E|Z(t)| = 0 
E[Z(t)Z(t—u)] = 0(u)o% (6.32) 
= E[Z(t)P = 6(0)0z, 
6(0) is Dirac’s delta function, 
when u=0 
O(u) = 
0 when u ~ 0, 
and f end =e (6.33) 
Accordingly, E [Z(t)]* => ©, which means that the white noise Z(t) is physically 
unrealizable. The output X(z) is a stochastic process with zero mean. Using the differen- 
tial operator D = d/dt, D” = (d/drt)”, 
X(t) = (D + ao)}Z(t). (6.34) 
When we express Green’s function as G(V), 
[-<) t 
X(t) = | G(v)- Z(t—v)dy = | G(t—v)Z(v)av. (6.35) 
0 —0o 
This equation is the orthogonal decomposition of X(f), since the Z(t)’s are uncorre- 
lated, or independent at different times. From Eq. 6.34 and Eq. 6.35, it is also clear that, 
when the forcing function is Dirac’s delta function, X(t) is the Green’s function, 
(D + ao)G(t) = d(¢) 
G(t) = | G(v)d(t—v)dv. (6.36) 
0 
Here, since the solution of the homogeneous equation 
(D + ao) 'X() = 0 (6.37) 
is 
OSE CG, (6.38) 
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