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In this paper, we focus on large intelligent reflecting surfaces (IRSs) andpropose a new codebook construction method to obtain a set of predesignedphase-shift configurations for the IRS unit cells. Since the overhead forchannel estimation and the complexity of online optimization for IRS-assistedcommunications scale with the size of the phase-shift codebook, the design ofsmall codebooks is of high importance. We show that there exists a fundamentaltradeoff between power efficiency and the size of the codebook. We firstanalyze this tradeoff for baseline designs that employ a linear phase-shiftacross the IRS. Subsequently, we show that an efficient design for smallcodebooks mandates higher-order phase-shift variations across the IRS.Consequently, we propose a quadratic phase-shift design, derive itscoefficients as a function of the codebook size, and analyze its performance.Our simulation results show that the proposed design yields a higher powerefficiency for small codebooks than the linear baseline designs.

Power Efficiency, Overhead, and Complexity Tradeoff in IRS-Assisted Communications -- Quadratic Phase-Shift Design