, _ R(1) R(2)—R(O) RA) 
2s ye EN ee 
R*(0) —R*(1) 
. _ R°(1)—R(O)R(2) 
TPO Ska) (5.103’) 
’ 
By solving the linear equation on dj, 22, we can easily determine the 4}, @> from the 
autocorrelation R(0), R(1), and R(2). Equations 5.83 and 5.84 can be written in vector 
form as 
R24 =-r, (5.104) 
where 
in ee (5.105) 
R(1) RO) 
a=[d),do’, r=(R(),R(2)]'. (5.106) 
ee 5.107 
therefore a = ( ) 
Thus & = [4;,42]' can be determined from r =[R(1),R(2)]' and 
R(O) RO) 
R2= 1 a) RO) | 
namely, from R(0), R(1) and R(2) . 
After a}, G2 are obtained, a2 can be estimated from Eq. 5.82, 
G2 = R(O) + 4,R(1)— 42R(2). (5.108) 
5.2.3.6 Spectrum of AR(2). From Eq. 5.59 for AR(2) 
X; + a)X;_1 + 22X12 = €;, 
Xe. + Q4Xj-y7-1 + 2X j-+-2 = Err . (5.109) 
Taking the product of both sides of these two equations, respectively, and then taking 
expectation of each term, 
E[X,-X,,]+af E[X1-Xi-1) +f ElX2°Xe+2] 
+a, E[X,- X;,-1] + aya E[X1 -X,,_5] + a2 E[X: X,_,_2] 
+a, E[X;, X-1]+a@3a2 E[X1~-1X1-2] + a2 E[X-X1-2] = Eles: 1+). 
(5.110) 
Therefore, for r = 0, 
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