1 1 
SAMO) = | Reade) ac ==— | | rucenRe—zp edt at 
1 ? t 
oe | hy(t)-e "dt ,- | Re-r) e OT —T1) gy 
= H;(@)sxx(w) (11.35) 
and = Hy) = syx(@)/sxx(w). (11.36) 
Eq. 11.36 shows even when Y(r) includes the nonlinear component, if the input X(¢) 
was a Gaussian process, the linear response frequency function H (w) is expressed by the 
ratio of syy (w) and sxy (w) as in the purely linear case. 
3. Autocovariance Ryy (7); linear spectrum syy (w). 
As in the linear process, the Fourier transform of second order moment or the auto- 
covariance function is called the linear spectrum, 
Ryy(t) = E[Y(t) — my] [Y(t + 7) — my] 
= E[Y(t)- Y(t+1)] — ms. (11.37) 
Here by the use of the functional expression of Y(t) and Y(t +t) as Eq. 11.11 and appro- 
priate variables, 
Ryy(t) = | hy(s1) hy(r1) E[X(¢—5s)) -X(t+t—-11)] dsydry 
8-—, 8 
ioe] 
: | 
—-o 
Generally, when X,, X2, X3, X4 are the probability variables that are from a joint Gaussian 
distribution, then 
E[X, X2 X3 X4] = Miz Maa4+M 13 Mo4+ My4 M23. (11.39) 
| | ho(uy, u2)h2(v1, v2) E[X(t—uy) X(t—uz) X(t+ T-Vv1) X(¢+T—v2)] 
_-@® —-o 
dujduzdv,dv7 — mé. (11.38) 
Here 
Miz = E[X; - Xx). 
From this relation, 
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