  This paper focuses on the minimum mean squared error (MMSE) channel estimator
for multiple-input multiple-output (MIMO) systems with one-bit quantization at
the receiver side. Despite its optimality and significance in estimation
theory, the MMSE channel estimator has not been fully investigated in this
context due to its general non-linearity and computational complexity. Instead,
the typically suboptimal Bussgang linear MMSE (BLMMSE) estimator has been
widely adopted. In this work, we develop a new framework to compute the MMSE
channel estimator that hinges on computation of the orthant probability of the
multivariate normal distribution. Based on this framework, we determine a
necessary and sufficient condition for the BLMMSE channel estimator to be
optimal and equivalent to the MMSE estimator. Under the assumption of specific
channel correlation or pilot symbols, we further utilize the framework to
derive analytical expressions for the MMSE channel estimator that are
particularly convenient for computation when certain system dimensions become
large, thereby enabling a comparison between the BLMMSE and MMSE channel
estimators in these cases.
