Beamforming Optimization and Capacity Scaling Analysis for Double Hybrid Active-Passive RISs Aided Wireless Communications

Reconfigurable intelligent surface (RIS) has emerged as a promising technology to enhance the coverage and capacity in future wireless networks due to its capability of reshaping the wireless propagation environment. In order to flexibly integrate the low-cost advantage of the fully-passive RIS and the high amplification gain of the fully-active RIS, a novel hybrid active-passive RIS (HRIS) composed of both passive and active elements has recently been proposed. In this paper, we aim to maximize the ergodic capacity of a double-HRIS aided communication system by jointly optimizing the BS transmit beamformer and the reflection matrices at the two HRISs. To tackle this challenging problem, we first derive an accurate analytical approximation of the ergodic capacity, and then propose an efficient alternating optimization (AO) algorithm to find a locally optimal solution. For characterizing system capacity scaling explicitly, we focus on a simplified double-reflection-only system, where the optimal BS transmit beamformer and the optimal phase shifts at two HRISs are proved to achieve beam alignment, and the optimal amplification factors of all active HRIS elements are shown to be the same. Based on these facts, we derive the capacity scaling law with respect to an asymptotically large number of HRIS active/passive elements. It is revealed that under a fixed HRIS power budget, the capacity scales faster with the increasing number of passive elements than with that of active elements. Numerical simulation results demonstrate the superior performance of our proposed algorithm over existing baselines, and also verify the capacity scaling law.