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

翻译：在智能反射面辅助通信中，由于IRS的多输入多输出（MIMO）规模通常非常庞大，波束搜索往往耗时严重。分层码本被广泛采用以降低搜索复杂度，其性能强烈依赖于不同波束宽度的波束赋形设计方案。本文提出一种非恒定相位差（NCPD）波束赋形算法。为实现NCPD算法，我们首先将IRS的相移建模为连续函数，并通过分析其阵列因子确定连续函数参数。基于NCPD算法可灵活调控IRS阵列所形成波束的宽度、方向与形状的特性，我们进一步提出分层码本及两种波束训练方案：联合搜索（JS）方案与方向性搜索（DWS）方案。仿真结果表明，相较其他宽波束算法，NCPD算法具有更高的精准度、更低的旁瓣水平，且在不同规模的IRS上表现出更强的稳定性。NCPD方法形成的波束失配率显著降低。该算法的时间复杂度为常数阶，因此更适用于解决实际大规模IRS场景中的波束赋形设计问题。