When dealing with windowed speech we need to take into account the boundary effects in order to avoid large prediction errors at the edges. Can refine the area over which we perform least squares minimisation in equation 67 and make use of the fact that samples are zero outside of the window to rewrite as:

Now is only dependent on the difference, *i*-*j*, and may be
written in terms of the autocorrelation function, :

Now is Toeplitz:

Efficient methods exist to invert such matrices, one of which is
Durbin's algorithm. Denoting the values of the LP parameters at
iteration *i* by and the residual energy by
() for i = 1, 2, ...

For example, for the signal of figure 21:

Therefore on the first iteration:

And on the second iteration:

The parameters are known as the reflection parameters. Note that:

- All intermediate solutions are calculated
- This method also provides the reflection coefficients
- The resulting filter is guaranteed to be stable
- The value of the squared prediction residual, is also computed and is guaranteed to decrease (or remain constant) on each iteration

Speech Vision Robotics group/Tony Robinson