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.

翻译：本文研究了在高斯联合输入的信道估计背景下，单比特量化系统中均方误差最优的条件均值估计器。我们分析了通常非线性的条件均值估计器与基于巴桑定理的经典线性巴桑估计器之间的关系。我们提出一项新发现：在多种特殊情形下，巴桑估计器等价于条件均值估计器，包括单变量高斯输入以及量化前无加性噪声的多导频信号情形。对于一般情形，我们通过数值仿真量化了巴桑估计器与条件均值估计器之间的差距——该差距随维度升高和导频序列增长而增大。受条件均值估计器启发，我们提出一种最优导频序列，并推导了该情形下方差的闭合表达式。随后，我们找到了渐进大导频数条件下方差的闭合极限，该结论同样适用于巴桑估计器。最后，我们针对不同系统参数和性能指标进行数值实验，揭示了量化体制中最优信道估计器的行为特征。在此背景下，可量化量化系统中著名的随机共振效应。