Optimal Beamforming for Multi-Target Multi-User ISAC Exploiting Prior Information: How Many Sensing Beams Are Needed?

This paper studies a multi-target multi-user integrated sensing and communication (ISAC) system where a multi-antenna base station (BS) communicates with multiple single-antenna users in the downlink and senses the unknown and random angle information of multiple targets based on their reflected echo signals at the BS receiver as well as their prior probability information. We focus on a general beamforming structure with both communication beams and dedicated sensing beams, whose design is highly non-trivial as more sensing beams provide more flexibility in sensing, but introduce extra interference to communication. To resolve this trade-off, we first characterize the periodic posterior Cram\'er-Rao bound (PCRB) as a lower bound of the mean-cyclic error (MCE) in multi-target sensing. Then, we optimize the beamforming to minimize the maximum periodic PCRB among all targets to ensure fairness, subject to individual communication rate constraints at multiple users. Despite the non-convexity of this problem, we propose a general construction method for the optimal solution by leveraging semi-definite relaxation (SDR), and derive a general bound on the number of sensing beams needed. Moreover, we unveil specific structures of the optimal solution in various cases, where tighter bounds on the number of sensing beams needed are derived (e.g., no or at most one sensing beam is needed under stringent rate constraints or with homogeneous targets). Next, we study the beamforming optimization to minimize the sum periodic PCRB under user rate constraints. By applying SDR, we propose a general construction method for the optimal solution and its specific structures which yield lower computational complexities. We derive a general bound and various tighter bounds on the number of sensing beams needed. Numerical results validate our analysis and effectiveness of our proposed beamforming designs.