Fundamental MMSE-Rate Performance Limits of Integrated Sensing and Communication Systems

Integrated sensing and communication (ISAC) demonstrates promise for 6G networks; yet its performance limits, which require addressing functional Pareto stochastic optimizations, remain underexplored. Existing works either overlook the randomness of ISAC signals or approximate ISAC limits from sensing and communication (SAC) optimum-achieving strategies, leading to loose bounds. In this paper, ISAC limits are investigated by considering a random ISAC signal designated to simultaneously estimate the sensing channel and convey information over the communication channel, adopting the modified minimum-mean-square-error (MMSE), a metric defined in accordance with the randomness of ISAC signals, and the Shannon rate as respective SAC metrics. First, conditions for optimal channel input and output distributions on the MMSE-Rate limit are derived employing variational approaches, leading to high-dimensional convolutional equations. Second, leveraging variational conditions, a Blahut-Arimoto-type algorithm is proposed to numerically determine optimal distributions and SAC performance, with its convergence to the limit proven. Third, closed-form SAC-optimal waveforms are derived, characterized by power allocation according to channel statistics/realization and waveform selection; existing methods to establish looser ISAC bounds are rectified. Finally, a compound signaling strategy is introduced for coincided SAC channels, which employs sequential SAC-optimal waveforms for channel estimation and data transmission, showcasing significant rate improvements over non-coherent "capacity". This study systematically investigates ISAC performance limits from joint estimation- and information-theoretic perspectives, highlighting key SAC tradeoffs and potential ISAC design benefits. The methodology readily extends to various metrics, such as estimation rate and the Cramer-Rao Bound.