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<?xml version="1.0" encoding="UTF-8"?>
<refentry xmlns="http://docbook.org/ns/docbook" xmlns:xlink="http://www.w3.org/1999/xlink"
xmlns:svg="http://www.w3.org/2000/svg" xmlns:ns4="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="pspect" xml:lang="en">
<refnamediv>
<refname>pspect</refname>
<refpurpose>two sided cross-spectral estimate between 2 discrete time signals using
the Welch's average periodogram method.
</refpurpose>
</refnamediv>
<refsynopsisdiv>
<title>Syntax</title>
<synopsis>
[sm [,cwp]]=pspect(sec_step,sec_leng,wtype,x [,y] [,wpar])
[sm [,cwp]]=pspect(sec_step,sec_leng,wtype,nx [,ny] [,wpar])
</synopsis>
</refsynopsisdiv>
<refsection>
<title>Arguments</title>
<variablelist>
<varlistentry>
<term>x</term>
<listitem>
<para>vector, the time-domain samples of the first signal.</para>
</listitem>
</varlistentry>
<varlistentry>
<term>y</term>
<listitem>
<para>
vector, the time-domain samples of the second signal. If <literal>y</literal>
is omitted it is supposed to be equal to <literal>x</literal>
(auto-correlation). If it is present, it must have the same number of
element than <literal>x</literal>.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>nx</term>
<listitem>
<para>
a scalar : the number of samples in the <literal>x</literal>
signal. In this case the segments of the <literal>x</literal> signal are loaded by a
user defined function named <literal>getx</literal> (see
below).
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>ny</term>
<listitem>
<para>a scalar : the number of samples in the
<literal>y</literal> signal. In this case the segments of
the y signal are loaded by a user defined function named
<literal>gety</literal> (see below). If present
<literal>ny</literal> must be equal to
<literal>nx</literal>.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>sec_step</term>
<listitem>
<para>
offset of each data window. The overlap <literal>D
</literal>
is given by sec_leng -sec_step<literal>. if
sec_step==sec_leng/2
</literal>
50% overlap is made. The
overlap
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>sec_leng</term>
<listitem>
<para>Number of points of the window.</para>
</listitem>
</varlistentry>
<varlistentry>
<term>wtype</term>
<listitem>
<para>The window type</para>
<itemizedlist>
<listitem>
<para>
<literal>'re'</literal>: rectangular
</para>
</listitem>
<listitem>
<para>
<literal>'tr'</literal>: triangular
</para>
</listitem>
<listitem>
<para>
<literal>'hm'</literal>: Hamming
</para>
</listitem>
<listitem>
<para>
<literal>'hn'</literal>: Hann
</para>
</listitem>
<listitem>
<para>
<literal>'kr'</literal>: Kaiser,in this case the wpar
argument must be given
</para>
</listitem>
<listitem>
<para>
<literal>'ch'</literal>: Chebyshev, in this case the wpar
argument must be given
</para>
</listitem>
</itemizedlist>
</listitem>
</varlistentry>
<varlistentry>
<term>wpar</term>
<listitem>
<para>
optional parameters for <literal>Kaiser and Chebyshev
windows:
</literal>
</para>
<itemizedlist>
<listitem>
<para>
'kr': <literal>wpar must be a strictly positive
number
</literal>
</para>
</listitem>
<listitem>
<para>
'ch': <literal>wpar</literal> must be a 2 element vector
<literal> [main_lobe_width,side_lobe_height]with
</literal>
<literal>0<main_lobe_width<.5</literal>, and
<literal>side_lobe_height>0</literal>
</para>
</listitem>
</itemizedlist>
</listitem>
</varlistentry>
<varlistentry>
<term>sm</term>
<listitem>
<para>
Two sided power spectral estimate in the interval <literal>[0,1]</literal> of the
normalized frequencies. It is a row array with <literal>sec_len</literal>
elements . The array is real in case of auto-correlation and
complex in case of cross-correlation.
</para>
<para>The associated normalized frequencies array is
<literal>linspace(0,1,sec_len)</literal>.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>cwp</term>
<listitem>
<para>unspecified Chebyshev window parameter in case of Chebyshev
windowing, or an empty matrix.
</para>
</listitem>
</varlistentry>
</variablelist>
</refsection>
<refsection>
<title>Description</title>
<para>Computes the cross-spectrum estimate of two signals
<literal>x</literal> and <literal>y</literal> if both are given and the
auto-spectral estimate of <literal>x</literal> otherwise. Spectral
estimate obtained using the modified periodogram method.
