where denotes the th data frame from the signal . To obtain the th-order linear predictor coefficients , we solve the following system of linear normal equations (also called Yule-Walker or Wiener-Hopf equations):
In matlab syntax, the solution is given by `` '', where , and . Since the covariance matrix is symmetric and Toeplitz by construction,11.4 an solution exists using the Durbin recursion.11.5
If the rank of the autocorrelation matrix is , then the solution to (10.12) is unique, and this solution is always minimum phase  (i.e., all roots of are inside the unit circle in the plane , so that is always a stable all-pole filter). In practice, the rank of is (with probability 1) whenever includes a noise component. In the noiseless case, if is a sum of sinusoids, each (real) sinusoid at distinct frequency adds 2 to the rank. A dc component, or a component at half the sampling rate, adds 1 to the rank of .
The choice of time window for forming a short-time sample autocorrelation and its weighting also affect the rank of . Equation (10.11) applied to a finite-duration frame yields what is called the autocorrelation method of linear prediction . 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 stays within the frame--a so-called ``unwindowed'' method. The autocorrelation method sums over the whole frame and replaces by zero when points outside the frame--a so-called ``windowed'' method (windowed by the rectangular window).
Linear Prediction Order Selection
Linear Prediction Methods