Cramér-Rao Bound Analysis and Beamforming Design for 3D Extended Target in ISAC: From Optimization to Learning Approaches

This paper considers an integrated sensing and communication (ISAC) system where a multi-antenna base station transmits a common signal for joint multi-user communication and extended target (ET) sensing. We first propose a second-order truncated Fourier series surface model for an arbitrarily-shaped three-dimensional ET. Utilizing this model, we derive novel closed-form Cram{\'e}r-Rao bounds (CRBs) for the ET kinematic parameter estimation, such as the center range, azimuth angle, elevation angle, and orientation. Further, we formulate and solve two transmit beamforming design problems with optimization algorithms. The first one, named the CRB minimization problem, minimizes the CRB under constraints of communication signal-to-interference-plus-noise ratio (SINR) requirement, transmit power, and ET-specific beam coverage requirement, which is solved using the semidefinite relaxation technique. The second one, named the weighted-ISAC-metric problem, targets an weighted objective function combining the communication sum rate and sensing CRB, and is solved using the successive convex approximation technique. Additionally, by exploiting the penalty method, we introduce an unsupervised learning-based approach and propose a unique ISAC graph neural network (ISACGNN), composed of separate communication, sensing, and integration modules, to address both problems. Numerical results reveal the diverse CRB characteristics for different radar targets. The proposed beamforming designs are superior to existing baselines with better trade-off between communication and sensing performance, and a more appropriate beampattern for sensing the 3D ET. Besides, our proposed ISACGNN can effectively mimic the dynamic structures of the SINR, sum rate, and CRB, demonstrating remarkable scalability.