in Eq. 11.41 gives 
Syy(W) = lH\(w)I*sxx(w) 
oOo w 
+2 | | H2(@1, 02) sxx(@1) Sxx(@2)6[@1 — @ — @2)}dw dw? 
—-O -—-O 
= H,(w)?sxx(w) + 2 | \H>(@ — ©, @2)°Sxx(@ — W2)sxx(@2)dw2. (11.43) 
This shows that the linear spectrum syy(w) is the sum of the spectrum of the linear 
part IH, (w)P?sxx(@) and the modifying term in Eq. 11.43 defined by the second nonlinear 
response function H2(w —w2, 2) and the product of two linear spectra syx(w —@2) and 
Sxx(@2). 
4. Cross correlation of third moment Ryxx; cross bispectrum, syxx (w), and the 
second order frequency response function Hz (w),@2) . 
The cross correlation of the third moment, for example Ryxx (T),T2), 1S 
Ryxx(T1, T2) 
E{{Y(t)— my} X(t—1) X(t-12)] 
E[Y(t)X(t—1) X(t—12)] — myE[X(t—11) X(t—1)]. (11.44) 
Here expressing Y(t) by its functional expression Eq. 11.11 and using the same relation as 
Eg. 11.39’ and Eq. 11.33 for my gives 
Ryxx(t1,T2) = | | ho(uy, U2) E[X(t—uy) X(t—u2) X(t—T1) X(t—T2)] dujduz 
—my Rxx(T —7T2) 
= | | ho(uy, u2)[Rxx(u1 — U2) - Rxx(t1 —T2) 
+ Ryx(t;-—u 1) Ryx(t2 — U2) + Rxx(t2— uy) Rxx(t1 —uz)] dujduz 
co oo 
= | | h2(v1, v2) Rxx(V1, v2) dvidv2 Rxx(T1 —T2). (11.45) 
In this equation, the first term and the last term are the same and cancel each other. The 
second and the third terms are the same, as h(u, uz) is symmetrical with u, and v2, and 
the range of integrals is from — © — + © for both. 
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