THE CUMULANT THEORY OF CYCLOSTATIONARY TIME-SERIES .1. FOUNDATION

The problem of characterizing the sine-wave components in the output o f a polynomial nonlinear system with a cyclostationary random time-ser ies input is investigated. The concept of a pure nth-order sine wave i s 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 highe r order moments and cumulants of narrowband spectral components of tim e-series are defined and then idealized to the case of infinitesimal b andwidth. Such spectral moments and cumulants are shown to be characte rized by the Fourier transforms of the temporal moments and cumulants of the time-series. It is established that the temporal and spectral c umulants have certain mathematical and practical advantages over their moment counterparts. To put the contributions of the paper in perspec tive, a uniquely comprehensive historical survey that traces the devel opment of the ideas of temporal and spectral cumulants from their ince ption is provided.