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






In matlab syntax, the solution is given by ``





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).
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Linear Prediction Order Selection
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Linear Prediction Methods