Active reconfigurable intelligent surfaces (RISs) have recently been proposed to compensate for the severe multiplicative fading effect of conventional passive RIS-aided systems. Each reflecting element of active RISs is assisted by an amplifier such that the incident signal can be reflected and amplified instead of only being reflected as in passive RIS-aided systems. This work addresses the practical challenge that, on the one hand, in active RIS-aided systems the perfect individual CSI of the RIS-aided channels cannot be acquired due to the lack of signal processing power at the active RISs, but, on the other hand, this CSI is required to calculate the expected system data rate and RIS transmit power needed for transceiver design. To address this issue, we first derive closed-form expressions for the average achievable rate and the average RIS transmit power based on partial CSI of the RIS-aided channels. Then, we formulate an average achievable rate maximization problem for jointly optimizing the active beamforming at both the base station (BS) and the RIS. This problem is then tackled using the majorization--minimization (MM) algorithm framework, and, for each iteration, semi-closed-form solutions for the BS and RIS beamforming are derived based on the Karush-Kuhn-Tucker (KKT) conditions. To ensure the quality of service (QoS) of each user, we further formulate a rate outage constrained beamforming problem, which is solved using the Bernstein-Type inequality (BTI) and semidefinite relaxation (SDR) techniques. Numerical results show that the proposed algorithms can efficiently overcome the challenges imposed by imperfect CSI in active RIS-aided wireless systems.

翻译：近期提出的主动可重构智能表面（RIS）旨在弥补传统无源RIS辅助系统中严重的乘性衰落效应。主动RIS的每个反射单元由放大器辅助，使入射信号不仅能像无源RIS系统那样被反射，还能被放大。本文针对以下实际挑战展开研究：一方面，在主动RIS辅助系统中，由于主动RIS缺乏信号处理能力，无法获取RIS辅助信道的完美个体信道状态信息（CSI）；另一方面，收发机设计需要该CSI来计算预期系统数据速率和RIS发射功率。为解决这一问题，我们首先基于RIS辅助信道的部分CSI，推导了平均可达速率和平均RIS发射功率的闭式表达式。随后，我们构建了一个平均可达速率最大化问题，以联合优化基站（BS）和RIS处的主动波束成形。该问题采用最大最小化（MM）算法框架求解，每轮迭代中基于卡罗需-库恩-塔克（KKT）条件导出BS和RIS波束成形的半闭式解。为保障各用户的服务质量（QoS），我们进一步构建了速率中断约束下的波束成形问题，并利用伯恩斯坦型不等式（BTI）和半定松弛（SDR）技术求解。数值结果表明，所提算法能有效克服主动RIS辅助无线系统中非完美CSI带来的挑战。