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author Daniele Nicolodi <nicolodi@science.unitn.it>
date Wed, 07 Dec 2011 17:24:36 +0100
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1 <!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"
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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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9 "text/html; charset=us-ascii">
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10
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11 <title>Noise whitening (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;">&nbsp;</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 "gapfill.html"><img src="b_prev.gif" border="0" align=
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30 "bottom" alt="Data gap filling"></a>&nbsp;&nbsp;&nbsp;<a href=
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31 "sigproc.html"><img src="b_next.gif" border="0" align=
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32 "bottom" alt="Signal Processing in LTPDA"></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>Noise whitening</h1>
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37 <hr>
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38
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39 <p>
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40
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41 <!-- ================================================== -->
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42 <!-- BEGIN CONTENT FILE -->
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43 <!-- ================================================== -->
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44 <!-- ===== link box: Begin ===== -->
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45 <p>
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46 <table border="1" width="80%">
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47 <tr>
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48 <td>
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49 <table border="0" cellpadding="5" class="categorylist" width="100%">
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50 <colgroup>
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51 <col width="37%"/>
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52 <col width="63%"/>
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53 </colgroup>
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54 <tbody>
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55 <tr valign="top">
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56 <td>
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57 <a href="#WhitenIntro">Introduction</a>
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58 </td>
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59 <td>Noise whitening in LTPDA.</td>
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60 </tr>
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61 <tr valign="top">
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62 <td>
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63 <a href="#WhitenAlgo">Algorithm</a>
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64 </td>
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65 <td>Whitening Algorithms.</td>
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66 </tr>
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67 <tr valign="top">
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68 <td>
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69 <a href="#Whiten1D">1D data</a>
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70 </td>
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71 <td>Whitening noise in one-dimensional data.</td>
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72 </tr>
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73 <tr valign="top">
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74 <td>
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75 <a href="#Whiten2D">2D data</a>
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76 </td>
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77 <td>Whitening noise in two-dimensional data.</td>
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78 </tr>
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79 </tbody>
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80 </table>
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81 </td>
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82 </tr>
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83 </table>
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84 </p>
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85 <!-- ===== link box: End ====== -->
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86
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87
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88 <!-- ===== Intro ====== -->
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89 <h2><a name="WhitenIntro">Noise whitening in LTPDA</a></h2>
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90 <p>
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91 A random process <i>w(t)</i> is considered white if it is zero mean
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92 and uncorrelated:
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93 </p>
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94 <p>
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95 <IMG src="images/whitening01.gif" align="center" border="0">
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96 </p>
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97 <p>
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98 As a consequence, the power spectral density of a white process is a
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99 constant at every frequency:
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100 </p>
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101 <p>
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102 <IMG src="images/whitening02.gif" align="center" border="0">
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103 </p>
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104 <p>
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105 In other words, The power per unit of frequency associated to a white noise
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106 process is uniformly distributed on the whole available frequency range.
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107 An example is reported in figure 1.
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108 </p>
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109
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110 <div align="center">
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111 <table border="0">
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112 <caption align="bottom">
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113 <b> Figure 1:</b> Power spectral density (estimated with the welch method)
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114 of a gaussian unitary variance zero mean random process.
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115 The process <i>w(t)</i> is assumed to have
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116 physical units of <tt>m</tt> therefore its power spectral density has
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117 physical units of <tt>m^2/Hz</tt>. Note that the power spectral density
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118 average value is 2 instead of the expected 1 (unitary variance process)
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119 since we calculated one-sided power spectral density.
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120 </caption>
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121 <tr>
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122 <td>
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123 <IMG src="images/whitening03.png" align="center" border="0">
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124 </td>
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125 </tr>
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126 </table>
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127 </div>
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128
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129 <p>
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130 </p>
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131
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132 <p>
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133 A non-white (colored) noise process is instead characterized by a given
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134 distribution of the power per unit of frequency along the available frequency
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135 bandwidth. <br>
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136 Whitening operation on a given non-white process corresponds to force
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137 such a process to satisfy the conditions described above for a white process.
