Graduate Thesis Or Dissertation
 

Fourier Analysis of Temporal Logic

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https://ir.library.oregonstate.edu/concern/graduate_thesis_or_dissertations/5q47rw579

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  • Signal Temporal logic(STL) provides a convenient framework to describe various system properties over time. While there has been extensive research on verification and control synthesis from STL requirements, a lot of them revolve around the time-domain analysis. So, there are still some questions properly unaddressed like what is the frequency content of temporal logic formulas? To expand on this, when we monitor a signal against a formula, which frequency bands of the signal are relevant to the logic and should be preserved, and which can be safely discarded? In any Cyber-physical systems, the signals being mostly analog are filtered or compressed in some form before being monitored, so these questions become very relevant. To answer these questions, we focus on monitors that measure the robustness of a signal relative to a specification in STL. Since conventional Fourier transform only holds for linear-time invariant systems, we prove that robustness monitors, which are mostly nonlinear can be modeled using Volterra series. We then study the Fourier transforms of these Volterra representations, and provide a series composition method to derive the Fourier transforms of entire formulas. We also redefine the measurement process in temporal logic on the basis of distributions to make it compatible with the practical way of doing measurements in signal processing. We provide some experiments that illustrate these results. This work enables a common formalism to study both logical and non-logical operations in the frequency domain and also integrates temporal logic monitoring into common signal processing toolchains as just another signal processing operation.
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