11.2 HIGHER ORDER RESPONSE FUNCTION, 
hy(%j,T2,° * - Tn); An(W1,@2,- > -Wn) 
When h,(T;,72,° * :T,) are smooth and absolutely integratable, the nth order Fourier 
pair exists as was stated at the beginning, 
1 
hy(T1,T2,° ° Tr) = ope |. ao | expifon+0: T2 +++ ++Op Tn| H,(@1,@2,° - Mn) 
—-@ —_o 
dw, aW2---dW,, (11.4) 
H,(@1,@2,° + -@n) = | 26 | expi-ivo, TT +2 T+: - -+@r,Tn)] 
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h,(T1,T2,° - -T,) are real functions and are symmetrical fort,,T2-- -T,, 
thus 
h,(T1,T2,° > - Tr) = hn(02,T1,° - Tr) =- - + =Mn(t2,73,- > - t,T1) (11.6) 
H,(@1,@2,° > - Wn) = H,(@2,@1,° > -@p)=- > + = Ay(W2,03,- - - Wp, @}). (11.7) 
If we regard Y(t) as the output and X(t) as the input process, h,(T),T2,- - - Tn) 18 
called the nth order impulse response function and H,(@,@2,- - :@,) is the nth order 
frequency response function. 
When the processes are not continuous but discrete, then the Voltera—Wiener 
expression for the nonlinear response Y, and linear input X, is 
Y, = yy 8u Xpu + y y 8nv KGa, Xtv 
uUu=o u=ov=o0 
Hoi Mita Xen, Nebr Xa ce in (11.8) 
u=Ov=0W=0 
= yi >: ; use tu, Xt Xe? * + Xtuys (11.8’) 
od tl 
n 
and the following discrete Fourier pair is assumed to exist: 
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