The spectral correlation theory of cyclostationary time-series

Abstract

A spectral correlation theory for cyclostationary time-series is introduced. It is established that a time-series is cyclostationary if and only if there exists a quadratic time-invariant transformation that generates spectral lines, and this is so if and only if the time-series exhibits spectral correlation. Fundamental properties of a characterizing spectral correlation function are developed. These include the effects of periodic modulation and periodically time-variant linear filtering. Relationships between the spectral correlation function and the radar ambiguity function and the Wigner-Ville distribution are explained. The spectral correlation properties of Rice's representation for bandpass time-series are derived. A generalization of the Wiener relation from the spectral density function to the spectral correlation function is developed, and generalizations of the aliasing formula for periodic time-sampling, and the frequency conversion formula for amplitude modulation, from the spectral density function to the spectral correlation function are developed.