Quasi-Static Phase Shift Design for A Double-IRS Cooperatively Assisted System

This paper investigates a double-IRS cooperatively assisted system, where a multi-antenna BS serves a single-antenna user with the help of two multi-element IRSs connected by the inter-IRS channel. The channel between any two nodes is modeled with Rician fading. The BS adopts the instantaneous CSI-adaptive maximum-ratio transmission (MRT) beamformer, and the two IRSs adopt a cooperative quasi-static phase shift design. The goal is to maximize the average achievable rate, which can be reflected by the average channel power of the equivalent channel, at low channel estimation cost and phase adjustment costs and computational complexity. First, we obtain a tractable expression of the average channel power of the equivalent channel. Then, we jointly optimize the phase shifts of the two IRSs to maximize the average channel power of the equivalent channel. We propose a computationally efficient iterative algorithm to obtain a stationary point of the non-convex problem. We show that the optimal quasi-static phase shift design for the double-IRS cooperatively assisted system achieves an average channel power gain in order identical to that of the optimal instantaneous CSI-adaptive phase shift design for the same system and higher than that of the optimal quasi-static phase shift design for a counterpart single-IRS assisted system. Finally, we numerically demonstrate notable gains of the proposed cooperative quasi-static phase shift design over the existing solutions. To our knowledge, this is the first work that optimizes the quasi-static phase shift design of a double-IRS cooperatively assisted system and characterizes its advantage over the optimal quasi-static phase shift design of the counterpart single-IRS-assisted system.