Probabilistic Shaping-Based ISAC Systems With Finite Constellations: Analysis and Optimization

Integrated sensing and communication (ISAC) has been recognized as a key technology of the next-generation wireless networks. This paper focuses on the ISAC system with discrete signal inputs. Compared with the ISAC systems with continuous inputs in most prior works, it can provide useful insights for ISAC networks with digital modulation at the expense of compensating for the performance gap caused by the finite constellations. To tackle the above problem, a unified analytical framework is proposed to analyze and optimize the ISAC systems with finite constellations in this paper. Based on the analytical framework, both the communication ergodic rate and sensing estimation rate considering non-uniform constellations are derived to characterize the sensing and communication (S&C) performance and quantify the performance gap caused by discrete signals in ISAC networks. To compensate for the performance gap, the probabilistic shaping (PS)-based waveform design is formulated as a multi-objective optimization problem by jointly optimizing the discrete symbol distribution and the power scaling factor for achieving the Pareto-optimal S&C performance. Given the non-convexity of the above problem, an alternating algorithm is proposed to obtain the PS-based waveform. Moreover, the properties of the sensing-optimal and communication-optimal PS-based waveform are analyzed. It is proved that the sensing function prefers amplitude-deterministic signals, and the probabilities of outermost constellations are equal if they are distributed in a circularly symmetric manner. It is indicated that communication functionality requires quasi-Gaussian distribution, which is consistent with the results of our proposed algorithm. Numerical results characterize the tradeoff between S&C and validate the superiority of the proposed PS scheme in comparison with other input distributions.