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.

翻译：雷达系统通常采用精心设计的确定性信号进行目标感知，而集成感知与通信（ISAC）系统必须采用随机信号来传输有用信息。本文分析了多天线系统中依赖随机信号的感知与ISAC性能。为此，我们定义了一种新的感知性能指标，即遍历线性最小均方误差（ELMMSE），该指标刻画了随机ISAC信号的平均估计误差。随后，我们研究了一种数据相关预编码（DDP）方案以在仅感知场景中最小化ELMMSE，该方案以高实现开销为代价达到最优性能。为降低成本，我们通过随机梯度投影（SGP）提出了一种替代的数据无关预编码（DIP）方案。此外，我们揭示了仅感知DDP与DIP预编码器的最优结构。进一步地，我们将所提出的DDP与DIP方法扩展到ISAC场景，并通过一种定制的基于惩罚的交替优化算法求解。数值结果表明，相比将信号样本协方差矩阵视为确定性的传统ISAC信令方案，所提出的DDP与DIP方法实现了显著性能增益，这证明了随机ISAC信号值得专用预编码设计。