  In millimeter wave (mmWave) massive multiple-input multiple-output (MIMO)
systems, one-bit analog-to-digital converters (ADCs) are employed to reduce the
impractically high power consumption, which is incurred by the wide bandwidth
and large arrays. In practice, the mmWave band consists of a small number of
paths, thereby rendering sparse virtual channels. Then, the resulting maximum a
posteriori (MAP) channel estimation problem is a sparsity-constrained
optimization problem, which is NP-hard to solve. In this paper, iterative
approximate MAP channel estimators for mmWave massive MIMO systems with one-bit
ADCs are proposed, which are based on the gradient support pursuit (GraSP) and
gradient hard thresholding pursuit (GraHTP) algorithms. The GraSP and GraHTP
algorithms iteratively pursue the gradient of the objective function to
approximately optimize convex objective functions with sparsity constraints,
which are the generalizations of the compressive sampling matching pursuit
(CoSaMP) and hard thresholding pursuit (HTP) algorithms, respectively, in
compressive sensing (CS). However, the performance of the GraSP and GraHTP
algorithms is not guaranteed when the objective function is ill-conditioned,
which may be incurred by the highly coherent sensing matrix. In this paper, the
band maximum selecting (BMS) hard thresholding technique is proposed to modify
the GraSP and GraHTP algorithms, namely the BMSGraSP and BMSGraHTP algorithms,
respectively. The BMSGraSP and BMSGraHTP algorithms pursue the gradient of the
objective function based on the band maximum criterion instead of the naive
hard thresholding. In addition, a fast Fourier transform-based (FFT-based) fast
implementation is developed to reduce the complexity. The BMSGraSP and
BMSGraHTP algorithms are shown to be both accurate and efficient, whose
performance is verified through extensive simulations.
