What is the frequency content of temporal logic formulas? That is, 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? This question is relevant whenever signals are filtered or compressed before being monitored, which is almost always the case for analog signals. To answer this question, we focus on monitors that measure the robustness of a signal relative to a specification in Signal Temporal Logic. We prove that robustness monitors can be modeled using Volterra series. We then study the Fourier transforms of these Volterra representations, and provide a method to derive the Fourier transforms of entire formulas. We also make explicit the measurement process in temporal logic and re-define it on the basis of distributions to make it compatible with measurements in signal processing. Experiments illustrate these results. Beyond compression, this work enables the integration of temporal logic monitoring into common signal processing toolchains as just another signal processing operation, and enables a common formalism to study both logical and non-logical operations in the frequency domain, which we refer to as Logical Signal Processing.

中文翻译：

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逻辑信号处理：时间逻辑的傅立叶分析

时序逻辑公式的频率内容是什么？也就是说，当我们根据公式监控信号时，信号的哪些频段与逻辑相关应该保留，哪些可以安全地丢弃？每当信号在被监控之前被过滤或压缩时，这个问题就很重要，模拟信号几乎总是如此。为了回答这个问题，我们重点关注测量信号相对于信号时间逻辑中的规范的稳健性的监视器。我们证明可以使用 Volterra 系列对稳健性监视器进行建模。然后我们研究这些 Volterra 表示的傅里叶变换，并提供一种方法来推导出整个公式的傅里叶变换。我们还明确了时间逻辑中的测量过程，并根据分布对其进行了重新定义，使其与信号处理中的测量兼容。实验说明了这些结果。除了压缩之外，这项工作还可以将时间逻辑监控集成到通用信号处理工具链中，作为另一种信号处理操作，并支持通用形式来研究频域中的逻辑和非逻辑操作，我们将其称为逻辑信号加工。