The cumulant theory of cyclostationary time-series. I. Foundation

The problem of characterizing the sine-wave components in the output of a polynomial nonlinear system with a cyclostationary random time-series input is investigated. The concept of a pure nth-order sine wave is introduced, and it is shown that pure nth-order sine-wave strengths in the output time-series are given by scaled Fourier coefficients of the polyperiodic temporal cumulant of the input time-series. The higher order moments and cumulants of narrowband spectral components of time-series are defined and then idealized to the case of infinitesimal bandwidth. Such spectral moments and cumulants are shown to be characterized by the Fourier transforms of the temporal moments and cumulants of the time-series. It is established that the temporal and spectral cumulants have certain mathematical and practical advantages over their moment counterparts. To put the contributions of the paper in perspective, a uniquely comprehensive historical survey that traces the development of the ideas of temporal and spectral cumulants from their inception is provided