#### Computation of Linear Prediction Coefficients

In the autocorrelation method of linear prediction, the linear
prediction coefficients
are computed from the
Bartlett-window-biased *autocorrelation function*
(Chapter 6):

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* [162] (*i.e.*, all roots of
are inside the unit circle in the
plane [263], 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 [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 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).

**Next Section:**

Linear Prediction Order Selection

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Linear Prediction Methods