  This paper tackles the problem of millimeter-Wave (mmWave) channel estimation
in massive MIMO communication systems. A new Bayes-optimal channel estimator is
derived using recent advances in the approximate belief propagation (BP)
Bayesian inference paradigm. By leveraging the inherent sparsity of the mmWave
MIMO channel in the angular domain, we recast the underlying channel estimation
problem into that of reconstructing a compressible signal from a set of noisy
linear measurements. Then, the generalized approximate message passing (GAMP)
algorithm is used to find the entries of the unknown mmWave MIMO channel
matrix. Unlike all the existing works on the same topic, we model the
angular-domain channel coefficients by Laplacian distributed random variables.
Further, we establish the closed-form expressions for the various statistical
quantities that need to be updated iteratively by GAMP. To render the proposed
algorithm fully automated, we also develop an expectation-maximization (EM)
based procedure that can be easily embedded within GAMP's iteration loop in
order to learn all the unknown parameters of the underlying Bayesian inference
problem. Computer simulations show that the proposed combined EM-GAMP algorithm
under a Laplacian prior exhibits improvements both in terms of channel
estimation accuracy, achievable rate, and computational complexity as compared
to the Gaussian mixture prior that has been advocated in the recent literature.
In addition, it is found that the Laplacian prior speeds up the convergence
time of GAMP over the entire signal-to-noise ratio (SNR) range.
