Fundamental CRB-Rate Tradeoff in ISAC: the Pareto Boundary with Arbitrary Input Distribution

Integrated sensing and communication (ISAC) is a promising technique for future wireless applications that demand both high-quality communication and precise sensing. Characterizing the fundamental tradeoff between sensing and communication (S&C) is essential for guiding cost-effective ISAC designs. However, the exact performance boundary characterizing the tradeoff between the communication rate and the sensing estimation accuracy remains unknown. To tackle this issue, this paper studies the exact Pareto boundary of the Cramér-Rao bound (CRB) and communication rate performance region, as well as the optimal input signal distributions that achieve this boundary. First, the explicit analytical expression of the expected CRB is derived with arbitrary random inputs, which quantify the accuracy of sensing estimation. It is proved that the bound-achieving distribution, which maximizes communication rate while ensuring CRB and power constants, is both timeindependent and circular symmetric. Leveraging these properties, a modified Blahut-Arimoto algorithm is developed based on functional analysis to numerically determine the optimal distribution on the CRB-rate boundary and correspondingly characterize the exact CRB-rate region. Numerical results demonstrate that ISAC systems enable a flexible CRB-rate tradeoff by adjusting the input signal distribution, outperforming separated S&C design benchmark. It is revealed that the communication benefits from greater signal amplitude randomness, while sensing accuracy improves with higher instantaneous power.