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.

翻译：本文针对单比特量化系统中高斯混合模型分布的目标信号（可能受到加性高斯白噪声干扰）的均方误差最优条件均值估计器，提出了新的基础性研究成果。我们首先推导了Bussgang估计器（量化系统中著名的线性最小均方误差估计器）的闭式解析表达式。随后给出了特殊情况下条件均值估计器的闭式解析表达式，揭示了最优估计器在单比特量化观测中呈线性特性——这与更高分辨率情况截然不同。通过与近期研究的高斯情形对比，我们建立了新的均方误差不等式，并证明目标信号与辅助量化噪声存在相关性。我们将分析拓展至多观测场景，研究了均方误差最优发射序列并进行了渐近分析，从而推导出均方误差及其极限的解析表达式。这些成果对各类信号处理应用的分析与设计具有广泛影响。