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Yule Walker Equations for LPC

Started by naebad January 8, 2006
for a random signal (AR)

y(k)=-a1y(k-1)-a2y(k-2)-...-any(k-n)+e(k)  (e(k) is white noise)

I am a bit confused - I understood the YW normal  equations were of the
form


Ra = b

where R is the correlation matrix


 a=[a1,a2...an] is the parameter vector and

b=[-r(2),-r(3)...-r(n)]

However I see many web refs to the above which has the same except

b=[sigma^2,0,0...0]

where sigma^2 is the variance of  y eg

http://www.cbi.dongnocchi.it/glossary/YuleWalker.html

?

Naebad

You get the Yule-Walker equations by multiplying the equation with
itself and taking the expectations.

The multiplication yields the following kind of products
 y(k-m)y(k-r) , which on taking the expectation give you the
coefficients of the correlation matrix.
y(k-m)e(k), On expectation...this just goes to 0. Signal and noise are
supposed to be uncorrelated.
One term with e(k)e(k)...This gives you the sigma^2 (noise power) on
expectation.

Work it out. You should be able to get it.

Raja


naebad wrote:
> for a random signal (AR) > > y(k)=-a1y(k-1)-a2y(k-2)-...-any(k-n)+e(k) (e(k) is white noise) > > I am a bit confused - I understood the YW normal equations were of the > form > > > Ra = b > > where R is the correlation matrix > > > a=[a1,a2...an] is the parameter vector and > > b=[-r(2),-r(3)...-r(n)] > > However I see many web refs to the above which has the same except > > b=[sigma^2,0,0...0] > > where sigma^2 is the variance of y eg > > http://www.cbi.dongnocchi.it/glossary/YuleWalker.html > > ? > > Naebad