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                <td>
                  <a href="#WhitenIntro">Introduction</a>
                </td>
                <td>Noise whitening in LTPDA.</td>
              </tr>
              <tr valign="top">
                <td>
                  <a href="#WhitenAlgo">Algorithm</a>
                </td>
                <td>Whitening Algorithms.</td>
              </tr>
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                <td>
                  <a href="#Whiten1D">1D data</a>
                </td>
                <td>Whitening noise in one-dimensional data.</td>
              </tr>
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                <td>
                  <a href="#Whiten2D">2D data</a>
                </td>
                <td>Whitening noise in two-dimensional data.</td>
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  <!-- ===== Intro ====== -->
  <h2><a name="WhitenIntro">Noise whitening in LTPDA</a></h2>
  <p>
    A random process <i>w(t)</i> is considered white if it is zero mean 
    and uncorrelated:
  </p>
  <p>
    <IMG src="images/whitening01.gif" align="center" border="0">
  </p>
  <p>
    As a consequence, the power spectral density of a white process is a 
    constant at every frequency:
  </p>
  <p>
    <IMG src="images/whitening02.gif" align="center" border="0">
  </p>
  <p>
    In other words, The power per unit of frequency associated to a white noise
    process is uniformly distributed on the whole available frequency range.
    An example is reported in figure 1.
  </p>
  
  <div align="center">
    <table border="0">
      <caption align="bottom">
        <b> Figure 1:</b> Power spectral density (estimated with the welch method)
        of a gaussian unitary variance zero mean random process. 
        The process <i>w(t)</i> is assumed to have 
        physical units of <tt>m</tt> therefore its power spectral density has
        physical units of <tt>m^2/Hz</tt>. Note that the power spectral density
        average value is 2 instead of the expected 1 (unitary variance process)
        since we calculated one-sided power spectral density.
      </caption>
      <tr>
        <td>
          <IMG src="images/whitening03.png" align="center" border="0">
        </td>
      </tr>
    </table>
  </div>
  
  <p>
  </p>
  
  <p>
    A non-white (colored) noise process is instead characterized by a given
    distribution of the power per unit of frequency along the available frequency
    bandwidth. <br>
    Whitening operation on a given non-white process corresponds to force 
    such a process to satisfy the conditions described above for a white process.
  </p>
  <p>
    In LTPDA there are different methods for noise whitening:
    <ul>
      <li> <a href="matlab:doc('ao/buildWhitener1D')"> buildWhitener1D.m</a> 
      <li> <a href="matlab:doc('ao/whiten1D')"> whiten1D.m</a> 
      <li> <a href="matlab:doc('ao/firwhiten')">firwhiten.m</a>
      <li> <a href="matlab:doc('ao/whiten2D')">whiten2D.m</a>
    </ul>
    They accept time series analysis objects as an input and they output noise 
    whitening filters or whitened time series analysis objects.
  </p>
  
  <!-- ===== Algorithm ====== -->
  <h2><a name="WhitenAlgo">Whitening Algorithms</a></h2>

  <h3>buildWhitener1D</h3>
  <p>
    <tt>buildWhitener1D</tt> performs a frequency domain identification of the system
    in order to extract the proper whitening filter. The function needs a model
    for the one-sided power spectral density of the given process. If no model
    is provided, the power spectral density of the process is calculated with
    the <a href="matlab:doc('ao/psd')">psd</a> and <a href="matlab:doc('ao/bin_data')">bin_data</a> algorithm. <br>
    <ol>
      <li> The inverse of the square root of the model for the power spectral 
      density is fit in z-domain in order to determine a whitening 
      filter.
      <li> Unstable poles are removed by an all-pass stabilization procedure.
      <li> Whitening filter is provided at the output.
    </ol>
  </p>
  
  <h3>Whiten1D</h3>
  <p>
    <tt>whiten1D</tt> implements the same functionality of <tt>buildWhitener1D</tt>
    but it adds the filtering step so input data are filtered with the identified filter
    internally to the method.
  </p>
  
