Linear and Nonlinear MMSE Estimation in One-Bit Quantized Systems under a Gaussian Mixture Prior

TL;DR

This paper derives new closed-form expressions for the MMSE estimator in one-bit quantized systems with Gaussian mixture signals, revealing linearity of the optimal estimator and providing insights for system design.

Contribution

It introduces novel analytic formulas for the CME in one-bit systems with GMM signals and compares them to Gaussian cases, advancing understanding of quantized estimation.

Findings

Optimal estimator is linear in one-bit quantized observations.

Derived MSE bounds and inequalities for GMM signals.

Extended analysis to multiple observations and asymptotic regimes.

Abstract

We present new fundamental results for the mean square error (MSE)-optimal conditional mean estimator (CME) in one-bit quantized systems for a Gaussian mixture model (GMM) distributed signal of interest, possibly corrupted by additive white Gaussian noise (AWGN). We first derive novel closed-form analytic expressions for the Bussgang estimator, the well-known linear minimum mean square error (MMSE) estimator in quantized systems. Afterward, closed-form analytic expressions for the CME in special cases are presented, revealing that the optimal estimator is linear in the one-bit quantized observation, opposite to higher resolution cases. Through a comparison to the recently studied Gaussian case, we establish a novel MSE inequality and show that that the signal of interest is correlated with the auxiliary quantization noise. We extend our analysis to multiple observation scenarios, examining the MSE-optimal transmit sequence and conducting an asymptotic analysis, yielding analytic expressions for the MSE and its limit. These contributions have broad impact for the analysis and design of various signal processing applications.