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138 </p>
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139 <p>
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140 In LTPDA there are different methods for noise whitening:
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141 <ul>
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142 <li> <a href="matlab:doc('ao/buildWhitener1D')"> buildWhitener1D.m</a>
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143 <li> <a href="matlab:doc('ao/whiten1D')"> whiten1D.m</a>
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144 <li> <a href="matlab:doc('ao/firwhiten')">firwhiten.m</a>
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145 <li> <a href="matlab:doc('ao/whiten2D')">whiten2D.m</a>
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146 </ul>
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147 They accept time series analysis objects as an input and they output noise
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148 whitening filters or whitened time series analysis objects.
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149 </p>
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150
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151 <!-- ===== Algorithm ====== -->
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152 <h2><a name="WhitenAlgo">Whitening Algorithms</a></h2>
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153
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154 <h3>buildWhitener1D</h3>
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155 <p>
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156 <tt>buildWhitener1D</tt> performs a frequency domain identification of the system
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157 in order to extract the proper whitening filter. The function needs a model
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158 for the one-sided power spectral density of the given process. If no model
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159 is provided, the power spectral density of the process is calculated with
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160 the <a href="matlab:doc('ao/psd')">psd</a> and <a href="matlab:doc('ao/bin_data')">bin_data</a> algorithm. <br>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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161 <ol>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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162 <li> The inverse of the square root of the model for the power spectral
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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163 density is fit in z-domain in order to determine a whitening
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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164 filter.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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165 <li> Unstable poles are removed by an all-pass stabilization procedure.
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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166 <li> Whitening filter is provided at the output.
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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167 </ol>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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168 </p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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169
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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170 <h3>Whiten1D</h3>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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171 <p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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172 <tt>whiten1D</tt> implements the same functionality of <tt>buildWhitener1D</tt>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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173 but it adds the filtering step so input data are filtered with the identified filter
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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174 internally to the method.
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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175 </p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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176
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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177 <h3>Firwhiten</h3>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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178 <p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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179 <tt>firwhiten</tt> whitens the input time-series by building an FIR
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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180 whitening filter. <br>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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181 <ol>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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182 <li> Make ASD of time-series.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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183 <li> Perform running median to get noise-floor estimate <a href="matlab:doc('ao/smoother')">ao/smoother</a>.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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184 <li> Invert noise-floor estimate.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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185 <li> Call <a href="matlab:doc('mfir')">mfir()</a> on noise-floor estimate to produce whitening filter.
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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186 <li> Filter data.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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187 </ol>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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188 </p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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189
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Daniele Nicolodi <nicolodi@science.unitn.it>
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190 <h3>Whiten2D</h3>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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191 <p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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192 <tt>whiten2D</tt> whitens cross-correlated time-series. Whitening
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Daniele Nicolodi <nicolodi@science.unitn.it>
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193 filters are constructed by a fitting procedure to the models
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Daniele Nicolodi <nicolodi@science.unitn.it>
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194 for the corss-spectral matrix provided.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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195 In order to work with <tt>whiten2D</tt> you must provide
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Daniele Nicolodi <nicolodi@science.unitn.it>
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196 a model (frequency series analysis objects) for the cross-spectral density
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Daniele Nicolodi <nicolodi@science.unitn.it>
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197 matrix of the process.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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198 <ol>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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199 <li> Whitening filters frequency response is calculated by the
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Daniele Nicolodi <nicolodi@science.unitn.it>
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200 eigendecomposition of the cross-spectral matrix.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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201 <li> Calculated responses are fit in z-domain in order to identify
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Daniele Nicolodi <nicolodi@science.unitn.it>
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202 corresponding autoregressive moving average filters.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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203 <li> Input time-series is filtered. The filtering process corresponds to:<br>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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204 w(1) = Filt11(a(1)) + Filt12(a(2))<br>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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205 w(2) = Filt21(a(1)) + Filt22(a(2))
