This paper studies a challenging scenario in a multiple-input multiple-output (MIMO) integrated sensing and communication (ISAC) system where the locations of the sensing target and the communication user are both unknown and random, while only their probability distribution information is known. In this case, how to fully utilize the spatial resources by designing the transmit beamforming such that both sensing and communication can achieve satisfactory performance statistically is a difficult problem, which motivates the study in this paper. Moreover, we aim to reveal if it is desirable to have similar probability distributions for the target and user locations in terms of the ISAC performance. Firstly, based on only probability distribution information, we establish communication and sensing performance metrics via deriving the expected rate or posterior Cramér-Rao bound (PCRB). Then, we formulate the transmit beamforming optimization problem to minimize the PCRB subject to the expected rate constraint, for which the optimal solution is derived. It is unveiled that the rank of the optimal transmit covariance matrix is upper bounded by the summation of MIMO communication channel matrices for all possible user locations. Furthermore, due to the need to cater to multiple target/user locations, we investigate whether dynamically employing different beamforming designs over different time slots improves the performance. It is proven that using a static beamforming strategy is sufficient for achieving the optimal performance. Numerical results validate our analysis, show that ISAC performance improves as the target/user location distributions become similar, and provide useful insights on the BS-user/-target association strategy.

翻译：本文研究多输入多输出集成感知与通信系统中一个具有挑战性的场景：感知目标与通信用户的位置均未知且随机，仅知其概率分布信息。在此情况下，如何通过设计发射波束成形以充分利用空间资源，使得感知与通信在统计意义上均能获得满意性能，是一个难题，这也构成了本文的研究动机。此外，我们旨在从ISAC性能角度揭示目标与用户位置具有相似概率分布是否具有优势。首先，基于概率分布信息，我们通过推导期望速率或后验克拉美-罗下界建立了通信与感知性能度量。随后，我们构建了在期望速率约束下最小化PCRB的发射波束成形优化问题，并推导出其最优解。研究揭示，最优发射协方差矩阵的秩受限于所有可能用户位置对应的MIMO通信信道矩阵之和。进一步地，由于需兼顾多个目标/用户位置，我们探究了在不同时隙动态采用不同波束成形设计是否能提升性能。理论证明采用静态波束成形策略即可实现最优性能。数值结果验证了我们的理论分析，表明当目标/用户位置分布趋于相似时ISAC性能得到提升，并为基站-用户/目标关联策略提供了有益启示。