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Sensing performance is typically evaluated by classical metrics, such asCramer-Rao bound and signal-to-clutter-plus-noise ratio. The recent developmentof the integrated sensing and communication (ISAC) framework motivated theefforts to unify the metric for sensing and communication, where researchershave proposed to utilize mutual information (MI) to measure the sensingperformance with deterministic signals. However, the need to communicate inISAC systems necessitates the use of random signals for sensing applicationsand the closed-form evaluation for the sensing mutual information (SMI) withrandom signals is not yet available in the literature. This paper investigatesthe achievable performance and precoder design for sensing applications withrandom signals. For that purpose, we first derive the closed-form expressionfor the SMI with random signals by utilizing random matrix theory. The resultreveals some interesting physical insights regarding the relation between theSMI with deterministic and random signals. The derived SMI is then utilized tooptimize the precoder by leveraging a manifold-based optimization approach. Theeffectiveness of the proposed methods is validated by simulation results.

Sensing Mutual Information with Random Signals in Gaussian Channels