Optimal Beamforming for Integrated Sensing and Communication Exploiting Prior Distribution Information: How Many 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 from reflected echo signals and prior distribution information. We focus on a general transmit beamforming structure with both communication beams and dedicated sensing beams, whose design is highly non-trivial as more sensing beams provide more flexibility in multi-target sensing, but introduce extra interference to multi-user communication. We aim to unveil bounds on the minimum number of dedicated sensing beams needed, and devise numerical algorithms that find optimal beamforming solutions with small numbers of sensing beams which yield low algorithmic and implementational complexity. To this end, we first characterize the sensing performance via deriving the periodic posterior Cram\'er-Rao bound (PCRB) as a lower bound of the mean-cyclic error (MCE). Then, we optimize the beamforming to minimize the maximum periodic PCRB among all targets to ensure fairness, subject to individual rate constraints at multiple users. Despite the non-convexity of this problem, we propose a general construction method for the optimal solution via semi-definite relaxation (SDR), and derive a general bound on the number of dedicated sensing beams needed. Moreover, we unveil the specific structures of the optimal solution in various practical cases, where tighter bounds on the number of sensing beams needed are derived together with lower-complexity algorithms that find the optimal solution with small numbers of sensing beams. Next, we extend the above studies to the beamforming optimization problem for minimizing the sum periodic PCRB among all targets. Numerical results validate our bounds and the effectiveness of our beamforming designs.