ISAC 
Ox 
Li) (12.131’) 
The linear, quadratic, and cubic frequency response functions connecting the white 
noise W(t) with the output Y(t) will be 
0,H\(w) = 0,L(w)G(w) 
07H2(w1,@2) = 02L(w1)L(w2)Gx(@1, @2) (12.132) 
03H3(@1, 0203) = 02L(@1)L(w2)L(W3)G3(@1, @2, W3). 
Then Y(t) can be related to the white noise input W(t) as 
Y(t) = Ox | hy(t)W(t—7))dry 
+02 | | hy(T1, T2)W(t— 71) W(t — T2)dt ar 
+o; | | | h3(T1,T2, 73) W(t — 71) W(t — T2)W(t —13)dt\dt2dt3, (12.133) 
where 
h(t) = = | H\(w) expliwtjdw 
1 
h2(T1,T2) = so | | H12(@ 1,2) exp[i(@ 17 + W2T2) dw dw (12.134) 
1 
h3(T1, 72,73) = Om: | | | #01,02,03 exp[i(@ 17] + W2T2 + W373) ]dw\dw2dws. 
The derivatives of the output are then 
Y(t) = 0, | hy(t)W(t —1))dr1 
+05 | | hy(t1, T2)W(t— 7) W(t — 12)dr dt 
+o; | | | h3(T,72,73)W(t—1;)W(t —T2)W(t—13)dt\dt2dt3, (12.135) 
where 
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