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author Daniele Nicolodi <nicolodi@science.unitn.it>
date Mon, 05 Dec 2011 16:20:06 +0100
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1 <h2>Description</h2>
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2 <p>
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3 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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4 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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5 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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6 by means of a linear interpolation whereas the latter uses a smoother curve ---see examples below.
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7 <br>
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8 <br>
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9 <h2>Syntax</h2>
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10 </p>
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11 <div class="fragment"><pre>
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12 <br>
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13 b = gapfilling(a1, a2, pl)
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14 </pre>
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15 </div>
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16 <p>
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17 where <tt>a1</tt> and <tt>a2</tt> are the two segments to join.
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18 </p>
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19 <h2>Parameters</h2>
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20 The following parameters apply for this method:
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21 <ul>
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22 <li><tt>'method'</tt> - method used to interpolate missing data (see below for details)
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23 <ul>
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24 <li> <tt>'linear'</tt> (default option) </li>
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25 <li> <tt>'spline'</tt> </li>
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26 </ul>
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27 <li><tt>'addnoise'</tt> - with this option <i>noise</i> can be
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28 added to the interpolated data.
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29 This noise is defined as a <br>
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30 random variable with
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31 zero mean and variance equal to the high-frequency
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32 noise of the input AO. </li>
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33
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34 </ul>
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35 </p>
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36 <p>
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37 <h2>Algorithm</h2>
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38 </p>
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39 <p>
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40 <b>Linear :</b> The gap is filled using a linear approximation.
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41 </p>
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42 <p>
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43 <div align="center">
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44 <IMG src="images/spike50x.png" width="205" height="27" border="0">
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45 </div>
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46 </p>
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47 <br>
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48 the notation used in the previous expression is schematically explained in the figure below
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49 <br>
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50 <p>
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51 <div align="center">
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52 <img src="images/interplinear1.png" border="1" width="400px">
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53 </div>
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54 </p>
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55 <br>
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56 <br>
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57 <br>
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58 <p>
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59 <b>Spline :</b> a third order interpolation is used in this case
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60 </p>
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61 <p>
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62 <br>
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63 <div align="center">
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64 <IMG src="images/spike51x.png" width="237" height="16" border="0">
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65 </div>
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66 <br>
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67 </p>
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68 <p>
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69 The parameters <tt>a</tt>, <tt>b</tt>, <tt>c</tt> and <tt>d</tt> are calculated
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70 by solving next system of equations:
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71 </p>
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72 <p>
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73 <div align="center">
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74 <IMG src="images/spike52x.png" width="205" height="117" border="0">
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75 </div>
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76 </p>
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77 <br>
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78 The result is schematically shown in the figure below
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79 <br>
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80 <p>
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81 <div align="center">
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82 <img src="images/interpspline.png" border="0" width="400px">
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83 </div>
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84 </p>
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85 <br>
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86 <p>
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87 <h2>Examples</h2>
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88 </p>
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89 <p>
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90 1. Missing data between two vectors of random data interpolated with the <tt>linear</tt> method.
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91 </p>
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92 <p>
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93 <div class="fragment"><pre>
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94 <br>
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95 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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96 xs = split(x1,plist(<span class="string">'chunks'</span>,3)); <span class="comment">% split in three segments</span>
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97 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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98 y = gapfilling(xs(1),xs(3), pl); <span class="comment">% fill between 2nd and 3rd segment</span>
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99 iplot(x1,y)
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100 </pre>
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101 </div>
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102 </p>
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103 <p>
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104 <div align="center">
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105 <img src="images/gapexample1.png" border="1" >
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106 </div>
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107 </p>
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108 <br>
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109 <br>
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110 <p>
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111 2. Missing data between two data vectors interpolated with the <tt>spline</tt> method.
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112 </p>
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113 <p>
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114 <div class="fragment"><pre>
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115 <br>
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116 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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117 xs = split(x1,plist(<span class="string">'chunks'</span>,3)); <span class="comment">% split in three segments</span>
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118 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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119 y = gapfilling(xs(1),xs(3), pl); <span class="comment">% fill between 2nd and 3rd segment</span>
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120 iplot(x1,y)
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121 </pre>
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122 </div>
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123 </p>
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124 <p>
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125 <div align="center">
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126 <img src="images/gapexample2.png" border="1">
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127 </div>
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128 </p>
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129
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