Using G(r) in Eq. 6.56 gives 
ew _ ow 
xX vt 
(t)= (eee eae ————_Z(t-v)dy 
0 
t 
1) _ Ar(t-v) 
= | eee ed Z(v)dv. (6.69) 
A, -A2 
—-o 
Equation 6.69 is the orthogonal decomposition of X(t) and also shows the general 
solution of the nonhomogeneous Eq. 6.54. 
The autocovariance function is then, by Eq. 6.56, 
R(s) = E[X(t) X(t-s)] 
= /3 [oonze—vrdv' | Gorze—s—vyd 
0 0 
| | G(v')GW)E[Z(t —v')Z(t—s —v)]dvdv' 
0 0 
| [ Gorrcose + s—v'av dv 
On0 
= 0% | G(v)G(v + s)dv. (6.70) 
0 
Inserting in Eq. 6.70 the expression G(v) = - 7 and manipulating gives 
1-42 
2 
B Ares Bee rate Tt EL 2. (671) 
O7 
SO Cee Ce) eee) 
Then the variance is 
218 
