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| author | Daniele Nicolodi <nicolodi@science.unitn.it> |
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| date | Mon, 05 Dec 2011 16:20:06 +0100 |
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<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/1999/REC-html401-19991224/loose.dtd"> <html lang="en"> <head> <meta name="generator" content= "HTML Tidy for Mac OS X (vers 1st December 2004), see www.w3.org"> <meta http-equiv="Content-Type" content= "text/html; charset=us-ascii"> <title>Noise whitening (LTPDA Toolbox)</title> <link rel="stylesheet" href="docstyle.css" type="text/css"> <meta name="generator" content="DocBook XSL Stylesheets V1.52.2"> <meta name="description" content= "Presents an overview of the features, system requirements, and starting the toolbox."> </head> <body> <a name="top_of_page" id="top_of_page"></a> <p style="font-size:1px;"> </p> <table class="nav" summary="Navigation aid" border="0" width= "100%" cellpadding="0" cellspacing="0"> <tr> <td valign="baseline"><b>LTPDA Toolbox</b></td><td><a href="../helptoc.html">contents</a></td> <td valign="baseline" align="right"><a href= "gapfill.html"><img src="b_prev.gif" border="0" align= "bottom" alt="Data gap filling"></a> <a href= "sigproc.html"><img src="b_next.gif" border="0" align= "bottom" alt="Signal Processing in LTPDA"></a></td> </tr> </table> <h1 class="title"><a name="f3-12899" id="f3-12899"></a>Noise whitening</h1> <hr> <p> <!-- ================================================== --> <!-- 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> </p> <br> <br> <table class="nav" summary="Navigation aid" border="0" width= "100%" cellpadding="0" cellspacing="0"> <tr valign="top"> <td align="left" width="20"><a href="gapfill.html"><img src= "b_prev.gif" border="0" align="bottom" alt= "Data gap filling"></a> </td> <td align="left">Data gap filling</td> <td> </td> <td align="right">Signal Processing in LTPDA</td> <td align="right" width="20"><a href= "sigproc.html"><img src="b_next.gif" border="0" align= "bottom" alt="Signal Processing in LTPDA"></a></td> </tr> </table><br> <p class="copy">©LTP Team</p> </body> </html>