From Cramér–Rao To Barankin: Fundamental Trade-Off in OFDM Integrated Sensing and Communication Systems

Integrated sensing and communication (ISAC) has emerged as a key technology for future communication systems. In this paper, we provide a general framework to reveal the fundamental trade-off between sensing and communication in OFDM systems, where a unified ISAC signal is exploited to perform both tasks. To evaluate the sensing performance, we introduce two representative performance metrics: The Cramér-Rao Bound (CRB) and the Barakin Bound (BRB). For the asymptotic case when the number of subcarriers is large, we show that the asymptotically optimal input distribution that achieves the Pareto boundary point of the Capacity-CRB\BRB region is Gaussian and the entire Pareto boundary can be obtained by solving a power allocation problem. We prove that the power allocation problem for obtaining the Capacity-CRB region can be calculated by solving a convex optimization problem. However, the Capacity-BRB region is more difficult to be characterized due to the non-convexity of the optimization problem. Therefore, we propose an iterative algorithm to obtain an inner bound of the Capacity-BRB region. Moreover, we derive the sufficient conditions under which the Gaussian distribution remains asymptotically optimal when the number of subcarriers approaches infinity and the delay gap between two targets approaches zero simultaneously. For the non-asymptotic case, an optimization problem is formulated for sensing-optimal input distribution and the capacity-achieving distribution is derived under the sensing-optimal constraints. Finally, numerical simulations are conducted to verify the theoretical analysis and provide useful insights.