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<?xml version="1.0" encoding="UTF-8"?>
<!--
*
* This help file was generated from levinson.sci using help_from_sci().
*
-->
<refentry version="5.0-subset Scilab" xml:id="levinson" xml:lang="en"
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<refnamediv>
<refname>levinson</refname>
<refpurpose>Levinson-Durbin Recurssion Algorithm</refpurpose>
<para> </para>
</refnamediv>
<refsynopsisdiv>
<title>Calling Sequence</title>
<synopsis>
a = levinson(r)
a = levinson(r,n)
[a,e] = levinson(r,n)
[a,e,k] = levinson(r,n)
</synopsis>
<para> </para>
</refsynopsisdiv>
<refsection>
<title>Parameters</title>
<variablelist>
<varlistentry><term>r</term>
<listitem><para> Real or complex deterministic autocorrelation sequence input </para></listitem></varlistentry>
<varlistentry><term>a</term>
<listitem><para> Coefficients of length(r)-1 order Autoregressive linear process </para></listitem></varlistentry>
<varlistentry><term>n</term>
<listitem><para> Order of autoregressive model (default value is length(r)-1 , if n is not given)</para></listitem></varlistentry>
<varlistentry><term>e</term>
<listitem><para> Prediction error of order n</para></listitem></varlistentry>
<varlistentry><term>k</term>
<listitem><para> Reflection coefficient vector of length n</para></listitem></varlistentry>
</variablelist>
<para> </para>
</refsection>
<refsection>
<title>Description</title>
<para> The Levinson-Durbin recursion algorithm is used for finding an all-pole IIR filter with a given deterministic autocorrelation sequence (r) </para>
<para>
<latex>
\begin{eqnarray}
H(z) = \frac{1}{1+a(2)z^{-1}+a(3)z^{-2} + ... +a(n+1)z^{-n}}
\end{eqnarray}
</latex>
</para>
<para> </para>
</refsection>
<refsection>
<title>Examples </title>
<para> Estimate the coefficients of an autoregressive process given by equation </para>
<para>
<latex>
\begin{eqnarray}
x(n) = 0.1x(n-1) - 0.8x(n-2) + w(n)
\end{eqnarray}
</latex>
</para>
<para> </para>
<programlisting role="example"><![CDATA[
a = [1 0.1 -0.8];
v = 0.4; //variance, v=0.4
w = sqrt(v)*rand(15000,1,"normal");
x = filter(1,a,w);
[r,lg] = xcorr(x,'biased');
r(lg<0) = [];
ar = levinson(r,length(a)-1) //coefficients of autoregressive process
]]></programlisting>
</refsection>
</refentry>
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