This actor uses the Levinson-Durbin algorithm to compute the linear
predictor coefficients of a random process, given its autocorrelation
function as an input. These coefficients are produced both in
tapped delay line form (on the <i>linearPredictor</i> output) and in
lattice filter form (on the <i>reflectionCoefficients</i> output).
The <i>order</i> of the predictor (the number of <i>linearPredictor</i>
and coefficients <i>reflectionCoefficients</i> produced) is the
number of lags of the supplied autocorrelation.
The <i>errorPower</i> output is the power of the prediction error
as a function of the predictor order.
The inputs and outputs are all arrays of doubles.
<p>
The autocorrelation estimates provided as inputs can be generated
by the Autocorrelation actor. It the Autocorrelation actor is set
so that its <i>biased</i> parameter is true, then the combined
effect of that actor and this one is called the autocorrelation
method. The <i>order</i> of the predictor is the value of the
<i>numberOfLags</i> parameter of the Autocorrelation actor.
If the length of the autocorrelation input is odd, then it is assumed
to be a symmetric autocorrelation function, and the <i>order</i> of the
predictor calculated by this actor is (length + 1)/2. Otherwise,
the <i>order</i> is 1 + (length/2), which assumes that discarding the last
sample of the autocorrelation would make it symmetric.
<p>
Three output signals are generated by this actor. On the
<i>errorPower</i> output port, an array of length <i>order</i> + 1
gives the prediction error power for each predictor order from zero
to <i>order</i>. The first value in this array, which corresponds
to the zeroth-order predictor, is simply the zero-th lag of the
input autocorrelation, which is the power of the random process
with that autocorrelation. Note that for signals without noise
whose autocorrelations are estimated by the Autocorrelation actor,
the <i>errorPower</i> output can get small. If it gets close
to zero, or goes negative, this actor fixes it at zero.
"Close to" is determined by the close() method of the
ptolemy.math.SignalProcessing class.
<p>
The <i>linearPredictor</i> output gives the coefficients of an
FIR filter that performs linear prediction for the random process.
This set of coefficients is suitable for directly feeding a
VariableFIR actor, which accepts outside coefficients.
The number of coefficients produced is equal to the <i>order</i>.
The predictor coefficients produced by this actor can be
used to create a maximum-entropy spectral estimate of the input
to the Autocorrelation actor. They can also be used for
linear-predictive coding, and any number of other applications.
<p>
The <i>reflectionCoefficients</i> output is the reflection
coefficients, suitable for feeding directly to a VariableLattice
actor, which will then generate the forward and backward prediction error.
The number of coefficients produced is equal to the <i>order</i>.
<p>
Note that the definition of reflection coefficients is not quite
universal in the literature. The reflection coefficients in
reference [2] is the negative of the ones generated by this actor,
which correspond to the definition in most other texts,
and to the definition of partial-correlation (PARCOR)
coefficients in the statistics literature.
<p>
<b>References</b>
<p>[1]
J. Makhoul, "Linear Prediction: A Tutorial Review",
<i>Proc. IEEE</i>, vol. 63, pp. 561-580, Apr. 1975.
<p>[2]
S. M. Kay, <i>Modern Spectral Estimation: Theory & Application</i>,
Prentice-Hall, Englewood Cliffs, NJ, 1988.
Edward A. Lee
$Id: LevinsonDurbin.java 70398 2014-10-22 23:44:32Z cxh $
Ptolemy II 1.0
Yellow (eal)
Red (cxh)
The autocorrelation input, which is an array.
The output for the error power, as a function of the predictor
order. This produces an array.
The output for linear predictor coefficients.
This produces an array.
The output for lattice filter coefficients for a prediction
error filter. This produces an array.