Using the differential operator D = d/dr gives 
(D? + 2xaa,D + w2)X(t) = — ft, 
Or 
(D? +a,D + a)X(t) = — fit), 
where @; = 2KW,, Qo =>. 
Eq. 6.53’ is analogous to the AR model of the discrete process AR(2) as given by 
Eq. 5.59 or Eq. 5.62, 
X,+4)X1+@2X.2=€, or (1+a,;B+aB7)X,=€,. 
(6.53) 
(6.53’) 
If we assume (1/M)f(r) is a continuous pure random process Z(t), then we can say 
that X(r) is an autoregressive process of continuous time of order 2 and express this 
process as A(2) as did Pandit and Wu.*° 
Then A(2) can be expressed as 
(D? + 2kw,D + w2)X(t) = Z(t) 
o% when r=?’ 
where E[Z()Z(t')] = 03+ d(t-t') = 
0 when tr?’ 
which corresponds to Eq. 5.62 for AR (2), and 
X(t) = (D? + 2kw,D + w2)!Z(t) = (D? + aD + a) !Z(0). 
Then using the Green’s function G(t), 
X(t) = (D? + 2kw,D + w2)'Z(t) 
= | G(v)Z(t—v)dy, 
0 
and 
(D* + ayD + ao)G(t) = A(t) 
214 
(6.54) 
(6.55) 
(6.56) 
(6.57) 
