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The autocorrelation method

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 tex2html_wrap_inline3095 as:


 eqnarray982

Now tex2html_wrap_inline3095 is only dependent on the difference, i-j, and may be written in terms of the autocorrelation function, tex2html_wrap_inline3101:


eqnarray993

Now tex2html_wrap_inline3083 is Toeplitz:


eqnarray999

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 tex2html_wrap_inline3107 and the residual energy by tex2html_wrap_inline3109 (tex2html_wrap_inline3111) for i = 1, 2, ...


eqnarray1019

For example, for the signal of figure 21:


eqnarray1034

Therefore on the first iteration:
eqnarray1036

And on the second iteration:


eqnarray1042

The parameters tex2html_wrap_inline3113 are known as the reflection parameters. Note that:



Speech Vision Robotics group/Tony Robinson