€;/ = a, X,2+€,. (5.58) 
The residual part €, =€, —a3 X;_2 is now a purely random process. Then inserting 
Eq. 5.57 into Eq. 5.58 gives X;—a’X;_) — a3 X;-2 = €;. 
X12 
=> 
Fig. 5.14. €, vs. X,_. for AR(2). 
Here changing —a' = a), —a3 > ap, 
Xp t+ QyX7_1 + Q2X72 = €;. (5.59) 
The process, which satisfies Eq. 5.59, is called the autoregressive process of the sec- 
ond order. Here, a; and a2 are constant, {€,} is a stationary pure random process, and we 
assume E[e,] = 0. 
5.23.1 Solution of AR(2), Green’s Function for AR(2). Starting from Eq. 5.59 
X,+ @,X;-1 + A2X1-2 = € 1, 
we can solve X; as the general solution of a second order difference equation, i.e., a 
general solution equals a complementary function plus a particular solution. A comple- 
mentary function is the solution of the homogeneous equation 
XxX; + Q4X1-1 + a2X 1-2 = 0. (5.60) 
The solution of this homogeneous equation is in the shape of 
Ay yi +A2 1. (5.61) 
(41,42, being the roots of the characterisitc equation f(Z) = 0. The particular solution can 
be obtained, using the backward shifting operator B, as 
(1 +a,B + a>B?)X, = €;. (5.62) 
Setting 
Q(Z) =1+a\Z+a7Z’, (5.63) 
a(B)X,= €;. (5.64) 
If the quadrant fD=Z +a\Z+a,Z =0 (5.65) 
has two roots 41,2, 
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