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<?xml version="1.0" encoding="UTF-8"?>
<!--
* Scilab ( http://www.scilab.org/ ) - This file is part of Scilab
* Copyright (C) 2006-2008 - INRIA
*
* This file must be used under the terms of the CeCILL.
* This source file is licensed as described in the file COPYING, which
* you should have received as part of this distribution. The terms
* are also available at
* http://www.cecill.info/licences/Licence_CeCILL_V2-en.txt
*
-->
<refentry xmlns="http://docbook.org/ns/docbook" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:svg="http://www.w3.org/2000/svg" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:db="http://docbook.org/ns/docbook" version="5.0-subset Scilab" xml:lang="en" xml:id="sfact">
<refnamediv>
<refname>sfact</refname>
<refpurpose> discrete time spectral factorization</refpurpose>
</refnamediv>
<refsynopsisdiv>
<title>Calling Sequence</title>
<synopsis>F=sfact(P)</synopsis>
</refsynopsisdiv>
<refsection>
<title>Arguments</title>
<variablelist>
<varlistentry>
<term>P</term>
<listitem>
<para>real polynomial matrix</para>
</listitem>
</varlistentry>
</variablelist>
</refsection>
<refsection>
<title>Description</title>
<para>
Finds <literal>F</literal>, a spectral factor of
<literal>P</literal>. <literal>P</literal> is a polynomial matrix such that
each root of <literal>P</literal> has a mirror image w.r.t the unit
circle. Problem is singular if a root is on the unit circle.
</para>
<para>
<literal>sfact(P)</literal> returns a polynomial matrix
<literal>F(z)</literal> which is antistable and such that
</para>
<para>
<literal>P = F(z)* F(1/z) *z^n</literal>
</para>
<para>
For scalar polynomials a specific algorithm is implemented.
Algorithms are adapted from Kucera's book.
</para>
</refsection>
<refsection>
<title>Examples</title>
<programlisting role="example"><![CDATA[
//Simple polynomial example
z=poly(0,'z');
p=(z-1/2)*(2-z)
w=sfact(p);
w*numer(horner(w,1/z))
//matrix example
F1=[z-1/2,z+1/2,z^2+2;1,z,-z;z^3+2*z,z,1/2-z];
P=F1*gtild(F1,'d'); //P is symmetric
F=sfact(P)
roots(det(P))
roots(det(gtild(F,'d'))) //The stable roots
roots(det(F)) //The antistable roots
clean(P-F*gtild(F,'d'))
//Example of continuous time use
s=poly(0,'s');
p=-3*(s+(1+%i))*(s+(1-%i))*(s+0.5)*(s-0.5)*(s-(1+%i))*(s-(1-%i));p=real(p);
//p(s) = polynomial in s^2 , looks for stable f such that p=f(s)*f(-s)
w=horner(p,(1-s)/(1+s)); // bilinear transform w=p((1-s)/(1+s))
wn=numer(w); //take the numerator
fn=sfact(wn);f=numer(horner(fn,(1-s)/(s+1))); //Factor and back transform
f=f/sqrt(horner(f*gtild(f,'c'),0));f=f*sqrt(horner(p,0)); //normalization
roots(f) //f is stable
clean(f*gtild(f,'c')-p) //f(s)*f(-s) is p(s)
]]></programlisting>
</refsection>
<refsection role="see also">
<title>See Also</title>
<simplelist type="inline">
<member>
<link linkend="gtild">gtild</link>
</member>
<member>
<link linkend="fspecg">fspecg</link>
</member>
</simplelist>
</refsection>
</refentry>
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