Closed-Form Approximations of Range Mutual Information for Integrated Sensing and Communication Systems

Sensing mutual information (SMI) is widely adopted as a performance metric for integrated sensing and communication (ISAC) to enhance both sensing and communication capabilities. However, conventional approaches derive SMI from amplitude and phase, whereas an explicit evaluation of range mutual information (RMI) remains absent. In this paper, we investigate a novel closed-form approximation of RMI for ISAC. We first derive an explicit expression for the posterior probability density function (PDF) of the target range, which is formulated as a function of the signal's autocorrelation and cross-correlation. Furthermore, we show that under high signal-to-noise ratio (SNR), the estimated range PDF approximates a Gaussian distribution in the sensing-unconstrained scenario and a truncated Gaussian distribution in the sensing-constrained scenario. Finally, we derive closed-form approximations of the RMI in both scenarios under high SNR. In the sensing-unconstrained scenario, the RMI is proportional to the delay interval, root-mean-square bandwidth, and SNR. In the constrained scenario, we obtain a closed-form RMI approximation by introducing an entropy correction term that quantifies the impact of boundary constraints. Additionally, we employ a maximum likelihood estimation (MLE) method to assess range estimation performance. Simulation results validate the accuracy of the theoretical results and the effectiveness of the proposed approximations.