  <h3>Firwhiten</h3>
  <p>
    <tt>firwhiten</tt> whitens the input time-series by building an FIR
    whitening filter. <br>
    <ol>
      <li> Make ASD of time-series.
      <li> Perform running median to get noise-floor estimate <a href="matlab:doc('ao/smoother')">ao/smoother</a>.
      <li> Invert noise-floor estimate.
      <li> Call <a href="matlab:doc('mfir')">mfir()</a> on noise-floor estimate to produce whitening filter.
      <li> Filter data.
    </ol>
  </p>
  
  <h3>Whiten2D</h3>
  <p>
    <tt>whiten2D</tt> whitens cross-correlated time-series. Whitening
    filters are constructed by a fitting procedure to the models
    for the corss-spectral matrix provided.
    In order to work with <tt>whiten2D</tt> you must provide
    a model (frequency series analysis objects) for the cross-spectral density
    matrix of the process.
    <ol>
      <li> Whitening filters frequency response is calculated by the 
      eigendecomposition of the cross-spectral matrix.
      <li> Calculated responses are fit in z-domain in order to identify
      corresponding autoregressive moving average filters.
      <li> Input time-series is filtered. The filtering process corresponds to:<br>
      w(1) = Filt11(a(1)) + Filt12(a(2))<br>
      w(2) = Filt21(a(1)) + Filt22(a(2))
    </ol>
  </p>
  
  
  <!-- ===== 1D Examples ====== -->
  <h2><a name="buildWhitener1D">Whitening noise in one-dimensional data</a></h2>
  <p>
    We can now test an example of the one-dimensinal whitening filters capabilities.
    With the following commands we can generate a colored noise data series
    for parameters description please refer to the
    <a href="matlab:doc('ao')">ao</a>,
    <a href="matlab:doc('miir')">miir</a> and
    <a href="matlab:doc('ao/filter')">filter</a>
    documentation pages.
  </p>
  <div class="fragment"><pre>
    
    fs = 1; <span class="comment">% sampling frequency</span>
    
    <span class="comment">% Generate gaussian white noise</span>
    pl = plist(<span class="string">'tsfcn'</span>, <span class="string">'randn(size(t))'</span>, ...
      <span class="string">'fs'</span>, fs, ...
      <span class="string">'nsecs'</span>, 1e5, ...
      <span class="string">'yunits'</span>, <span class="string">'m'</span>);
    a = ao(pl);
    
    <span class="comment">% Get a coloring filter</span>
    pl = plist(<span class="string">'type'</span>, <span class="string">'bandpass'</span>, ...
      <span class="string">'fs'</span>, fs, ...
      <span class="string">'order'</span>, 3, ...
      <span class="string">'gain'</span>, 1, ...
      <span class="string">'fc'</span>, [0.03 0.1]);
    ft = miir(pl);
    
    <span class="comment">% Coloring noise</span>
    af = filter(a, ft);
    
  </pre></div>
  
  <p>
    Now we can try to white colored noise.
  </p>

  
  <h3>buildWhitener1D</h3>
  <p>
    If you want to try <tt>buildWhitener1D</tt> to get a whitening filter for
    the present colored noise, you can try the following code. Please refer to the
    <a href="matlab:doc('ao/buildWhitener1D')">buildWhitener1D</a> documentation page
    for the meaning of any parameter. The result of the whitening procedure 
    is reported in figure 2.
  </p>
  
  <div class="fragment"><pre>
      
    pl = plist(...
      <span class="string">'MaxIter'</span>, 30, ...
      <span class="string">'MinOrder'</span>, 9, ...
      <span class="string">'MaxOrder'</span>, 15, ...
      <span class="string">'FITTOL'</span>, 5e-2);
  
    wfil = buildWhitener1D(af,pl);

    aw = filter(af,wfil);
    
