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In this paper, we studied the problem of beam alignment for millimeter wave(mmWave) communications, in which we assume a hybrid analog and digitalbeamforming structure is employed at the transmitter (i.e. base station), andan 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 beamalignment problem is formulated as a sparse encoding and phaseless decodingproblem. More specifically, the problem of interest involves finding a sparsesensing matrix and an efficient recovery algorithm to recover the support andmagnitude of the sparse signal from compressive phaseless measurements. Asparse bipartite graph coding (SBG-Coding) algorithm is developed for sparseencoding and phaseless decoding. Our theoretical analysis shows that, in thenoiseless case, our proposed algorithm can perfectly recover the support andmagnitude of the sparse signal with probability exceeding a pre-specified valuefrom $\mathcal{O}(K^2)$ measurements, where $K$ is the number of nonzeroentries of the sparse signal. The proposed algorithm has a simple decodingprocedure which is computationally efficient and noise-robust. Simulationresults show that our proposed method renders a reliable beam alignment in thelow and moderate signal-to-noise ratio (SNR) regimes and presents a clearperformance advantage over existing methods.

Fast Beam Alignment for Millimeter Wave Communications: A Sparse Encoding and Phaseless Decoding Approach