</para>
<para>
The cross spectrum of two signal <literal>x</literal> and <literal>y</literal> is defined to be
</para>
<para>
<inlinemediaobject>
<imageobject>
<imagedata>
<mml:math>
<mml:semantics>
<mml:mrow>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi mml:fontstyle="italic">xy</mml:mi>
</mml:msub>
<mml:mrow>
<mml:mrow>
<mml:mo mml:stretchy="false">(</mml:mo>
<mml:mo mml:stretchy="false">ω</mml:mo>
<mml:mo mml:stretchy="false">)</mml:mo>
</mml:mrow>
<mml:mo mml:stretchy="false">=</mml:mo>
<mml:mfrac>
<mml:mn>1</mml:mn>
<mml:mi>N</mml:mi>
</mml:mfrac>
</mml:mrow>
<mml:mrow>
<mml:mo mml:stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mrow>
<mml:munderover>
<mml:mo mml:stretchy="false">∑</mml:mo>
<mml:mrow>
<mml:mi>n</mml:mi>
<mml:mo mml:stretchy="false">=</mml:mo>
<mml:mn>0</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mi>N</mml:mi>
<mml:mo mml:stretchy="false">−</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:munderover>
<mml:mi>x</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mo mml:stretchy="false">(</mml:mo>
<mml:mi>n</mml:mi>
<mml:mo mml:stretchy="false">)</mml:mo>
</mml:mrow>
<mml:msup>
<mml:mi>e</mml:mi>
<mml:mrow>
<mml:mrow>
<mml:mo mml:stretchy="false">−</mml:mo>
<mml:mi>i</mml:mi>
</mml:mrow>
<mml:mo mml:stretchy="false">ω</mml:mo>
<mml:mi>n</mml:mi>
</mml:mrow>
</mml:msup>
</mml:mrow>
<mml:mo mml:stretchy="false">)</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mo mml:stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mrow>
<mml:munderover>
<mml:mo mml:stretchy="false">∑</mml:mo>
<mml:mrow>
<mml:mi>n</mml:mi>
<mml:mo mml:stretchy="false">=</mml:mo>
<mml:mn>0</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mi>N</mml:mi>
<mml:mo mml:stretchy="false">−</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:munderover>
<mml:mover mml:accent="true">
<mml:mi>y</mml:mi>
<mml:mo mml:stretchy="false">ˉ</mml:mo>
</mml:mover>
</mml:mrow>
<mml:mrow>
<mml:mo mml:stretchy="false">(</mml:mo>
<mml:mi>n</mml:mi>
<mml:mo mml:stretchy="false">)</mml:mo>
</mml:mrow>
<mml:msup>
<mml:mi>e</mml:mi>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mo mml:stretchy="false">ω</mml:mo>
<mml:mi>n</mml:mi>
</mml:mrow>
</mml:msup>
</mml:mrow>
<mml:mo mml:stretchy="false">)</mml:mo>
</mml:mrow>
</mml:mrow>
<mml:annotation mml:encoding="StarMath 5.0"> S_{xy}(%omega)={1}
over {N} (sum from{n=0} to{N-1} x(n)e^{-i %omega n}) ( sum
from{n=0} to{N-1} bar y(n)e^{i %omega n})
</mml:annotation>
</mml:semantics>
</mml:math>
</imagedata>
</imageobject>
</inlinemediaobject>
</para>
<para>The modified periodogram method of spectral estimation repeatedly
calculates the periodogram of windowed sub-sections of the data contained
in <literal>x</literal> and <literal>y</literal> . These periodograms are
then averaged together and normalized by an appropriate constant to obtain
the final spectral estimate. It is the averaging process which reduces the
variance in the estimate.
</para>
<para>
For batch processing, the <literal>x</literal> and
<literal>y</literal> data may be read segment by segment using the
<literal>getx </literal>and <literal>gety</literal> user defined
functions. These functions have the following syntax:
</para>
<para>
<literal>xk=getx(ns,offset)</literal> and
<literal>yk=gety(ns,offset)</literal> where <literal>ns</literal> is the
segment size and <literal>offset</literal> is the index of the first
element of the segment in the full signal.
</para>
</refsection>
<refsection>
<title>Reference</title>
<para>Oppenheim, A.V., and R.W. Schafer. Discrete-Time Signal Processing,
Upper Saddle River, NJ: Prentice-Hall, 1999
</para>
</refsection>
<refsection>
<title>Examples</title>
<programlisting role="example"><![CDATA[
rand('normal');rand('seed',0);
x=rand(1:1024-33+1);
//make low-pass filter with eqfir
nf=33;bedge=[0 .1;.125 .5];des=[1 0];wate=[1 1];
h=eqfir(nf,bedge,des,wate);
//filter white data to obtain colored data
h1=[h 0*ones(1:max(size(x))-1)];
x1=[x 0*ones(1:max(size(h))-1)];
hf=fft(h1,-1); xf=fft(x1,-1);y=real(fft(hf.*xf,1));
//plot magnitude of filter
h2=[h 0*ones(1:968)];hf2=fft(h2,-1);hf2=real(hf2.*conj(hf2));
hsize=max(size(hf2));fr=(1:hsize)/hsize;plot(fr,log(hf2));
//pspect example
sm=pspect(100,200,'tr',y);smsize=max(size(sm));fr=(1:smsize)/smsize;
plot(fr,log(sm));
rand('unif');
]]></programlisting>
</refsection>
<refsection role="see also">
<title>See also</title>
<simplelist type="inline">
<member>
<link linkend="cspect">cspect</link>
</member>
<member>
<link linkend="pspect">pspect</link>
</member>
<member>
<link linkend="mese">mese</link>
</member>
<member>
<link linkend="window">window</link>
</member>
</simplelist>
</refsection>
</refentry>
|
