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In intelligent reflecting surface (IRS) assisted communication, beam searchis usually time-consuming as the multiple-input multiple-output (MIMO) of IRSis usually very large. Hierarchical codebooks is a widely accepted method forreducing the complexity of searching time. The performance of this methodstrongly depends on the design scheme of beamforming of different beamwidths.In this paper, a non-constant phase difference (NCPD) beamforming algorithm isproposed. To implement the NCPD algorithm, we first model the phase shift ofIRS as a continuous function, and then determine the parameters of thecontinuous function through the analysis of its array factor. Then, we proposea hierarchical codebook and two beam training schemes, namely the jointsearching (JS) scheme and direction-wise searching (DWS) scheme by using theNCPD algorithm which can flexibly change the width, direction and shape of thebeam formed by the IRS array. Simulation results show that the NCPD algorithmis more accurate with smaller side lobes, and also more stable on IRS ofdifferent sizes compared to other wide beam algorithms. The misalignment rateof the beam formed by the NCPD method is significantly reduced. The timecomplexity of the NCPD algorithm is constant, thus making it more suitable forsolving the beamforming design problem with practically large IRS.

Hierarchical Codebook Design and Analytical Beamforming Solution for IRS Assisted Communication