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The prediction coefficients $\{a_i\}_{i=1}^M$ are computable from the autocorrelation function:

$\displaystyle r_{y_m}(l) \mathrel{\stackrel{\Delta}{=}}\sum_{n=-\infty}^\infty y_m(n)y_m(n+l) = \hbox{\sc DFT}^{-1}\left\vert Y_m\right\vert^2$

To obtain the th-order linear predictor coefficents $\{a_1,\ldots,a_M\}$ , solve the $M\times M$ system of linear equations:

$\displaystyle \sum_{i=1}^M a_i r_{y_m}(\vert i-j\vert) = -r_{y_m}(j), \qquad j=1,2,\ldots,M$

In Matlab, `` $\verb+a=R\p+$ '', where $\verb+p(j)+ = r_{y_m}(j)$ , and $\verb+R(i,j)+=r_{y_m}(\vert i-j\vert)$ .

Solution always exists
If rank is , solution is unique
Unique solution is always stable
(roots of are inside the unit circle in the plane)
Since is Toeplitz, an solution exists

written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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