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Daniele Nicolodi <nicolodi@science.unitn.it>
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Mon, 05 Dec 2011 16:20:06 +0100 (2011-12-05)
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+ − <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>
+ −
+ −
+ −