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 6):

$$r_{y_m}(l) \isdefs \sum_{n=-\infty}^\infty y_m(n)y_m(n+l) \eqsp \hbox{\sc DFT}^{-1}\left\vert Y_m\right\vert^2 \protect$$ (11.11)

where $y_m$ denotes the $m$ th data frame from the signal $y$ . To obtain the $M$ th-order linear predictor coefficients $\{a_1,\ldots,a_M\}$ , we solve the following $M\times M$ system of linear normal equations (also called Yule-Walker or Wiener-Hopf equations):

$$\sum_{i=1}^M a_i r_{y_m}(\vert i-j\vert) \eqsp -r_{y_m}(j), \qquad j=1,2,\ldots,M \protect$$ (11.12)

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,^11.4 an $O(M^2)$ solution exists using the Durbin recursion.^11.5

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 (10.12) is unique, and this solution is always minimum phase [162] (i.e., all roots of $A(z)$ are inside the unit circle in the $z$ plane [263], 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 (10.11) applied to a finite-duration frame yields what is called the autocorrelation method of linear prediction [162]. 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 §10.3.3 below.

The classic covariance method computes an unbiased sample covariance matrix by limiting the summation in (10.11) 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).