fZ) =Z-a=0, (5.16) 
namely 
M=a. (5.17) 
A is an arbitrary constant and is determined by the initial conditions. From Eqs. 
5.15 and 5.17, 
X,= Aa’. (5.18) 
One particular solution of Eq. 5.7 is obtained as follows using the backward shifting 
operator B. Equation 5.7 is 
(1 —aB) X;=€;, (5.19) 
X,=(1-aBye,= 5 (@/ Bye, = > ai €,;. (5.20) 
j=0 j=0 
This equation is the same as Eq. 5.10 or 5.13 expressed by Green’s function. Therefore, 
the general solution of Eq. 5.7 is 
[e.e} 
X,=Aa'+ >. a! €1.; (5.21) 
j=0 
The first term that is the solution of the homogenous equation is the free oscillation of 
this system. When lal < 1, this free oscillation decays, and only the second term remains 
as a Stationary oscillation as in Eq. 5.20. Equation 5.19 is, more generally, 
a(B)X, = €;, (5.22) 
X,=a7 (Bye, (5.23) 
where 
a(Z) = 1-aZ. (5.24) 
In order to have the free oscillation damp out, lal < 1 is necessary and, with the character- 
istic Eq. 5.16 equated to zero, {Z) = 0 must have its root inside the unit circle. Then from 
the equation 
a(Z) = 0, (5.25) 
or 1—aZ =0, its root is the reverse of the root of the equation Z—a = 0, or fiZ) = 0. We 
find a(Z)=0 must have its root outside the unit circle. 
522.3 Inverse Function of AR(1). Green’s function can be considered as an indi- 
cation of how X; can be expressed by the MA (©) process, because X; is expressed as the 
summation of infinite €, at preceding time points. 
106 
