  Radar systems typically employ well-designed deterministic signals for target
sensing, while integrated sensing and communications (ISAC) systems have to
adopt random signals to convey useful information. This paper analyzes the
sensing and ISAC performance relying on random signaling in a multi-antenna
system. Towards this end, we define a new sensing performance metric, namely,
ergodic linear minimum mean square error (ELMMSE), which characterizes the
estimation error averaged over random ISAC signals. Then, we investigate a
data-dependent precoding (DDP) scheme to minimize the ELMMSE in sensing-only
scenarios, which attains the optimized performance at the cost of high
implementation overhead. To reduce the cost, we present an alternative
data-independent precoding (DIP) scheme by stochastic gradient projection
(SGP). Moreover, we shed light on the optimal structures of both sensing-only
DDP and DIP precoders. As a further step, we extend the proposed DDP and DIP
approaches to ISAC scenarios, which are solved via a tailored penalty-based
alternating optimization algorithm. Our numerical results demonstrate that the
proposed DDP and DIP methods achieve substantial performance gains over
conventional ISAC signaling schemes that treat the signal sample covariance
matrix as deterministic, which proves that random ISAC signals deserve
dedicated precoding designs.
