  In this paper, we studied the problem of beam alignment for millimeter wave
(mmWave) communications, in which we assume a hybrid analog and digital
beamforming structure is employed at the transmitter (i.e. base station), and
an omni-directional antenna or an antenna array is used at the receiver (i.e.
user). By exploiting the sparse scattering nature of mmWave channels, the beam
alignment problem is formulated as a sparse encoding and phaseless decoding
problem. More specifically, the problem of interest involves finding a sparse
sensing matrix and an efficient recovery algorithm to recover the support and
magnitude of the sparse signal from compressive phaseless measurements. A
sparse bipartite graph coding (SBG-Coding) algorithm is developed for sparse
encoding and phaseless decoding. Our theoretical analysis shows that, in the
noiseless case, our proposed algorithm can perfectly recover the support and
magnitude of the sparse signal with probability exceeding a pre-specified value
from $\mathcal{O}(K^2)$ measurements, where $K$ is the number of nonzero
entries of the sparse signal. The proposed algorithm has a simple decoding
procedure which is computationally efficient and noise-robust. Simulation
results show that our proposed method renders a reliable beam alignment in the
low and moderate signal-to-noise ratio (SNR) regimes and presents a clear
performance advantage over existing methods.
