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+ − 3
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+ − 4 <html lang="en">
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+ − 5 <head>
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+ − 6 <meta name="generator" content=
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+ − 7 "HTML Tidy for Mac OS X (vers 1st December 2004), see www.w3.org">
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+ − 10
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+ − 11 <title>Whitening noise (LTPDA Toolbox)</title>
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+ − 12 <link rel="stylesheet" href="docstyle.css" type="text/css">
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+ − 13 <meta name="generator" content="DocBook XSL Stylesheets V1.52.2">
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+ − 14 <meta name="description" content=
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+ − 15 "Presents an overview of the features, system requirements, and starting the toolbox.">
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+ − 16 </head>
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+ − 17
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+ − 18 <body>
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+ − 19 <a name="top_of_page" id="top_of_page"></a>
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+ − 20
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+ − 21 <p style="font-size:1px;"> </p>
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+ − 22
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+ − 23 <table class="nav" summary="Navigation aid" border="0" width=
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+ − 24 "100%" cellpadding="0" cellspacing="0">
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+ − 25 <tr>
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+ − 26 <td valign="baseline"><b>LTPDA Toolbox</b></td><td><a href="../helptoc.html">contents</a></td>
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+ − 27
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+ − 28 <td valign="baseline" align="right"><a href=
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+ − 29 "ltpda_training_topic_2_5.html"><img src="b_prev.gif" border="0" align=
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+ − 30 "bottom" alt="Remove trends from a time-series AO"></a> <a href=
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+ − 31 "ltpda_training_topic_2_7.html"><img src="b_next.gif" border="0" align=
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+ − 32 "bottom" alt="Select and find data from an AO"></a></td>
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+ − 33 </tr>
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+ − 34 </table>
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+ − 35
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+ − 36 <h1 class="title"><a name="f3-12899" id="f3-12899"></a>Whitening noise</h1>
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+ − 37 <hr>
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+ − 38
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+ − 39 <p>
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+ − 40 <p>
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+ − 41 The LTPDA toolbox offers various ways in which you could whiten data. Perhaps you know the whitening
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+ − 42 filter you want to use, in which case you can build the filter and filter the data. Alternatively, you
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+ − 43 may have a model for the spectral content of the data, in which case you can use the method <tt>ao/whiten1D</tt>
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+ − 44 if you are dealing with single, uncorrelated data streams, or <tt>ao/whiten2D</tt> if you have a pair of
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+ − 45 correlated data streams. You can also use <tt>ao/whiten1D</tt> in the case where you don't have a model for
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+ − 46 the spectral content of the data. In this case, the method
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+ − 47 calculates the spectrum of the data, re-bins the spectrum so to
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+ − 48 reduce the individual points fluctuations, and fits a model
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+ − 49 of the spectrum as a series of partial fractions z-domain filters.
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+ − 50 </p>
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+ − 51 <p>
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+ − 52 The whitening algorithms are highly configurable and accept a large number of parameters. The main ones that
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+ − 53 we will change from the defaults in the following examples are
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+ − 54 <table cellspacing="0" class="body" cellpadding="2" border="0" width="80%">
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+ − 55 <colgroup>
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+ − 56 <col width="25%"/>
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+ − 57 <col width="75%"/>
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+ − 58 </colgroup>
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+ − 59 <thead>
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+ − 60 <tr valign="top">
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+ − 61 <th class="categorylist">Key</th>
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+ − 62 <th class="categorylist">Description</th>
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+ − 63 </tr>
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+ − 64 </thead>
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+ − 65 <tbody>
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+ − 66 <!-- Key 'Plot' -->
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+ − 67 <tr valign="top">
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+ − 68 <td bgcolor="#f3f4f5">
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+ − 69 <p><tt>PLOT</tt></p>
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+ − 70 </td>
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+ − 71 <td bgcolor="#f3f4f5">
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+ − 72 <p>Plot the result of the fitting as it proceeds.</p>
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+ − 73 </td>
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+ − 74 </tr>
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+ − 75 <!-- Key 'MAXORDER' -->
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+ − 76 <tr valign="top">
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+ − 77 <td bgcolor="#f3f4f5">
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+ − 78 <p><tt>MAXORDER</tt></p>
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+ − 79 </td>
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+ − 80 <td bgcolor="#f3f4f5">
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+ − 81 <p>Specify the maximum allowed model order that can be fit.</p>
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+ − 82 </td>
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+ − 83 </tr>
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+ − 84 <!-- Key 'weights' -->
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+ − 85 <tr valign="top">
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+ − 86 <td bgcolor="#f3f4f5">
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+ − 87 <p><tt>WEIGHTS</tt></p>
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+ − 88 </td>
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+ − 89 <td bgcolor="#f3f4f5">
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+ − 90 <p>Choose the way the data is weighted in the fitting procedure.</p>
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+ − 91 </td>
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+ − 92 </tr>
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+ − 93 <!-- Key 'RMSVAR' -->
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+ − 94 <tr valign="top">
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+ − 95 <td bgcolor="#f3f4f5">
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+ − 96 <p><tt>RMSVAR</tt></p>
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+ − 97 </td>
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+ − 98 <td bgcolor="#f3f4f5">
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+ − 99 <p>Check if the variation of the RMS error is smaller than 10^(-b),
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+ − 100 where b is the value given in the plist.</p>
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+ − 101 </td>
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+ − 102 </tr>
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+ − 103 </tbody>
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+ − 104 </table>
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+ − 105 </p>
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+ − 106 <p>
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+ − 107 We will start by whitening some data using this last method, i.e., allowing <tt>whiten1D</tt> to determine
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+ − 108 the whitening filter from the data itself.
