# Yule Walker Equations for LPC

Started by 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

?

```
```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

> 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
>
> ?
>