R(O) = —a;R(1) —apR(2)- - -—a, R(n) +a2 
R(1) =—a}R(0) —a@2R(1)- - -—a,R(n-1)- 
(5.220) 
R(k) = —ayR(k—- 1) —a2R(k—2)- - -—a,R(k—n) 
g= Iho o 970, 
Using the backward shift operator B gives 
BIR(k) = R(k-j). (5.221) 
Then the last equation in Eq. 5.220 is 
R= ( >) =a; Bi Jn (5.222) 
j=l 
or 
n 
SG BR = 0. (5.222) 
jJ=0 
Equation 5.222 is very similar to the expression of the homogeneous equation of the 
process itself in Eqs. 5.211’, 5.211” when €,=0 as 
a(B)R(k) = 0. G:222) 
Therefore, as was Eq. 5.166 for AR(2), the general form of R(&) is 
R(k) = By wh + Bo ub+-- +B, uk (Goze) 
Here “),42- : -M, are the eigenvalues, and B,, Bz, - - - B, are constants that can be 
obtained theoretically from the boundary conditions 
R(O)=B,+B2+---+B, 
R(1) = By fi + Bz fat: > + Br Mn 
(5.224) 
The spectrum is, as given by Eq. 5.209, 
2 2 
S(@) = AGN = Uae In ISITE aN a (5.225) 
21 | a(e*)|? 27 | 1+ ajen" tae + + apex |" 
188 
