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Computation of Linear Prediction Coefficients

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

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

To obtain the $ M$ 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_{x_m}(\vert i-j\vert) = -r_{x_m}(j), \qquad j=1,2,\ldots,M
$

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


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``Cross Synthesis Using Cepstral Smoothing or Linear Prediction for Spectral Envelopes'', by Julius O. Smith III, (From Lecture Overheads, Music 421).
Copyright © 2014-03-24 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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