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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:


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:


Now tex2html_wrap_inline3083 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 tex2html_wrap_inline3107 and the residual energy by tex2html_wrap_inline3109 (tex2html_wrap_inline3111) for i = 1, 2, ...


For example, for the signal of figure 21:


Therefore on the first iteration:

And on the second iteration:


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

Speech Vision Robotics group/Tony Robinson