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+ − 2 "http://www.w3.org/TR/1999/REC-html401-19991224/loose.dtd">
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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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+ − 8 <meta http-equiv="Content-Type" content=
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+ − 10
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+ − 11 <title>Data gap filling (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 "spikeclean.html"><img src="b_prev.gif" border="0" align=
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+ − 30 "bottom" alt="Spikes reduction in data"></a> <a href=
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+ − 31 "whitening.html"><img src="b_next.gif" border="0" align=
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+ − 32 "bottom" alt="Noise whitening"></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>Data gap filling</h1>
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+ − 37 <hr>
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+ − 38
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+ − 39 <p>
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+ − 40 <h2>Description</h2>
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+ − 41 <p>
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+ − 42 Gaps in data can be filled with <i>interpolated</i> data if desired. LTPDA <a href="matlab:doc('ao/gapfilling')"><tt>gapfilling</tt></a> joins two AOs
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+ − 43 by means of a segment of <i>synthetic</i> data. This segment is calculated from the two input AOs. Two different
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+ − 44 methods are possible to fill the gap in the data: <tt>linear</tt> and <tt>spline</tt>. The former fills the data
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+ − 45 by means of a linear interpolation whereas the latter uses a smoother curve ---see examples below.
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+ − 46 <br>
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+ − 47 <br>
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+ − 48 <h2>Syntax</h2>
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+ − 49 </p>
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+ − 50 <div class="fragment"><pre>
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+ − 51 <br>
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+ − 52 b = gapfilling(a1, a2, pl)
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+ − 53 </pre>
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+ − 54 </div>
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+ − 55 <p>
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+ − 56 where <tt>a1</tt> and <tt>a2</tt> are the two segments to join.
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+ − 57 </p>
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+ − 58 <h2>Parameters</h2>
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+ − 59 The following parameters apply for this method:
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+ − 60 <ul>
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+ − 61 <li><tt>'method'</tt> - method used to interpolate missing data (see below for details)
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+ − 62 <ul>
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+ − 63 <li> <tt>'linear'</tt> (default option) </li>
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+ − 64 <li> <tt>'spline'</tt> </li>
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+ − 65 </ul>
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+ − 66 <li><tt>'addnoise'</tt> - with this option <i>noise</i> can be
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+ − 67 added to the interpolated data.
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+ − 68 This noise is defined as a <br>
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+ − 69 random variable with
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+ − 70 zero mean and variance equal to the high-frequency
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+ − 71 noise of the input AO. </li>
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+ − 72
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+ − 73 </ul>
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+ − 74 </p>
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+ − 75 <p>
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+ − 76 <h2>Algorithm</h2>
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+ − 77 </p>
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+ − 78 <p>
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+ − 79 <b>Linear :</b> The gap is filled using a linear approximation.
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+ − 80 </p>
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+ − 81 <p>
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+ − 82 <div align="center">
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+ − 83 <IMG src="images/spike50x.png" width="205" height="27" border="0">
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+ − 84 </div>
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+ − 85 </p>
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+ − 86 <br>
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+ − 87 the notation used in the previous expression is schematically explained in the figure below
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+ − 88 <br>
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+ − 89 <p>
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+ − 90 <div align="center">
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+ − 91 <img src="images/interplinear1.png" border="1" width="400px">
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+ − 92 </div>
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+ − 93 </p>
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+ − 94 <br>
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+ − 95 <br>
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+ − 96 <br>
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+ − 97 <p>
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+ − 98 <b>Spline :</b> a third order interpolation is used in this case
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+ − 99 </p>
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+ − 100 <p>
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+ − 101 <br>
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+ − 102 <div align="center">
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+ − 103 <IMG src="images/spike51x.png" width="237" height="16" border="0">
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+ − 104 </div>
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+ − 105 <br>
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+ − 106 </p>
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+ − 107 <p>
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+ − 108 The parameters <tt>a</tt>, <tt>b</tt>, <tt>c</tt> and <tt>d</tt> are calculated
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+ − 109 by solving next system of equations:
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+ − 110 </p>
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+ − 111 <p>
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+ − 112 <div align="center">
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+ − 113 <IMG src="images/spike52x.png" width="205" height="117" border="0">
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+ − 114 </div>
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+ − 115 </p>
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+ − 116 <br>
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+ − 117 The result is schematically shown in the figure below
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+ − 118 <br>
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+ − 119 <p>
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+ − 120 <div align="center">
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+ − 121 <img src="images/interpspline.png" border="0" width="400px">
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+ − 122 </div>
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+ − 123 </p>
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+ − 124 <br>
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+ − 125 <p>
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+ − 126 <h2>Examples</h2>
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+ − 127 </p>
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+ − 128 <p>
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+ − 129 1. Missing data between two vectors of random data interpolated with the <tt>linear</tt> method.