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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206 </ol>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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207 </p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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208
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Daniele Nicolodi <nicolodi@science.unitn.it>
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209
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Daniele Nicolodi <nicolodi@science.unitn.it>
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210 <!-- ===== 1D Examples ====== -->
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Daniele Nicolodi <nicolodi@science.unitn.it>
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211 <h2><a name="buildWhitener1D">Whitening noise in one-dimensional data</a></h2>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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212 <p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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213 We can now test an example of the one-dimensinal whitening filters capabilities.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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214 With the following commands we can generate a colored noise data series
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Daniele Nicolodi <nicolodi@science.unitn.it>
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215 for parameters description please refer to the
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Daniele Nicolodi <nicolodi@science.unitn.it>
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216 <a href="matlab:doc('ao')">ao</a>,
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Daniele Nicolodi <nicolodi@science.unitn.it>
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217 <a href="matlab:doc('miir')">miir</a> and
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Daniele Nicolodi <nicolodi@science.unitn.it>
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218 <a href="matlab:doc('ao/filter')">filter</a>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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219 documentation pages.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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220 </p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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221 <div class="fragment"><pre>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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222
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Daniele Nicolodi <nicolodi@science.unitn.it>
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223 fs = 1; <span class="comment">% sampling frequency</span>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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224
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Daniele Nicolodi <nicolodi@science.unitn.it>
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225 <span class="comment">% Generate gaussian white noise</span>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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226 pl = plist(<span class="string">'tsfcn'</span>, <span class="string">'randn(size(t))'</span>, ...
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Daniele Nicolodi <nicolodi@science.unitn.it>
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227 <span class="string">'fs'</span>, fs, ...
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Daniele Nicolodi <nicolodi@science.unitn.it>
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228 <span class="string">'nsecs'</span>, 1e5, ...
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Daniele Nicolodi <nicolodi@science.unitn.it>
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229 <span class="string">'yunits'</span>, <span class="string">'m'</span>);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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230 a = ao(pl);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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231
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Daniele Nicolodi <nicolodi@science.unitn.it>
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232 <span class="comment">% Get a coloring filter</span>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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233 pl = plist(<span class="string">'type'</span>, <span class="string">'bandpass'</span>, ...
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Daniele Nicolodi <nicolodi@science.unitn.it>
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234 <span class="string">'fs'</span>, fs, ...
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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235 <span class="string">'order'</span>, 3, ...
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Daniele Nicolodi <nicolodi@science.unitn.it>
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236 <span class="string">'gain'</span>, 1, ...
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Daniele Nicolodi <nicolodi@science.unitn.it>
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237 <span class="string">'fc'</span>, [0.03 0.1]);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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238 ft = miir(pl);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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239
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Daniele Nicolodi <nicolodi@science.unitn.it>
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240 <span class="comment">% Coloring noise</span>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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241 af = filter(a, ft);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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242
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Daniele Nicolodi <nicolodi@science.unitn.it>
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243 </pre></div>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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244
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Daniele Nicolodi <nicolodi@science.unitn.it>
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245 <p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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246 Now we can try to white colored noise.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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247 </p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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248
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Daniele Nicolodi <nicolodi@science.unitn.it>
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249
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Daniele Nicolodi <nicolodi@science.unitn.it>
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250 <h3>buildWhitener1D</h3>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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251 <p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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252 If you want to try <tt>buildWhitener1D</tt> to get a whitening filter for
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Daniele Nicolodi <nicolodi@science.unitn.it>
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253 the present colored noise, you can try the following code. Please refer to the
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Daniele Nicolodi <nicolodi@science.unitn.it>
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254 <a href="matlab:doc('ao/buildWhitener1D')">buildWhitener1D</a> documentation page
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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255 for the meaning of any parameter. The result of the whitening procedure
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
256 is reported in figure 2.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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257 </p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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258
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
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diff changeset
259 <div class="fragment"><pre>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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260
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Daniele Nicolodi <nicolodi@science.unitn.it>
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261 pl = plist(...
