Go=1 
Goa Gr G, =i) 
a2G,+a,G; oP G2 =0 
a3G, + a2G; + a;G2+G3=0 (5.214) 
anGo + a,-1G} ape © oap(Gr, = 0, 
or 
G,=-a 
G2 =-a2+ a? 
G3 = —a3 + a0) + a\(a2 — a?) = — a3 + 2aja2 — a}. (5.215) 
Generally 
(a,B" +--+ -+1)G,=0. 
Explicitly as shown in Section 5.3.2 
Gj= gi Wi ter wht > + Bnbth G216) 
where jl; : - «4, are the eigenvalues or the roots of the characteristic equation equated to 
ZeTO as 
fi@)=Z+4a,z7""+---+a,=0 (5.217) 
and 
n—1 
mS ent (5.218) 
(Ui — 1) (1 — 2° > + i Mi) Ui Hiv) * «+ Ui Hn) 
where the denominator is the product of all terms w;- - -wjfor j=1,2---n, excluding 
the zero term uu; —;, as the special case where b} = b2 = b3 = -= b, = OmEg. 
5.207. 
5.4.3 Autocorrelation and Spectrum Function for AR(n) 
For AR(n) 
Xp = — 1X11 — A2Xp2° + + —GpXint €r. 
Therefore 
xeun =i QXi-7-1 = aX 7-2 — See ee AnXt-r-n +€r,. (5.219) 
Multiplying both sides of these two equations, respectively, taking the expected values 
gives 
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