How Many Reflecting Elements Are Needed for Energy- and Spectral-Efficient Intelligent Reflecting Surface-Assisted Communication

Abstract

This paper investigates and analyzes the number of reflecting elements for guaranteed energy- and spectral-efficient intelligent reflecting surface (IRS)-assisted communication systems. As opposed to previous works where the energy efficiency (EE)/the spectral efficiency (SE) maximization or the EE-SE tradeoff was considered, our goal is to minimize the number of reflecting elements in the IRS-assisted system. Besides, both the EE and the SE constraints are considered in the number minimization problem, which has not been addressed in existing works. However, both the EE and the SE performance do not admit exact closed-form expressions due to the non-convexity incurred by joint beamforming and phase shift design. In order to make the optimization problem tractable, we resort to their performance bounds for problem reformulation. By decomposing the problem into two sub-problems, we are able to derive closed-form expressions for the minimum number of reflecting elements, and both the coherent phase shift (CPS)-oriented solution and the random phase shift (RPS)-oriented solution are proposed for comparison. In order to shed light on the practical design, the relationship between the derived number of reflecting elements is established for both schemes, and the upper bounds on both the EE and the SE thresholds and the placement of the IRS to achieve only one reflecting element are obtained. Simulation results confirm the validity of our analysis on the minimum number of reflecting elements and effectiveness of both schemes.