谱相关的数学关系与物理意义

Mathematical Relationships and Physical Functions of Spectral Correlation

周期性是自然现象和工程现象的基本特性之一,从复杂多变的、含有噪声的观测数据中发现周期性并测量其参数是信号处理的重要主题.谱相关是分析循环平稳信号二阶周期性的有力工具,本文研究谱相关的周期性解析功能的缘由、过程与特点.首先,梳理了谱相关理论体系中诸多重要概念之间的内在联系,建立了相互之间转换或映射的数学关系图;然后提出了按定义和转换关系逐步求解谱相关的方法.通过单周期信号的分析和每步操作效果与周期的对应关系,诠释了循环自相关、谱相关、无限循环谱和循环频率等重要概念的物理意义.

The periodicity is a feature of science and engineering phenomena, so detecting periodicity from complex and noise-included samples and measuring its parameters are important topics in signal processing. Spectral correlation is a useful tool to find second-order periodicity of cyclostationary signals. This paper investigates the reasons, procedures and features using spectral correlation to analyze cyclicity.First, internal relations between main defmitions in spectral correlation theory are summarized and mathematic diagrams reflecting their transforming or mapping relations are built. Then a method to calculate spectral correlation based on the diagrams is proposed. By analysis of single periodic signals and connections between periods and results in each operation, physical functions of cyclic autocorrelalion, spectral correlation, limit periodic spectrum and cyclic fi＇equency, etc, are interpreted.