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<?xml version="1.0" encoding="UTF-8"?>
<!--
*
* This help file was generated from rlevinson.sci using help_from_sci().
*
-->
<refentry version="5.0-subset Scilab" xml:id="rlevinson" xml:lang="en"
xmlns="http://docbook.org/ns/docbook"
xmlns:xlink="http://www.w3.org/1999/xlink"
xmlns:svg="http://www.w3.org/2000/svg"
xmlns:ns3="http://www.w3.org/1999/xhtml"
xmlns:mml="http://www.w3.org/1998/Math/MathML"
xmlns:scilab="http://www.scilab.org"
xmlns:db="http://docbook.org/ns/docbook">
<refnamediv>
<refname>rlevinson</refname>
<refpurpose>This function computes the autocorrelation coefficients using prediction polynomial method.</refpurpose>
</refnamediv>
<refsynopsisdiv>
<title>Calling Sequence</title>
<synopsis>
R = rlevinson(a, efinal)
[R, U] = rlevinson(a, efinal)
[R, U, kr] = rlevinson(a, efinal)
[R, U, kr, e] = rlevinson(a, efinal)
</synopsis>
</refsynopsisdiv>
<refsection>
<title>Parameters</title>
<variablelist>
<varlistentry><term>a:</term>
<listitem><para>Input argument ,prediction polynomial coefficients.</para></listitem></varlistentry>
<varlistentry><term>efinal:</term>
<listitem><para>Input argument, final prediction error.</para></listitem></varlistentry>
<varlistentry><term>R:</term>
<listitem><para>Returns the auto-correlation coefficients.</para></listitem></varlistentry>
<varlistentry><term>U:</term>
<listitem><para>Returns upper triangular matrix of order length(a)*length(a).</para></listitem></varlistentry>
<varlistentry><term>kr:</term>
<listitem><para>Reflection coefficients.</para></listitem></varlistentry>
<varlistentry><term>e:</term>
<listitem><para>Prediction error.</para></listitem></varlistentry>
</variablelist>
</refsection>
<refsection>
<title>Description
</title>
<para>The reverse Levinson-Durbin recursion implements the step-down algorithm for solving the following symmetric Toeplitz system of linear equations for r, where r = [r(1) … r(p + 1)] and r(i)* denotes the complex conjugate of r(i).</para>
<para>
<latex>
\begin{eqnarray}
\begin{bmatrix}
r(1) r(2)* - - r(p)* \\
r(2) r(1) - - r(p-1)* \\
- - - - - \\
- - - - - \\
- - - - - \\
r(p) r(p-1) - r(2) r(1)
\end{bmatrix}\begin{bmatrix}
a(2)\\
a(3)\\
.\\
.\\
.\\
a(p+1)
\end{bmatrix}=\begin{bmatrix}
-r(2)\\
-r(3)\\
.\\
.\\
.\\
-r(p+1)
\end{bmatrix}
\end{eqnarray}
</latex>
</para>
<para>
Input vector a represents the polynomial coefficients of this prediction error filter in descending powers of z.
<para>
<latex>
\begin{eqnarray}
A(z)=1+a(2)^{z-1}+.............+a(n+1)^{z-p}
\end{eqnarray}
</latex></para>
</para>
<para>[r,u] = rlevinson(a,efinal) returns upper triangular matrix U from the UDU* decomposition
</para>
<latex>
\begin{eqnarray}
R^{-1}=UE^{-1}U*
\end{eqnarray}
</latex>
<para>
<latex>
</latex>
</para>
</refsection>
<refsection>
<title>Examples</title>
<programlisting role="example"><![CDATA[
X=[1 6/7 5/7 8/7 3/7 6/7];
[R U kr e] = rlevinson(X, 0.3)
]]></programlisting>
</refsection>
<refsection>
<title>See also</title>
<simplelist type="inline">
<member><link linkend="levinson">abs</link></member>
<member><link linkend="levdown">mean</link></member>
</simplelist>
</refsection>
</refentry>
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