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+ − 130 </p>
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+ − 131 <p>
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+ − 132 <div class="fragment"><pre>
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+ − 133 <br>
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+ − 134 x1 = ao(plist(<span class="string">'waveform'</span>,<span class="string">'noise'</span>,<span class="string">'nsecs'</span>,1e3,<span class="string">'fs'</span>,10,<span class="string">'name'</span>,<span class="string">'noise'</span>)); <span class="comment">% create an AO of random data</span>
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+ − 135 xs = split(x1,plist(<span class="string">'chunks'</span>,3)); <span class="comment">% split in three segments</span>
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+ − 136 pl = plist( <span class="string">'method'</span>, <span class="string"> 'linear'</span>, <span class="string">'addnoise'</span>, <span class="string">'no'</span>); <span class="comment">% linear gapfilling</span>
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+ − 137 y = gapfilling(xs(1),xs(3), pl); <span class="comment">% fill between 2nd and 3rd segment</span>
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+ − 138 iplot(x1,y)
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+ − 139 </pre>
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+ − 140 </div>
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+ − 141 </p>
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+ − 142 <p>
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+ − 143 <div align="center">
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+ − 144 <img src="images/gapexample1.png" border="1" >
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+ − 145 </div>
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+ − 146 </p>
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+ − 147 <br>
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+ − 148 <br>
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+ − 149 <p>
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+ − 150 2. Missing data between two data vectors interpolated with the <tt>spline</tt> method.
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+ − 151 </p>
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+ − 152 <p>
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+ − 153 <div class="fragment"><pre>
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+ − 154 <br>
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+ − 155 x1 = ao(plist(<span class="string">'tsfcn'</span>,<span class="string">'t.^1.8 + 1e6*rand(size(t))'</span>,<span class="string">'nsecs'</span>,5e3,<span class="string">'fs'</span>,10,<span class="string">'name'</span>,<span class="string">'noise'</span>)); <span class="comment">% create an AO</span>
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+ − 156 xs = split(x1,plist(<span class="string">'chunks'</span>,3)); <span class="comment">% split in three segments</span>
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+ − 157 pl = plist( <span class="string">'method'</span>, <span class="string"> 'spline'</span>, <span class="string">'addnoise'</span>, <span class="string">'no'</span>); <span class="comment">% cubic gapfilling</span>
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+ − 158 y = gapfilling(xs(1),xs(3), pl); <span class="comment">% fill between 2nd and 3rd segment</span>
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+ − 159 iplot(x1,y)
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+ − 160 </pre>
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+ − 161 </div>
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+ − 162 </p>
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+ − 163 <p>
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+ − 164 <div align="center">
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+ − 165 <img src="images/gapexample2.png" border="1">
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+ − 166 </div>
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+ − 167 </p>
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+ − 168
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+ − 169
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+ − 170
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+ − 171
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+ − 172 </p>
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+ − 173
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+ − 174 <br>
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+ − 175 <br>
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+ − 176 <table class="nav" summary="Navigation aid" border="0" width=
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+ − 177 "100%" cellpadding="0" cellspacing="0">
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+ − 178 <tr valign="top">
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+ − 179 <td align="left" width="20"><a href="spikeclean.html"><img src=
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+ − 180 "b_prev.gif" border="0" align="bottom" alt=
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+ − 181 "Spikes reduction in data"></a> </td>
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+ − 182
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+ − 183 <td align="left">Spikes reduction in data</td>
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+ − 187 <td align="right">Noise whitening</td>
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+ − 190 "whitening.html"><img src="b_next.gif" border="0" align=
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+ − 193 </table><br>
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+ − 195 <p class="copy">©LTP Team</p>
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+ − 196 </body>
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+ − 197 </html>