Sensing Mutual Information with Random Signals in Gaussian Channels: Bridging Sensing and Communication Metrics

Sensing performance is typically evaluated by classical radar metrics, such as Cramer-Rao bound and signal-to-clutter-plus-noise ratio. The recent development of the integrated sensing and communication (ISAC) framework motivated the efforts to unify the performance metric for sensing and communication, where mutual information (MI) was proposed as a sensing performance metric with deterministic signals. However, the need of communication in ISAC systems necessitates the transmission of random signals for sensing applications, whereas an explicit evaluation for the sensing mutual information (SMI) with random signals is not yet available in the literature. This paper aims to fill the research gap and investigate the unification of sensing and communication performance metrics. For that purpose, we first derive the explicit expression for the SMI with random signals utilizing random matrix theory. On top of that, we further build up the connections between SMI and traditional sensing metrics, such as ergodic minimum mean square error (EMMSE), ergodic linear minimum mean square error (ELMMSE), and ergodic Bayesian Cram\'{e}r-Rao bound (EBCRB). Such connections open up the opportunity to unify sensing and communication performance metrics, which facilitates the analysis and design for ISAC systems. Finally, SMI is utilized to optimize the precoder for both sensing-only and ISAC applications. Simulation results validate the accuracy of the theoretical results and the effectiveness of the proposed precoding designs.