Analysis and Optimization for Passive Detection in Multi-static Integrated Sensing and Communication

This paper investigates the passive detection problem in multi-static integrated sensing and communication (ISAC) systems where multiple spatially distributed receivers jointly detect a target without prior knowledge of the transmit signals from a base station (BS). While similar to passive radar, this problem differs in that the BS is cooperative and can optimize its transmit beamforming to balance the performance of detection and communication. Utilizing the generalized likelihood ratio test (GLRT) detector at sensing receivers, we first provide an asymptotic analysis of the GLRT statistic, and demonstrate how the signal-to-noise ratios (SNRs) of the direct and target paths affect the asymptotic detection probability. Then, we formulate a transmit beamforming optimization problem to maximize the asymptotic detection probability while satisfying the communication SNR under the transmit power constraint. Given the non-convex nature of the problem, we propose an optimization algorithm based on successive convex approximation and semi-definite relaxation methods. Numerical results verify the effectiveness of the asymptotic analysis and demonstrate the superiority of the proposed beamforming design in balancing the detection performance and the communication quality.