# On the Mean Square Error Optimal Estimator in One-Bit Quantized Systems

arXiv: https://arxiv.org/abs/2212.04470 · HTML: https://mlsanity.com/paper/2212.04470/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 61 references that are in the map.
- First posted on arXiv 8 Dec 2022.

## Builds on (most cited first, 8 of 8)

- [Channel Estimation for One-Bit Multiuser Massive MIMO Using Conditional GAN](https://mlsanity.com/paper/2006.11435/) (2020)
- [Deep Learning for Massive MIMO with 1-Bit ADCs: When More Antennas Need Fewer Pilots](https://mlsanity.com/paper/1910.06960/) (2019)
- [SVM-based Channel Estimation and Data Detection for One-Bit Massive MIMO Systems](https://mlsanity.com/paper/2003.10678/) (2020)
- [The Bussgang Decomposition of Non-Linear Systems: Basic Theory and MIMO Extensions](https://mlsanity.com/paper/2005.01597/) (2020)
- [Signal Recovery From 1-Bit Quantized Noisy Samples via Adaptive Thresholding](https://mlsanity.com/paper/1812.03977/) (2018)
- [Massive MIMO Channel Estimation with Low-Resolution Spatial Sigma-Delta ADCs](https://mlsanity.com/paper/2005.07752/) (2020)
- [Low-rank mmWave MIMO channel estimation in one-bit receivers](https://mlsanity.com/paper/1910.09141/) (2019)
- [Gridless Angular Domain Channel Estimation for mmWave Massive MIMO System With One-Bit Quantization Via Approximate Message Passing](https://mlsanity.com/paper/1909.10114/) (2019)

## Built on by (most cited first, 6 of 6)

- [On the Asymptotic Mean Square Error Optimality of Diffusion Models](https://mlsanity.com/paper/2403.02957/) (2024)
- [Channel Estimation for Quantized Systems based on Conditionally Gaussian Latent Models](https://mlsanity.com/paper/2309.04014/) (2023)
- [Linear and Nonlinear MMSE Estimation in One-Bit Quantized Systems under a Gaussian Mixture Prior](https://mlsanity.com/paper/2407.01305/) (2024)
- [Enhancing Channel Estimation in Quantized Systems with a Generative Prior](https://mlsanity.com/paper/2405.03542/) (2024)
- [MIMO Detection with Spatial Sigma-Delta ADCs: A Variational Bayesian Approach](https://mlsanity.com/paper/2410.03891/) (2024)
- [Weighted Bayesian Cramacuteer-Rao Bound for Mixed-Resolution Parameter Estimation](https://mlsanity.com/paper/2508.20761/) (2025)

## Abstract (arXiv)

This paper investigates the mean square error (MSE)-optimal conditional mean estimator (CME) in one-bit quantized systems in the context of channel estimation with jointly Gaussian inputs. We analyze the relationship of the generally nonlinear CME to the linear Bussgang estimator, a well-known method based on Bussgang's theorem. We highlight a novel observation that the Bussgang estimator is equal to the CME for different special cases, including the case of univariate Gaussian inputs and the case of multiple pilot signals in the absence of additive noise prior to the quantization. For the general cases we conduct numerical simulations to quantify the gap between the Bussgang estimator and the CME. This gap increases for higher dimensions and longer pilot sequences. We propose an optimal pilot sequence, motivated by insights from the CME, and derive a novel closed-form expression of the MSE for that case. Afterwards, we find a closed-form limit of the MSE in the asymptotically large number of pilots regime that also holds for the Bussgang estimator. Lastly, we present numerical experiments for various system parameters and for different performance metrics which illuminate the behavior of the optimal channel estimator in the quantized regime. In this context, the well-known stochastic resonance effect that appears in quantized systems can be quantified.