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author Daniele Nicolodi <nicolodi@science.unitn.it>
date Wed, 07 Dec 2011 17:29:47 +0100
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  <title>Data gap filling (LTPDA Toolbox)</title>
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  <h1 class="title"><a name="f3-12899" id="f3-12899"></a>Data gap filling</h1>
  <hr>
  
  <p>
	<h2>Description</h2>
<p>
  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
  by means of a segment of <i>synthetic</i> data. This segment is calculated from the two input AOs. Two different
  methods are possible to fill the gap in the data: <tt>linear</tt> and <tt>spline</tt>. The former fills the data
  by means of a linear interpolation whereas the latter uses a smoother curve ---see examples below.
  <br>
  <br>
  <h2>Syntax</h2>
</p>
<div class="fragment"><pre>
    <br>
    b = gapfilling(a1, a2, pl)
  </pre>
</div>
<p>
  where <tt>a1</tt> and <tt>a2</tt> are the two segments to join.
  </p>
  <h2>Parameters</h2>
  The following parameters apply for this method:
  <ul>
    <li><tt>'method'</tt> - method used to interpolate missing data (see below for details)
    <ul>
      <li> <tt>'linear'</tt> (default option) </li> 
      <li> <tt>'spline'</tt> </li>
    </ul>
    <li><tt>'addnoise'</tt> - with this option <i>noise</i> can be
      added to the interpolated data.
      This noise is defined as a <br>
      random variable with
      zero mean and variance equal to the high-frequency
    noise of the input AO. </li>
    
  </ul>
</p>
<p>
  <h2>Algorithm</h2>
</p>
<p>
  <b>Linear :</b> The gap is filled using a linear approximation.
</p>
<p>
  <div align="center">
    <IMG src="images/spike50x.png" width="205" height="27"  border="0">
  </div>
</p>
<br>
the notation used in the previous expression is schematically explained in the figure below
<br>
<p>
  <div align="center">
    <img src="images/interplinear1.png" border="1" width="400px">
  </div>
</p>
<br>
<br>
<br>
<p>
  <b>Spline :</b> a third order interpolation is used in this case
</p>
<p>
  <br>
  <div align="center">
    <IMG src="images/spike51x.png" width="237" height="16" border="0">
  </div>
  <br>
</p>
<p>
  The parameters <tt>a</tt>, <tt>b</tt>, <tt>c</tt> and <tt>d</tt> are calculated
  by solving next system of equations:
</p>
<p>
  <div align="center">
    <IMG src="images/spike52x.png" width="205" height="117" border="0">
  </div>
</p>
<br>
The result is schematically shown in the figure below
<br>
<p>
  <div align="center">
    <img src="images/interpspline.png" border="0" width="400px">
  </div>
</p>
<br>
<p>
  <h2>Examples</h2>
</p>
<p>
  1. Missing data between two vectors of random data interpolated with the <tt>linear</tt> method.
</p>
<p>
  <div class="fragment"><pre>
      <br>
  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>
  xs = split(x1,plist(<span class="string">'chunks'</span>,3));                              <span class="comment">% split in three segments</span>
  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>
  y = gapfilling(xs(1),xs(3), pl);                               <span class="comment">% fill between 2nd and 3rd segment</span>
  iplot(x1,y) 
    </pre>
  </div>
</p>
<p>
  <div align="center">
    <img src="images/gapexample1.png" border="1" >
  </div>
</p>
<br>
<br>
<p>
  2. Missing data between two data vectors interpolated with the <tt>spline</tt> method.
</p>
<p>
  <div class="fragment"><pre>
      <br>
  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>
  xs = split(x1,plist(<span class="string">'chunks'</span>,3));                               <span class="comment">% split in three segments</span>
  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>
  y = gapfilling(xs(1),xs(3), pl);                                <span class="comment">% fill between 2nd and 3rd segment</span>
  iplot(x1,y) 
    </pre>
  </div>
</p>
<p>
  <div align="center">
    <img src="images/gapexample2.png" border="1">
  </div>
</p>




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