Cramer-Rao Bounds for Activity Detection in Conventional and Fluid Antenna Systems

In this letter, we develop a unified Cramér-Rao bound (CRB) framework to characterize the fundamental performance limits of transmission activity detection in fluid antenna systems (FASs) and conventional multiple fixed-position antenna (FPA) systems. To facilitate CRB analysis applicable to activity indicators, we relax the binary activity states to continuous parameters, thereby aligning the bound-based evaluation with practical threshold-based detection decisions. Closed-form CRB expressions are derived for two representative detection formulations, namely covariance-oriented and coherent models. Moreover, for single-antenna FASs, we obtain a closed-form coherent CRB by leveraging random matrix theory. The results demonstrate that CRB-based analysis provides a tractable and informative benchmark for evaluating activity detection across architectures and detection schemes, and further reveal that FASs can deliver strong spatial-diversity gains with significantly reduced complexity.