This paper considers an active reconfigurable intelligent surface (RIS)-aided integrated sensing and communication (ISAC) system. We aim to maximize radar signal-to-interference-plus-noise-ratio (SINR) by jointly optimizing the beamforming matrix at the dual-function radar-communication (DFRC) base station (BS) and the reflecting coefficients at the active RIS subject to the quality of service (QoS) constraints of communication users (UE) and the transmit power constraints of active RIS and DFRC BS. To tackle the optimization problem, the majorization-minimization (MM) algorithm is applied to address the nonconvex radar SINR objective function, and the resulting quartic problem is solved by developing an semidefinite relaxation (SDR)-based approach. Moreover, we derive the scaling order of the radar SINR with a large number of reflecting elements. Next, the transmit power allocation problem and the deployment strategy of the active RIS are studied with a moderate number of reflecting elements. Finally, we validate the potential of the active RIS in ISAC systems compared to passive RIS. Additionally, we deliberate on several open problems that remain for future research.

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