CHAPTER 11 
VOLTERA EXPANSION AND APPLICATION OF POLYSPECTRA 
11.1 VOLTERA—WIENER EXPRESSION 
Suppose there is a weakly nonlinear response process Y(t) which is an output of a 
linear input process X(t). Both Y(t) and X(r) are stationary up to an appropriate order. 
Then when the sum of the absolute values of all kernel functions is not infinite, Y(t) can 
be expressed as 
foe) 
Y(t) = | 1 X(t—T) adt+ | 
—o 
h2(t1,T2) X(t—T) X(t—T) atyat2 
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8 
loo} 
-| | | hace, 12.23) X(t—T,) X(t—T2) X(t—73) dt\dt2 dt3- - - (11.1) 
-o 
n=0 
Dy | |. ae | ttes.t- > + Ty) X(f—-T1) X(t—T2) > - -X(C—-T) 
dT\dt2- - - dtp. (11.1’) 
Here h,(tT2- - -T,) is areal function of n variables —~© <T;< +0, j =1,2,---n 
and is called nth kernel function. All kernels are assumed to be smooth, absolutely inte- 
gratable, and to possess Fourier transforms. Besides, all these are supposed to be time 
invariant. When, for any 7; < 0,/,(T),T2,: - -T,) = 0, then the lower limit of these inte- 
grals can be 0. For n = 0, hy is the value of Y(t) when X(t) = 0. So we can include n = 0, 
generally; otherwise we assume h, = 0. Whenn = 1, Y(t) will be the linear system that 
has been treated in Part I and I and as Eq. 9.16 
Y(t) = | h(t) X(t—1) at. (11.2) 
Equation 11.1 is assumed to be a kind of Taylor expansion of Y(t) around the linear 
process expressed by Eq. 11.2. Accordingly, the terms form = 2 are regarded as modify- 
ing small terms. 
If we change the variables, 
Os yY | | Oi | tate—nit—ta, » + £-T,) X(T) X(T2)- > -XCn) 
at\dt.-- - aT, . (11.3) 
Equation 11.1 or 11.3 is called the Voltera—Wiener or functional expression. 
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