Environmental Engineering Reference
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where and are constants. Using Volterra series, equation (3.A.8) can be repre-
sented by the following set of linear periodically time-varying systems
where
and
are the 1st-, 2nd-, and third-order terms of the Volterra
series expansion of
respectively.
and
If
we have
Since
is periodic in
with period
we have
where is the Fourier series coefficient. The input of the second order Volterra
circuit, denoted by
is given by
Using (3.A.5) we obtain the zero-state response of the second order Volterra circuit
where
is a constant. If the system is asymptotically stable, then
as
Thus, the steady-state response is given by
Clearly,
is periodic in
with period
Comparing (3.A.11) with (3.A.15)
we conclude that
is also periodic in
with period
In a very like
manner, one can show that
is periodic in
with period
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