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

arXiv: https://arxiv.org/abs/2407.01305 · HTML: https://mlsanity.com/paper/2407.01305/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 39 references that are in the map.
- First posted on arXiv 1 Jul 2024.

## Builds on (most cited first, 7 of 7)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [On the Mean Square Error Optimal Estimator in One-Bit Quantized Systems](https://mlsanity.com/paper/2212.04470/) (2022)
- [Signal Recovery From 1-Bit Quantized Noisy Samples via Adaptive Thresholding](https://mlsanity.com/paper/1812.03977/) (2018)
- [Channel Estimation for Quantized Systems based on Conditionally Gaussian Latent Models](https://mlsanity.com/paper/2309.04014/) (2023)
- [On Optimal MMSE Channel Estimation for One-Bit Quantized MIMO Systems](https://mlsanity.com/paper/2404.05536/) (2024)
- [A Statistical Characterization of Wireless Channels Conditioned on Side Information](https://mlsanity.com/paper/2406.04282/) (2024)
- [Using the Best Linear Approximation With Varying Excitation Signals for Nonlinear System Characterization](https://mlsanity.com/paper/1805.04575/) (2018)

## Built on by (most cited first, 2 of 2)

- [Quantized Signal Recovery with Interference via Parametrized Look-Up Tables](https://mlsanity.com/paper/2507.18370/) (2025)
- [Weighted Bayesian Cramacuteer-Rao Bound for Mixed-Resolution Parameter Estimation](https://mlsanity.com/paper/2508.20761/) (2025)

## Abstract (arXiv)

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.