论文标题
逻辑信号处理：时间逻辑的傅立叶分析
Logical Signal Processing: a Fourier Analysis of Temporal Logic
论文作者
论文摘要
时间逻辑公式的频率内容是多少？也就是说，当我们针对公式监视信号时，信号的哪些频带与逻辑相关并应保留，并且可以安全丢弃哪些？每当对信号进行过滤或被压缩之前，在监视之前，这个问题都是相关的，对于模拟信号来说，这几乎总是这种情况。为了回答这个问题，我们专注于测量信号相对于信号时间逻辑规范的鲁棒性的监视器。我们证明可以使用Volterra系列对鲁棒性监测器进行建模。然后，我们研究这些伏特拉表示的傅立叶变换，并提供了一种推导整个公式的傅立叶变换的方法。我们还将在时间逻辑中明确规定测量过程，并根据分布重新定义它，以使其与信号处理中的测量值兼容。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视整合到通用信号处理工具链中，只是另一个信号处理操作，并使常见的形式主义能够研究频域中的逻辑和非逻辑操作，我们将其称为逻辑信号处理。
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.
