where A is the root of the characteristic equation 
A+Qo= 0, or A =-@Qo, 
X(6) —1Crenaes 
(6.39) 
(6.40) 
This coefficient C is determined from the initial conditions given by the characteris- 
tics of Dirac’s delta function as G(t) = 0, t< 0, G(0) = 1; therefore C = 1. Green’s 
function is obtained as 
Cae’ nia) 
G(t) = 
@ t<0 
Se) ete" 
here u(t) is the unit step function. 
The autocorrelation function is R(s) = E[X(t) X(t—s)]. 
Using X(t) = [ow Z(t—v') av’ 
0 
and X(t—s) = | oo Z(t—s—v) dv 
0 
gives 
R(s) ae | | Gv" GW )E[Z(t — v")Z(t— s—v) ]avdy' 
00 
00 
lo) cw s 
= 0% | G(v)G(v + s)dv = 0% | eA pAdlvts)p, — = eras 
0 0 
RCs) = R(s), 
thus 
211 
0 
(6.41) 
(6.41”) 
(6.42) 
