  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.
