逻辑信号处理：时序逻辑的傅立叶分析
Logical Signal Processing: a Fourier Analysis of Temporal Logic
摘要：
时态逻辑公式的频率内容是什么？也就是说，当我们根据公式监测信号时，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？这个问题与信号在被监测之前被滤波或压缩时有关，这几乎总是模拟信号的情况。为了回答这个问题，我们专注于测量信号相对于信号时序逻辑规范的鲁棒性的监视器。我们证明了鲁棒性监视器可以使用Volterra系列建模。然后研究了这些Volterra表示的傅里叶变换，并给出了一种推导整个公式的傅里叶变换的方法.我们还明确了时间逻辑的测量过程，并重新定义它的基础上的分布，使其与测量兼容的信号处理。实验证明了这些结果。除了压缩之外，这项工作还可以将时序逻辑监控集成到常见的信号处理工具链中，作为另一种信号处理操作，并实现一种常见的形式主义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处理。
英文摘要：
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.
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