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Integrated sensing and communications (ISAC) is recognized as a key enablingtechnology for future wireless networks. To shed light on the fundamentalperformance limits of ISAC systems, this paper studies the deterministic-randomtradeoff between sensing and communications (S&C) from a rate-distortionperspective under vector Gaussian channels. We model the ISAC signal as arandom matrix that carries information, whose realization is perfectly known tothe sensing receiver, but is unknown to the communication receiver. Wecharacterize the sensing mutual information conditioned on the random ISACsignal, and show that it provides a universal lower bound for distortionmetrics of sensing. Furthermore, we prove that the distortion lower bound isminimized if the sample covariance matrix of the ISAC signal is deterministic.We then offer our understanding of the main results by interpreting wirelesssensing as non-cooperative source-channel coding, and reveal thedeterministic-random tradeoff of S&C for ISAC systems. Finally, we providesufficient conditions for the achievability of the distortion bound byanalyzing a specific example of target response matrix estimation.

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