This work presents the first comprehensive framework for monitoring predicates in distributed cyber-physical systems under partial synchronization. The research addresses critical challenges in distributed runtime verification, including clock drift, network asynchrony, and the need for real-time monitoring of safety-critical properties across multiple agents. Our approach enables monitoring of global properties in systems where individual components only have local views, making it particularly valuable for autonomous vehicles, distributed control systems, and IoT networks. The framework provides theoretical guarantees while maintaining practical efficiency, representing a significant advancement in distributed systems monitoring.

This research introduces a novel mathematical framework that bridges signal processing and temporal logic through Fourier analysis. By applying frequency domain analysis to temporal logic formulas, we provide new insights into the robustness and satisfaction of temporal properties in continuous-time systems. The work establishes theoretical foundations for understanding how temporal logic properties behave in the frequency domain, enabling new approaches to robust monitoring and verification. This interdisciplinary approach has significant implications for cyber-physical systems, control theory, and signal processing applications where temporal logic specifications need to be satisfied in the presence of noise and uncertainty.