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Embedded SystemsFPGAElectronics

Spectral Audio Signal Processing
    Applications of the STFT
       Spectral Envelope Extraction
          Linear Prediction Spectral Envelope
             Computation of Linear Prediction Coefficients

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Computation of Linear Prediction Coefficients

In the autocorrelation method of linear prediction, the linear prediction coefficients $ \{a_i\}_{i=1}^M$ are computed from the Bartlett-window-biased autocorrelation function (Chapter 5):

$\displaystyle r_{y_m}(l) \isdefs \sum_{n=-\infty}^\infty y_m(n)y_m(n+l) \eqsp \hbox{\sc DFT}^{-1}\left\vert Y_m\right\vert^2 \protect$ (10.3)

To obtain the $ M$th-order linear predictor coefficents $ \{a_1,\ldots,a_M\}$, we solve the following $ M\times M$ system of linear normal equations (also called Yule-Walker or Wiener-Hopf equations):

$\displaystyle \sum_{i=1}^M a_i r_{y_m}(\vert i-j\vert) \eqsp -r_{y_m}(j), \qquad j=1,2,\ldots,M \protect$ (10.4)

In matlab syntax, the solution is given by `` $ \verb+a=R\p+$'', where $ \verb+p(j)+ = r_{y_m}(j)$, and $ \verb+R(i,j)+=r_{y_m}(\vert i-j\vert)$. Since the covariance matrix $ R$ is symmetric and Toeplitz by construction,10.4 an $ O(M^2)$ solution exists using the Durbin recursion.10.5

If the rank of the $ M\times M$ autocorrelation matrix $ R[i,j]=r_{y_n}(\vert i-j\vert)$ is $ M$, then the solution to (9.4) is unique, and this solution is always minimum phase [153] (i.e., roots of $ A(z)$ are inside the unit circle in the $ z$ plane [247], so that $ 1/A(z)$ is always a stable all-pole filter). In practice, the rank of $ R$ is $ M$ (with probability 1) whenever $ y(n)$ includes a noise component. In the noiseless case, if $ y(n)$ is a sum of sinusoids, each (real) sinusoid at distinct frequency $ 0<\omega_i T < \pi$ adds 2 to the rank. A dc component, or a component at half the sampling rate, adds 1 to the rank of $ R$.

The choice of time window for forming a short-time sample autocorrelation and its weighting also affect the rank of $ R$. Equation (9.3) applied to a finite-duration frame yields what is called the autocorrelation method of linear prediction [153]. Dividing out the Bartlett-window bias in such a sample autocorrelation yields a result closer to the covariance method of LP. A matlab example is given in §9.3.3 below.

The classic covariance method computes an unbiased sample covariance matrix by limiting the summation in (9.3) to a range over which $ y_m(n+l)$ stays within the frame--a so-called ``unwindowed'' method. The autocorrelation method sums over the whole frame and replaces $ y_m(n+l)$ by zero when $ n+l$ points outside the frame--a so-called ``windowed'' method (windowed by the rectangular window).

Previous: Linear Prediction Methods
Next: Linear Prediction Order Selection

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About the Author: 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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