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+ − <!-- $Id: whitening_content.html,v 1.5 2011/04/11 14:24:19 luigi Exp $ -->
+ −
+ − <!-- ================================================== -->
+ − <!-- BEGIN CONTENT FILE -->
+ − <!-- ================================================== -->
+ − <!-- ===== link box: Begin ===== -->
+ − <p>
+ − <table border="1" width="80%">
+ − <tr>
+ − <td>
+ − <table border="0" cellpadding="5" class="categorylist" width="100%">
+ − <colgroup>
+ − <col width="37%"/>
+ − <col width="63%"/>
+ − </colgroup>
+ − <tbody>
+ − <tr valign="top">
+ − <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>
+ − <tr valign="top">
+ − <td>
+ − <a href="#Whiten1D">1D data</a>
+ − </td>
+ − <td>Whitening noise in one-dimensional data.</td>
+ − </tr>
+ − <tr valign="top">
+ − <td>
+ − <a href="#Whiten2D">2D data</a>
+ − </td>
+ − <td>Whitening noise in two-dimensional data.</td>
+ − </tr>
+ − </tbody>
+ − </table>
+ − </td>
+ − </tr>
+ − </table>
+ − </p>
+ − <!-- ===== link box: End ====== -->
+ −
+ −
+ − <!-- ===== 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>
+ −
+ −
+ −
+ −
+ −
+ −
+ −
+ −