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author Daniele Nicolodi <nicolodi@science.unitn.it>
date Mon, 05 Dec 2011 16:20:06 +0100
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  <title>Whitening noise (LTPDA Toolbox)</title>
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  <p style="font-size:1px;">&nbsp;</p>

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      <td valign="baseline"><b>LTPDA Toolbox</b></td><td><a href="../helptoc.html">contents</a></td>

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  <h1 class="title"><a name="f3-12899" id="f3-12899"></a>Whitening noise</h1>
  <hr>
  
  <p>
	<p>
  The LTPDA toolbox offers various ways in which you could whiten data. Perhaps you know the whitening
  filter you want to use, in which case you can build the filter and filter the data. Alternatively, you
  may have a model for the spectral content of the data, in which case you can use the method <tt>ao/whiten1D</tt>
  if you are dealing with single, uncorrelated data streams, or <tt>ao/whiten2D</tt> if you have a pair of
  correlated data streams. You can also use <tt>ao/whiten1D</tt> in the case where you don't have a model for
  the spectral content of the data. In this case, the method
  calculates the spectrum of the data, re-bins the spectrum so to
  reduce the individual points fluctuations, and fits a model
  of the spectrum as a series of partial fractions z-domain filters.
</p>
<p>
  The whitening algorithms are highly configurable and accept a large number of parameters. The main ones that
  we will change from the defaults in the following examples are
  <table cellspacing="0" class="body" cellpadding="2" border="0" width="80%">
    <colgroup>
      <col width="25%"/>
      <col width="75%"/>
    </colgroup>
    <thead>
      <tr valign="top">
        <th class="categorylist">Key</th>
        <th class="categorylist">Description</th>
      </tr>
    </thead>
    <tbody>
      <!-- Key 'Plot' -->
      <tr valign="top">
        <td bgcolor="#f3f4f5">
          <p><tt>PLOT</tt></p>
        </td>
        <td bgcolor="#f3f4f5">
          <p>Plot the result of the fitting as it proceeds.</p>
        </td>
      </tr>
      <!-- Key 'MAXORDER' -->
      <tr valign="top">
        <td bgcolor="#f3f4f5">
          <p><tt>MAXORDER</tt></p>
        </td>
        <td bgcolor="#f3f4f5">
          <p>Specify the maximum allowed model order that can be fit.</p>
        </td>
      </tr>
      <!-- Key 'weights' -->
      <tr valign="top">
        <td bgcolor="#f3f4f5">
          <p><tt>WEIGHTS</tt></p>
        </td>
        <td bgcolor="#f3f4f5">
          <p>Choose the way the data is weighted in the fitting procedure.</p>
        </td>
      </tr>
      <!-- Key 'RMSVAR' -->
      <tr valign="top">
        <td bgcolor="#f3f4f5">
          <p><tt>RMSVAR</tt></p>
        </td>
        <td bgcolor="#f3f4f5">
          <p>Check if the variation of the RMS error is smaller than 10^(-b),
             where b is the value given in the plist.</p>
        </td>
      </tr>
    </tbody>
  </table>
</p>
<p>
  We will start by whitening some data using this last method, i.e., allowing <tt>whiten1D</tt> to determine
  the whitening filter from the data itself.
</p>
<p>
  The data we will whiten can be found in your data packet in the 'topic2' sub-directory.
</p>
<p>
  We start by loading the mat file:
</p>
<div class="fragment"><pre>
    a = ao(<span class="string">'topic2/whiten.mat'</span>);
  </pre>
</div>
<p>
  The AO stored in the variable <tt>a</tt> is a coloured noise time-series.
  Let's have a look at this times series using <tt>iplot</tt>.
</p>
<div class="fragment"><pre>
    >> iplot(a);
</pre></div>
<p>The result should be similar to: </p>
<img src="images/ltpda_training_1/topic2/coloured.png" alt="coloured" border="1">
<p>
  Before we can whiten the data, we have to define the parameter list for the whitening tool:
</p>
<div class="fragment"><pre>
    pl = plist(...
              <span class="string">'Plot'</span>, true, ...
              <span class="string">'MaxOrder'</span>, 9, ...
              <span class="string">'Weights'</span>, 2);</pre>
</div>
<p>
  Now we can call the whitening function <tt>whiten1D</tt> with our input AO, <tt>a</tt> and the
  parameter list <tt>pl</tt>:
</p>
<div class="fragment"><pre> >> aw = whiten1D(a,pl); </pre></div>
<p>
  To compare the whitened data with the coloured noise we compute the power spectrum (for details see <a
    href="ltpda_training_topic_3_2.html"><tt>Power spectral density estimation</tt></a>):
</p>
<div class="fragment"><pre>
    awxx = aw.lpsd;
    axx  = a.lpsd;
</pre></div>
<p>
  and finally plot our result in the frequency domain; in particular we plot the whitened data (<tt>awxx</tt>) compared to the coloured noise that was our input (<tt>axx</tt>).
</p>
</pre> </div>
<div class="fragment"><pre>
    iplot(axx, awxx);
  </pre>
</div>
<img src="images/ltpda_training_1/topic2/whiten.png" alt="white" border="1">

  </p>

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      <td align="left" width="20"><a href="ltpda_training_topic_2_5.html"><img src=
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      "Remove trends from a time-series AO"></a>&nbsp;</td>

      <td align="left">Remove trends from a time-series AO</td>

      <td>&nbsp;</td>

      <td align="right">Select and find data from an AO</td>

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