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+ − 109 </p>
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+ − 110 <p>
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+ − 111 The data we will whiten can be found in your data packet in the 'topic2' sub-directory.
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+ − 112 </p>
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+ − 113 <p>
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+ − 114 We start by loading the mat file:
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+ − 115 </p>
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+ − 116 <div class="fragment"><pre>
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+ − 117 a = ao(<span class="string">'topic2/whiten.mat'</span>);
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+ − 118 </pre>
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+ − 119 </div>
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+ − 120 <p>
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+ − 121 The AO stored in the variable <tt>a</tt> is a coloured noise time-series.
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+ − 122 Let's have a look at this times series using <tt>iplot</tt>.
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+ − 123 </p>
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+ − 124 <div class="fragment"><pre>
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+ − 125 >> iplot(a);
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+ − 126 </pre></div>
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+ − 127 <p>The result should be similar to: </p>
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+ − 128 <img src="images/ltpda_training_1/topic2/coloured.png" alt="coloured" border="1">
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+ − 129 <p>
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+ − 130 Before we can whiten the data, we have to define the parameter list for the whitening tool:
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+ − 131 </p>
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+ − 132 <div class="fragment"><pre>
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+ − 133 pl = plist(...
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+ − 134 <span class="string">'Plot'</span>, true, ...
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+ − 135 <span class="string">'MaxOrder'</span>, 9, ...
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+ − 136 <span class="string">'Weights'</span>, 2);</pre>
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+ − 137 </div>
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+ − 138 <p>
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+ − 139 Now we can call the whitening function <tt>whiten1D</tt> with our input AO, <tt>a</tt> and the
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+ − 140 parameter list <tt>pl</tt>:
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+ − 141 </p>
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+ − 142 <div class="fragment"><pre> >> aw = whiten1D(a,pl); </pre></div>
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+ − 143 <p>
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+ − 144 To compare the whitened data with the coloured noise we compute the power spectrum (for details see <a
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+ − 145 href="ltpda_training_topic_3_2.html"><tt>Power spectral density estimation</tt></a>):
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+ − 146 </p>
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+ − 147 <div class="fragment"><pre>
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+ − 148 awxx = aw.lpsd;
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+ − 149 axx = a.lpsd;
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+ − 150 </pre></div>
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+ − 151 <p>
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+ − 152 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>).
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+ − 153 </p>
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+ − 154 </pre> </div>
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+ − 155 <div class="fragment"><pre>
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+ − 156 iplot(axx, awxx);
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+ − 157 </pre>
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+ − 158 </div>
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+ − 159 <img src="images/ltpda_training_1/topic2/whiten.png" alt="white" border="1">
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+ − 160
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+ − 161 </p>
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+ − 162
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+ − 163 <br>
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+ − 164 <br>
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+ − 165 <table class="nav" summary="Navigation aid" border="0" width=
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+ − 166 "100%" cellpadding="0" cellspacing="0">
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+ − 167 <tr valign="top">
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+ − 168 <td align="left" width="20"><a href="ltpda_training_topic_2_5.html"><img src=
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+ − 169 "b_prev.gif" border="0" align="bottom" alt=
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+ − 170 "Remove trends from a time-series AO"></a> </td>
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+ − 171
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+ − 172 <td align="left">Remove trends from a time-series AO</td>
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+ − 173
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+ − 174 <td> </td>
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+ − 175
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+ − 176 <td align="right">Select and find data from an AO</td>
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+ − 177
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+ − 178 <td align="right" width="20"><a href=
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+ − 179 "ltpda_training_topic_2_7.html"><img src="b_next.gif" border="0" align=
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+ − 180 "bottom" alt="Select and find data from an AO"></a></td>
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+ − 181 </tr>
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+ − 182 </table><br>
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+ − 183
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+ − 184 <p class="copy">©LTP Team</p>
Daniele Nicolodi <nicolodi@science.unitn.it>
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+ − 185 </body>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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+ − 186 </html>