Fundamental CRB-Rate Trade-Off in ISAC Systems Under Correlated Communication-Sensing Channel

Integrated sensing and communication (ISAC) has been esteemed as a pivotal driver for the next-generation wireless networks in achieving dual-function spectrum efficiency. In typical ISAC systems, user equipment and targets are treated as distinct entities, operating without interaction between the communication and sensing functionalities. However, in scenarios where the user and target align as a unified entity, current solutions underperform due to the coupled performance metrics involving the achievable rate and the sensing Cramér–Rao Bound (CRB). To address this issue, focusing on this particular scenario, this paper delves into the design of optimal transmission precoder and conducts a fundamental trade-off analysis between the achievable rate and the target sensing CRB. Firstly, we derive a compact expression of the CRB, characterized by the angle and delay parameters of the sensing channel, thereby obtaining the subsequent position error bound (PEB) by exploiting the inherent connection between the PEB and CRB. Next, we aim to design the optimal precoder matrix to minimize the PEB, subject to constrains on the minimum communication rate and total transmit power budget. By leveraging the structural properties of the precoder covariance matrix, we develop an efficient algorithm to devise a closed-form optimal precoder design. Numerical results demonstrate that our proposal yields a favorable CRB-Rate trade-off across various scenarios, closely aligning with the performance of semi-definite relaxation (SDR)-based optimization scheme.