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Daniele Nicolodi <nicolodi@science.unitn.it>
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262 <span class="string">'MaxIter'</span>, 30, ...
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Daniele Nicolodi <nicolodi@science.unitn.it>
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263 <span class="string">'MinOrder'</span>, 9, ...
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Daniele Nicolodi <nicolodi@science.unitn.it>
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264 <span class="string">'MaxOrder'</span>, 15, ...
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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265 <span class="string">'FITTOL'</span>, 5e-2);
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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266
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Daniele Nicolodi <nicolodi@science.unitn.it>
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267 wfil = buildWhitener1D(af,pl);
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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268
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Daniele Nicolodi <nicolodi@science.unitn.it>
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269 aw = filter(af,wfil);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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270
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Daniele Nicolodi <nicolodi@science.unitn.it>
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271 </pre></div>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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272
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Daniele Nicolodi <nicolodi@science.unitn.it>
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273 <div align="center">
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Daniele Nicolodi <nicolodi@science.unitn.it>
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274 <table border="0">
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Daniele Nicolodi <nicolodi@science.unitn.it>
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275 <caption align="bottom">
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Daniele Nicolodi <nicolodi@science.unitn.it>
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276 <b> Figure 2:</b> Power spectral density (estimated with the welch method)
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Daniele Nicolodi <nicolodi@science.unitn.it>
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277 of colored and whitened processes.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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278 </caption>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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279 <tr>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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280 <td>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
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281 <IMG src="images/whitening04.png" align="center" border="0">
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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282 </td>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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283 </tr>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
284 </table>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
285 </div>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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286
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Daniele Nicolodi <nicolodi@science.unitn.it>
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287
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
288 <h3>Firwhiten</h3>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
289 <p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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290 As an alternative you can try <tt>firwhiten</tt> to whiten the present
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Daniele Nicolodi <nicolodi@science.unitn.it>
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291 colored noise. Please refer to the
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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292 <a href="matlab:doc('ao/firwhiten')">firwhiten</a> documentation page
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
293 for the meaning of any parameter. The result of the whitening procedure
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
294 is reported in figure 3.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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diff changeset
295 </p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
296
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
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diff changeset
297 <div class="fragment"><pre>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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298
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Daniele Nicolodi <nicolodi@science.unitn.it>
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299 pl = plist(...
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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300 <span class="string">'Ntaps'</span>, 5000, ...
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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301 <span class="string">'Nfft'</span>, 1e5, ...
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Daniele Nicolodi <nicolodi@science.unitn.it>
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302 <span class="string">'BW'</span>, 5);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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303
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Daniele Nicolodi <nicolodi@science.unitn.it>
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diff changeset
304 aw = firwhiten(af, pl);
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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305
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Daniele Nicolodi <nicolodi@science.unitn.it>
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306 </pre></div>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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307
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
308 <div align="center">
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
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diff changeset
309 <table border="0">
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Daniele Nicolodi <nicolodi@science.unitn.it>
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diff changeset
310 <caption align="bottom">
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
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diff changeset
311 <b> Figure 3:</b> Power spectral density (estimated with the welch method)
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
312 of colored and whitened processes.
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
313 </caption>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
314 <tr>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
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diff changeset
315 <td>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
316 <IMG src="images/whitening05.png" align="center" border="0">
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
317 </td>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
318 </tr>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
319 </table>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
320 </div>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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321
f0afece42f48 Import.
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322
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323 <!-- ===== 2D Examples ====== -->
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324 <h2><a name="Whiten2D">Whitening noise in two-dimensional data</a></h2>
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325
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326 <p>
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327 We consider now the problem of whitening cross correlated data series.
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328 As a example we consider a typical couple of x-dynamics LTP data series.
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329 <tt>a1</tt> and <tt>a2</tt> are interferometer output noise data series.
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330 In oreder to whiten data we must input a frequency response model of the
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331 cross spectral matrix of the cross-correlated process.
