Solving these linear equations gives 
R2) —R) 
EE Benne) O10 
ae RO)-ROR2  &@-62 ~ 
—R(2) —R(1) 
SR) eRe) 
OO) EO) Per —Aatey Ber ans 
oe _K@)-RORG) _ oF@ e(1)e3). (5.175) 
A A i D2 A 5 o 7 AQ ays 
“Ray -2@| B@=-RORQ®™” F)-€@ 
RO) = RO) 
After @, and a2 have been determined by and oz can be obtained by solving Eqs. 
5.155 and 5.156. From them, Pandit and Wu** showed that b; can be obtained as the 
solution of a quadratic equation 
b¢+Cb,+1=0 (5.176) 
where 
1+ 410 + 4202 
C=-a,+ A A A ’ 
—@,—(a2—1)Q1 
(5.177) 
although this is quadratic in terms of b, and is not a linear relation. 
5.2.4.6 Spectrum of ARMA(2.1). The expression of ARMA(2.1) is, by Eq. 5.120, 
X, + QyX,1 + 2X19 = E+) Ep} 
or, using the backward shift operator B, by Eq. 5.123, 
a(B)X,; = BiBye: 
(5.120’) 
a(Z) =1+a;Z+a2Z 
B(Z) =1+),Z. 
Therefore 
Xpor + AXj-y-1 + a2Xj-+-2 = Err + Dy€p-y-1 - (5.178) 
Taking the product of Eqs. 5.120 and 5.178 and then taking expected values of each term 
gives 
144 
