  Sensing performance is typically evaluated by classical 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 metric for sensing and communication, where researchers
have proposed to utilize mutual information (MI) to measure the sensing
performance with deterministic signals. However, the need to communicate in
ISAC systems necessitates the use of random signals for sensing applications
and the closed-form evaluation for the sensing mutual information (SMI) with
random signals is not yet available in the literature. This paper investigates
the achievable performance and precoder design for sensing applications with
random signals. For that purpose, we first derive the closed-form expression
for the SMI with random signals by utilizing random matrix theory. The result
reveals some interesting physical insights regarding the relation between the
SMI with deterministic and random signals. The derived SMI is then utilized to
optimize the precoder by leveraging a manifold-based optimization approach. The
effectiveness of the proposed methods is validated by simulation results.
