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odically time-varying systems differs fundamentally from that of linear
time-invariant systems.
To analyze nonlinear systems, we have shown that power series ap-
proaches are valid for characterizing nonlinear systems without memory
or general nonlinear systems at low frequencies at which the contribu-
tion of the past states of elements with memory is negligible. When
the effect of the past states of elements with memory can not be ne-
glected, Volterra functional series must be employed. We have intro-
duced multi-frequency time-varying network function
to characterize time-varying nonlinear systems in the time domain, and
multi-frequency network functions to characterize the
behavior of these systems in the frequency domain. The use of these
network functions allows us to derive the spectrum of nonlinear period-
ically time-varying systems to inputs of both single-tone and multiple
tones.
APPENDIX 3.A: Periodicity of
The Network Functions of
Nonlinear Periodically Time-Varying Systems
In this appendix, we show that the time-varying network functions of nonlinear
periodically time-varying systems are periodic in time with the same period.
Consider a linear periodically time-varying system characterized by
where is the input and is the response. The periodicity of the system is
characterized by where is the period of time variation. Note that
although we have chosen scalar form for the purpose of simplicity the results can be
readily extended to vector form. The solution of (3.A.1) is given by [46]
where
Without losing generality, let the input of the system be
. Representing
in Fourier series
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