Millimeter wave (mmWave) communication is expected to be widely deployed in fifth generation (5G) wireless networks due to the substantial bandwidth available at mmWave frequencies. To overcome the higher path loss observed at mmWave bands, most prior work focused on the design of directional beamforming using analog and/or hybrid beamforming techniques in largescale multiple-input multiple-output (MIMO) systems. Obtaining potential gains from highly directional beamforming in practical systems hinges on sufficient levels of channel estimation accuracy, where the problem of channel estimation becomes more challenging due to the substantial training overhead needed to sound all directions using a high-resolution narrow beam. In this work, we consider the design of multi-resolution beamforming sequences to enable the system to quickly search out the dominant channel direction for single-path channels. The resulting design generates a multilevel beamforming sequence that strikes a balance between minimizing the training overhead and maximizing beamforming gain, where a subset of multilevel beamforming vectors is chosen adaptively to provide an improved average data rate within a constrained time. We propose an efficient method to design a hierarchical multiresolution codebook utilizing a Butler matrix, a generalized discrete Fourier transform (DFT) matrix implemented using analog RF circuitry. Numerical results show the effectiveness of the proposed algorithm.
Millimeter wave, MIMO, multi-resolution beam alignment, Butler matrix
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