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332 </p>
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333 <p>
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334 <IMG src="images/whitening10.gif" align="center" border="0">
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335 </p>
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336 <p>
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337 Refer to <a href="matlab:doc('ao/firwhiten')">whiten2D</a> documentation page
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338 for the meaning of any parameter.
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339 </p>
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340
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341 <div class="fragment"><pre>
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342
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343 pl = plist(...
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344 <span class="string">'csd11'</span>, mod11, ...
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345 <span class="string">'csd12'</span>, mod12, ...
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346 <span class="string">'csd21'</span>, mod21, ...
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347 <span class="string">'csd22'</span>, mod22, ...
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348 <span class="string">'MaxIter'</span>, 75, ...
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349 <span class="string">'PoleType'</span>, 3, ...
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Daniele Nicolodi <nicolodi@science.unitn.it>
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350 <span class="string">'MinOrder'</span>, 20, ...
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351 <span class="string">'MaxOrder'</span>, 40, ...
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352 <span class="string">'Weights'</span>, 2, ...
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353 <span class="string">'Plot'</span>, false,...
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354 <span class="string">'Disp'</span>, false,...
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355 <span class="string">'MSEVARTOL'</span>, 1e-2,...
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Daniele Nicolodi <nicolodi@science.unitn.it>
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356 <span class="string">'FITTOL'</span>, 1e-3);
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357
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358 [aw1,aw2] = whiten2D(a1,a2,pl);
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359
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360 </pre></div>
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361
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362 <div align="center">
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363 <table border="0">
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364 <caption align="bottom">
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365 <b> Figure 4:</b> Power spectral density of the noisy data series
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366 before (left) and after (right) the whitening.
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367 </caption>
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368 <tr>
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369 <td>
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370 <IMG src="images/whitening06.png" align="center" border="0">
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371 </td>
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372 <td>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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373 <IMG src="images/whitening07.png" align="center" border="0">
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374 </td>
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375 </tr>
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376 </table>
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377 </div>
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378
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379 <div align="center">
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Daniele Nicolodi <nicolodi@science.unitn.it>
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380 <table border="0">
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Daniele Nicolodi <nicolodi@science.unitn.it>
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381 <caption align="bottom">
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382 <b> Figure 5:</b> Real (left) and Imaginary (right) part of the
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383 <a href="matlab:doc('ao/cohere')">coherence</a> function.
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384 Blue line refers to theoretical expectation for colored noise data.
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385 Red line refers to calculated values for colored noise data.
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386 Green line refers to calculated values for whitened noise data.
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387
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388 </caption>
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389 <tr>
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390 <td>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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391 <IMG src="images/whitening08.png" align="center" border="0">
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392 </td>
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393 <td>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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394 <IMG src="images/whitening09.png" align="center" border="0">
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395 </td>
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396 </tr>
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397 </table>
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398 </div>
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399
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400
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401
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402
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403
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404
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405
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406
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407
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408 </p>
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409
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Daniele Nicolodi <nicolodi@science.unitn.it>
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410 <br>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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411 <br>
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412 <table class="nav" summary="Navigation aid" border="0" width=
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413 "100%" cellpadding="0" cellspacing="0">
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414 <tr valign="top">
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415 <td align="left" width="20"><a href="gapfill.html"><img src=
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416 "b_prev.gif" border="0" align="bottom" alt=
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417 "Data gap filling"></a>&nbsp;</td>
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418
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419 <td align="left">Data gap filling</td>
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420
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421 <td>&nbsp;</td>
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422
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423 <td align="right">Signal Processing in LTPDA</td>
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424
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Daniele Nicolodi <nicolodi@science.unitn.it>
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425 <td align="right" width="20"><a href=
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426 "sigproc.html"><img src="b_next.gif" border="0" align=
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Daniele Nicolodi <nicolodi@science.unitn.it>
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427 "bottom" alt="Signal Processing in LTPDA"></a></td>
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428 </tr>
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429 </table><br>
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430
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431 <p class="copy">&copy;LTP Team</p>
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432 </body>
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433 </html>