Wa. Gardner et Cm. Spooner, THE CUMULANT THEORY OF CYCLOSTATIONARY TIME-SERIES .1. FOUNDATION, IEEE transactions on signal processing, 42(12), 1994, pp. 3387-3408
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.