IRS With Discrete Phase Shifts: When Is Quantization Optimal?

Abstract

Intelligent reflective surfaces (IRS) with discrete phase shifts are considered. While no analytical solutions for globally-optimal discrete phase shifts are known, quantization of optimized continuous phase shifts is often used in the literature instead but the optimality of this strategy remains unknown. It is known to be not optimal in some special cases, but does there exist a broad class of cases for which this strategy is globally optimal? A partial answer to this question is provided here. In particular, scalar minimum-distance quantization of optimized continuous phase shifts is shown to be a globally-optimal strategy under discrete phase shifts if all quantization errors do not exceed 50% of their maximum possible value. Under mild additional conditions, it is the only strategy achieving global optimum. This is further extended to the scenarios where all quantization errors belong to an interval not exceeding half of the quantization step size, including, as special cases, the scenarios where all quantization errors are either positive or negative.