Deterministic-Random Tradeoff of Integrated Sensing and Communications in Gaussian Channels: A Rate-Distortion Perspective

Integrated sensing and communications (ISAC) is recognized as a key enabling technology for future wireless networks. To shed light on the fundamental performance limits of ISAC systems, this paper studies the deterministic-random tradeoff between sensing and communications (S&C) from a rate-distortion perspective under vector Gaussian channels. We model the ISAC signal as a random matrix that carries information, whose realization is perfectly known to the sensing receiver, but is unknown to the communication receiver. We characterize the sensing mutual information conditioned on the random ISAC signal, and show that it provides a universal lower bound for distortion metrics of sensing. Furthermore, we prove that the distortion lower bound is minimized if the sample covariance matrix of the ISAC signal is deterministic. We then offer our understanding of the main results by interpreting wireless sensing as non-cooperative source-channel coding, and reveal the deterministic-random tradeoff of S&C for ISAC systems. Finally, we provide sufficient conditions for the achievability of the distortion bound by analyzing a specific example of target response matrix estimation.