scilab/modules/signal_processing/help/en_US/correlation_convolution/convol.xml

   1 <?xml version="1.0" encoding="UTF-8"?>
   2 <refentry 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:db="http://docbook.org/ns/docbook" xmlns:scilab="http://www.scilab.org" xml:id="convol" xml:lang="en">
   3     <refnamediv>
   4         <refname>convol</refname>
   5         <refpurpose>convolution</refpurpose>
   6     </refnamediv>
   7     <refsynopsisdiv>
   8         <title>Syntax</title>
   9         <synopsis>
  10             [y]=convol(h,x)
  11             [y,e1]=convol(h,x,e0)
  12         </synopsis>
  13     </refsynopsisdiv>
  14     <refsection>
  15         <title>Arguments</title>
  16         <variablelist>
  17             <varlistentry>
  18                 <term>h</term>
  19                 <listitem>
  20                     <para>a vector, first input sequence ("short" one) </para>
  21                 </listitem>
  22             </varlistentry>
  23             <varlistentry>
  24                 <term>x</term>
  25                 <listitem>
  26                     <para>a vector, second input sequence ( "long" one)</para>
  27                 </listitem>
  28             </varlistentry>
  29             <varlistentry>
  30                 <term>e0</term>
  31                 <listitem>
  32                     <para>a vector,old tail to overlap add (not used in first
  33                         call)
  34                     </para>
  35                 </listitem>
  36             </varlistentry>
  37             <varlistentry>
  38                 <term>y</term>
  39                 <listitem>
  40                     <para>a vector, the convolution. </para>
  41                 </listitem>
  42             </varlistentry>
  43             <varlistentry>
  44                 <term>e1</term>
  45                 <listitem>
  46                     <para>new tail to overlap add (not used in last call)</para>
  47                 </listitem>
  48             </varlistentry>
  49         </variablelist>
  50     </refsection>
  51     <refsection>
  52         <title>Description</title>
  53         <para>
  54             Calculates the convolution <literal>y= h*x</literal> of two discrete
  55             sequences by using the fft. The convolution is defined as follows:
  56         </para>
  57         <para>
  58             <inlinemediaobject>
  59                 <imageobject>
  60                     <imagedata>
  61                         <mml:math>
  62                             <mml:semantics>
  63                                 <mml:mrow>
  64                                     <mml:msub>
  65                                         <mml:mi>y</mml:mi>
  66                                         <mml:mi>k</mml:mi>
  67                                     </mml:msub>
  68                                     <mml:mo mml:stretchy="false">=</mml:mo>
  69                                     <mml:mrow>
  70                                         <mml:mrow>
  71                                             <mml:msub>
  72                                                 <mml:mo mml:stretchy="false">∑</mml:mo>
  73                                                 <mml:mi>j</mml:mi>
  74                                             </mml:msub>
  75                                             <mml:msub>
  76                                                 <mml:mi>h</mml:mi>
  77                                                 <mml:mi>j</mml:mi>
  78                                             </mml:msub>
  79                                         </mml:mrow>
  80                                         <mml:mo mml:stretchy="false">∗</mml:mo>
  81                                         <mml:msub>
  82                                             <mml:mi>x</mml:mi>
  83                                             <mml:mrow>
  84                                                 <mml:mrow>
  85                                                     <mml:mi>k</mml:mi>
  86                                                     <mml:mo mml:stretchy="false">+</mml:mo>
  87                                                     <mml:mn>1</mml:mn>
  88                                                 </mml:mrow>
  89                                                 <mml:mo mml:stretchy="false">−</mml:mo>
  90                                                 <mml:mi>j</mml:mi>
  91                                             </mml:mrow>
  92                                         </mml:msub>
  93                                     </mml:mrow>
  94                                 </mml:mrow>
  95                                 <mml:annotation mml:encoding="StarMath 5.0">y_k=sum_j h_j*x_{k+1-j}
  96                                 </mml:annotation>
  97                             </mml:semantics>
  98                         </mml:math>
  99                     </imagedata>
 100                 </imageobject>
 101             </inlinemediaobject>
 102         </para>
 103         <para>Overlap add method can be used.</para>
 104         <para>USE OF OVERLAP ADD METHOD: For
 105             <literal>x=[x1,x2,...,xNm1,xN]</literal> First call is
 106             <literal>[y1,e1]=convol(h,x1);</literal> Subsequent calls :
 107             <literal>[yk,ek]=convol(h,xk,ekm1)</literal>; Final call :
 108             <literal>[yN]=convol(h,xN,eNm1);</literal> Finally
 109             <literal>y=[y1,y2,...,yNm1,yN]</literal>.
 110         </para>
 111         <para>The algorithm based on the convolution definition is
 112             implemented for polynomial
 113             product: <literal>y=convol(h,x)</literal> is equivalent
 114             to <literal>y=coeff(poly(h,'z','c')*poly(x,'z','c')</literal> but
 115             much more efficient if <literal>x</literal> is a "long" array.
 116         </para>
 117     </refsection>
 118     <refsection>
 119         <title>Examples</title>
 120         <programlisting role="example"><![CDATA[
 121 x=1:3;
 122 h1=[1,0,0,0,0];h2=[0,1,0,0,0];h3=[0,0,1,0,0];
 123 x1=convol(h1,x),x2=convol(h2,x),x3=convol(h3,x),
 124 convol(h1+h2+h3,x)
 125 p1=poly(x,'x','coeff')
 126 p2=poly(h1+h2+h3,'x','coeff')
 127 p1*p2
 128  ]]></programlisting>
 129     </refsection>
 130     <refsection role="see also">
 131         <title>See also</title>
 132         <simplelist type="inline">
 133             <member>
 134                 <link linkend="corr">corr</link>
 135             </member>
 136             <member>
 137                 <link linkend="fft">fft</link>
 138             </member>
 139             <member>
 140                 <link linkend="pspect">pspect</link>
 141             </member>
 142         </simplelist>
 143     </refsection>
 144 </refentry>