNR criterion for estimator design, proves the MMSE and SNR estimator equivalence for all statistics, and derives bounds for quantized observation scenarios.

Findings

Proves the equivalence between MMSE and maximum SNR estimators.

Derives closed-form expressions for MSE and SNR of suboptimal estimators.

Shows that suboptimal estimators with finite resolution approach optimal MMSE as resolution increases.

Abstract

The minimum mean-squared error (MMSE) is one of the most popular criteria for Bayesian estimation. Conversely, the signal-to-noise ratio (SNR) is a typical performance criterion in communications, radar, and generally detection theory. In this paper we first formalize an SNR criterion to design an estimator, and then we prove that there exists an equivalence between MMSE and maximum-SNR estimators, for any statistics. We also extend this equivalence to specific classes of suboptimal estimators, which are expressed by a basis expansion model (BEM). Then, by exploiting an orthogonal BEM for the estimator, we derive the MMSE estimator constrained to a given quantization resolution of the noisy observations, and we prove that this suboptimal MMSE estimator tends to the optimal MMSE estimator that uses an infinite resolution of the observation. Besides, we derive closed-form expressions for the mean-squared error (MSE) and for the SNR of the proposed suboptimal estimators, and we show that these expressions constitute tight, asymptotically exact, bounds for the optimal MMSE and maximum SNR.