  </pre></div>
  
  <div align="center">
    <table border="0">
      <caption align="bottom">
        <b> Figure 2:</b> Power spectral density (estimated with the welch method)
        of colored and whitened processes.
      </caption>
      <tr>
        <td>
          <IMG src="images/whitening04.png" align="center" border="0">
        </td>
      </tr>
    </table>
  </div>
  
  
  <h3>Firwhiten</h3>
  <p>
    As an alternative you can try <tt>firwhiten</tt> to whiten the present 
    colored noise. Please refer to the
    <a href="matlab:doc('ao/firwhiten')">firwhiten</a> documentation page
    for the meaning of any parameter. The result of the whitening procedure 
    is reported in figure 3.
  </p>
  
  <div class="fragment"><pre>
      
    pl = plist(...
      <span class="string">'Ntaps'</span>, 5000, ...
      <span class="string">'Nfft'</span>, 1e5, ...
      <span class="string">'BW'</span>, 5);
  
    aw = firwhiten(af, pl);
    
  </pre></div>
  
  <div align="center">
    <table border="0">
      <caption align="bottom">
        <b> Figure 3:</b> Power spectral density (estimated with the welch method)
        of colored and whitened processes.
      </caption>
      <tr>
        <td>
          <IMG src="images/whitening05.png" align="center" border="0">
        </td>
      </tr>
    </table>
  </div>
  
  
  <!-- ===== 2D Examples ====== -->
  <h2><a name="Whiten2D">Whitening noise in two-dimensional data</a></h2>
  
  <p>
    We consider now the problem of whitening cross correlated data series.
    As a example we consider a typical couple of x-dynamics LTP data series.
    <tt>a1</tt> and <tt>a2</tt> are interferometer output noise data series.
    In oreder to whiten data we must input a frequency response model of the 
    cross spectral matrix of the cross-correlated process.
  </p>
  <p>
    <IMG src="images/whitening10.gif" align="center" border="0">
  </p>
  <p>
    Refer to <a href="matlab:doc('ao/firwhiten')">whiten2D</a> documentation page
    for the meaning of any parameter.
  </p>
  
  <div class="fragment"><pre>
      
    pl = plist(...
      <span class="string">'csd11'</span>, mod11, ...
      <span class="string">'csd12'</span>, mod12, ...
      <span class="string">'csd21'</span>, mod21, ...
      <span class="string">'csd22'</span>, mod22, ...
      <span class="string">'MaxIter'</span>, 75, ...
      <span class="string">'PoleType'</span>, 3, ...
      <span class="string">'MinOrder'</span>, 20, ...
      <span class="string">'MaxOrder'</span>, 40, ...
      <span class="string">'Weights'</span>, 2, ...
      <span class="string">'Plot'</span>, false,...
      <span class="string">'Disp'</span>, false,...
      <span class="string">'MSEVARTOL'</span>, 1e-2,...
      <span class="string">'FITTOL'</span>, 1e-3);
  
    [aw1,aw2] = whiten2D(a1,a2,pl);
    
  </pre></div>
  
  <div align="center">
    <table border="0">
      <caption align="bottom">
        <b> Figure 4:</b> Power spectral density of the noisy data series
        before (left) and after (right) the whitening.
      </caption>
      <tr>
        <td>
          <IMG src="images/whitening06.png" align="center" border="0">
        </td>
        <td>
          <IMG src="images/whitening07.png" align="center" border="0">
        </td>
      </tr>
    </table>
  </div>
  
  <div align="center">
    <table border="0">
      <caption align="bottom">
        <b> Figure 5:</b> Real (left) and Imaginary (right) part of the 
        <a href="matlab:doc('ao/cohere')">coherence</a> function. 
        Blue line refers to theoretical expectation for colored noise data.
        Red line refers to calculated values for colored noise data.
        Green line refers to calculated values for whitened noise data.
        
      </caption>
      <tr>
        <td>
          <IMG src="images/whitening08.png" align="center" border="0">
        </td>
        <td>
          <IMG src="images/whitening09.png" align="center" border="0">
        </td>
      </tr>
    </table>
  </div>