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.

翻译：本文构建了一个统一的克拉美-罗界（CRB）框架，用以刻画流体天线系统（FAS）与传统多固定位置天线（FPA）系统中传输活动检测的基本性能极限。为使CRB分析适用于活动指示器，我们将二元活动状态松弛为连续参数，从而使基于边界的评估与实际的基于阈值的检测决策相一致。针对两种代表性的检测模型——协方差导向模型与相干模型，推导了闭式CRB表达式。此外，针对单天线FAS，我们利用随机矩阵理论获得了闭式相干CRB。结果表明，基于CRB的分析为评估不同架构与检测方案下的活动检测性能提供了一个易于处理且信息丰富的基准，并进一步揭示了FAS能够以显著降低的复杂度实现强大的空间分集增益。