Digtal Signal ProcessingAnd Modeling

www.themegallery.com

The AutocorrelationExtension Problem

Chapter 5.2.8

2006. 09. 21 / KIM JEONG JOONG / 20067168

statistical

D.S.PAutocorrelation Extension

Given the first (p + 1) values of an autocorrelation sequence, r(k)

for k = 0, 1, . . . , p, how may we extend (extrapolate) this partial autocorrelation sequence for

k > p in such a way that the extrapolated sequence is a valid autocorrelation sequence?

the autocorrelation matrix formed from this sequence must be nonnegative definite, Rp>=0

Therefore, any extension must preserve this nonnegative definite property, i.e., Rp+1 >= 0, and Rp+2 >= 0, and so on.

statistical

D.S.PAutocorrelation Extension

This follows from the act that this extrapolation generates

the autocorrelation sequence of the AR(p) process that is consistent with the given autocorrelation values

given a partial autocorrelation sequence r, (k) for k = 0, 1 , . . . , p, what values of r(p+1) will produce a valid partial autocorrelation sequence with Rp+1 >= 0

The answers to these questions may be deduced from Property 7 by expressing rx ( p + 1 ) in terms of the reflection coefficient Г(p+1) may

place a bound on the allowable values for r(p + 1)

statistical

D.S.PAutocorrelation Extension

statistical

D.S.PExample 5. 2. 9

Given the partial autocorrelation sequence rx (0) = 1 and r, (1) = 0.5, let us find the set of allowable values for r, (2), assuming that rx (2) is real. For the first-order model

we have and a1 (1) = Г1 = -0.5. Thus, with a first-order modeling error of

€1 = rx(0)[l - |Г1|^2] = 0.75 we have

r,(2) = -Г2 € 1 - al(l)rx(l) = -0.75Г2 + 0.25 Therefore, with 1 rz 1 a 1 it follows that In the special case of rz = 0, rx(2) = 0.25 and, in the extreme cases of Г 2 = ±1, The autocorrelation values are r, (2) = -0.5 and r, (2) = 1.