  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.
