The term syy(w)/syx(w) is no longer equivalent to H,(w), but is modified by some term 
that includes the third order nonlinear frequency response function H3. 
3. Second order autocovariance function, linear spectrum 
Ryy(t) = E[Y(t)— my] [Y(t + T) — my] 
= E[Y(t)- Y(t+1)] —m} 
oOo ow 
= | [ huts phurDe{xt—oXte+1—ryldsidr 
—O —& 
fo 2) 
t {| | | frcuus hay, v2) E[X(t—uy) X(t—u2) X(t+T—-V}) 
ie xX X(t+T—V2)] dujduz dv\dv2 
rn | | | | | | h3(P1, P2, P3)h3(q1, 92, 93) 
x E|X(r— pi) X(¢—p2) X(¢—ps)-Xe+7- qu) Xe+t-q2) Xe+7—gs)| 
dp dp2dp3dq\dq2dq3 
—m>. (11.74) 
We take the Fourier transform, and the linear spectrum is, after manipulation, 
(oe) 
Syy(W = Sxx(w) | Hy\(w) +3 | H:@,03,-03) Sxx(w3) do | * 
—-o 
+2 | | H2(w — 2,02) | * syx(w —@2) Sxx(@2) doz 
{| H3(@ — @2,— 03, 2, W3) | 2 syx(W —@2— 03) Sxx(@3) dwrdw3, 
Fis (11.75) 
_ The first term, the term of syx(w) of this Eq. 11.75 is again modified by the cubic 
response H3, as was Eq. 11.73 that shows syx(w)/sxx(w) is not H;(w) anymore, but is 
modified by the effect of cubic response H3(@1,@2,w3). Bedrosian and Rice®° showed 
this modification clearly for the case when the input was the sum of sinusoidal waves, 
and showed the necessity to include the higher order terms in Voltera expansions. 
318 
