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author Daniele Nicolodi <nicolodi@science.unitn.it>
date Mon, 05 Dec 2011 16:20:06 +0100
parents f0afece42f48
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1 <!-- $Id: whitening_content.html,v 1.5 2011/04/11 14:24:19 luigi Exp $ -->
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2
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3 <!-- ================================================== -->
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4 <!-- BEGIN CONTENT FILE -->
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5 <!-- ================================================== -->
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6 <!-- ===== link box: Begin ===== -->
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7 <p>
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8 <table border="1" width="80%">
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9 <tr>
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10 <td>
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11 <table border="0" cellpadding="5" class="categorylist" width="100%">
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12 <colgroup>
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13 <col width="37%"/>
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14 <col width="63%"/>
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15 </colgroup>
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16 <tbody>
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17 <tr valign="top">
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18 <td>
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19 <a href="#WhitenIntro">Introduction</a>
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20 </td>
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21 <td>Noise whitening in LTPDA.</td>
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22 </tr>
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23 <tr valign="top">
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24 <td>
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25 <a href="#WhitenAlgo">Algorithm</a>
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26 </td>
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27 <td>Whitening Algorithms.</td>
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28 </tr>
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29 <tr valign="top">
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30 <td>
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31 <a href="#Whiten1D">1D data</a>
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32 </td>
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33 <td>Whitening noise in one-dimensional data.</td>
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34 </tr>
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35 <tr valign="top">
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36 <td>
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37 <a href="#Whiten2D">2D data</a>
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38 </td>
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39 <td>Whitening noise in two-dimensional data.</td>
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40 </tr>
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41 </tbody>
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42 </table>
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43 </td>
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44 </tr>
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45 </table>
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46 </p>
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47 <!-- ===== link box: End ====== -->
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48
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49
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50 <!-- ===== Intro ====== -->
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51 <h2><a name="WhitenIntro">Noise whitening in LTPDA</a></h2>
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52 <p>
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53 A random process <i>w(t)</i> is considered white if it is zero mean
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54 and uncorrelated:
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55 </p>
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56 <p>
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57 <IMG src="images/whitening01.gif" align="center" border="0">
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58 </p>
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59 <p>
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60 As a consequence, the power spectral density of a white process is a
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61 constant at every frequency:
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62 </p>
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63 <p>
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64 <IMG src="images/whitening02.gif" align="center" border="0">
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65 </p>
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66 <p>
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67 In other words, The power per unit of frequency associated to a white noise
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68 process is uniformly distributed on the whole available frequency range.
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69 An example is reported in figure 1.
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70 </p>
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71
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72 <div align="center">
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73 <table border="0">
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74 <caption align="bottom">
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75 <b> Figure 1:</b> Power spectral density (estimated with the welch method)
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76 of a gaussian unitary variance zero mean random process.
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77 The process <i>w(t)</i> is assumed to have
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78 physical units of <tt>m</tt> therefore its power spectral density has
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79 physical units of <tt>m^2/Hz</tt>. Note that the power spectral density
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80 average value is 2 instead of the expected 1 (unitary variance process)
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81 since we calculated one-sided power spectral density.
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82 </caption>
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83 <tr>
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84 <td>
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85 <IMG src="images/whitening03.png" align="center" border="0">
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86 </td>
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87 </tr>
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88 </table>
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89 </div>
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90
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91 <p>
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92 </p>
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93
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94 <p>
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95 A non-white (colored) noise process is instead characterized by a given
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96 distribution of the power per unit of frequency along the available frequency
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97 bandwidth. <br>
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98 Whitening operation on a given non-white process corresponds to force
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99 such a process to satisfy the conditions described above for a white process.
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100 </p>
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101 <p>
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102 In LTPDA there are different methods for noise whitening:
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103 <ul>
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104 <li> <a href="matlab:doc('ao/buildWhitener1D')"> buildWhitener1D.m</a>
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105 <li> <a href="matlab:doc('ao/whiten1D')"> whiten1D.m</a>
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106 <li> <a href="matlab:doc('ao/firwhiten')">firwhiten.m</a>
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107 <li> <a href="matlab:doc('ao/whiten2D')">whiten2D.m</a>
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108 </ul>
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109 They accept time series analysis objects as an input and they output noise
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110 whitening filters or whitened time series analysis objects.
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111 </p>
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112
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113 <!-- ===== Algorithm ====== -->
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114 <h2><a name="WhitenAlgo">Whitening Algorithms</a></h2>
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115
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116 <h3>buildWhitener1D</h3>
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117 <p>
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118 <tt>buildWhitener1D</tt> performs a frequency domain identification of the system
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119 in order to extract the proper whitening filter. The function needs a model
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120 for the one-sided power spectral density of the given process. If no model
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121 is provided, the power spectral density of the process is calculated with
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122 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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123 <ol>
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124 <li> The inverse of the square root of the model for the power spectral
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125 density is fit in z-domain in order to determine a whitening
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126 filter.
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127 <li> Unstable poles are removed by an all-pass stabilization procedure.
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128 <li> Whitening filter is provided at the output.
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129 </ol>
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130 </p>
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131
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132 <h3>Whiten1D</h3>
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133 <p>
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134 <tt>whiten1D</tt> implements the same functionality of <tt>buildWhitener1D</tt>
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135 but it adds the filtering step so input data are filtered with the identified filter
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136 internally to the method.
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137 </p>
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138
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139 <h3>Firwhiten</h3>
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140 <p>
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141 <tt>firwhiten</tt> whitens the input time-series by building an FIR
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142 whitening filter. <br>
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143 <ol>
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144 <li> Make ASD of time-series.
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145 <li> Perform running median to get noise-floor estimate <a href="matlab:doc('ao/smoother')">ao/smoother</a>.
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146 <li> Invert noise-floor estimate.
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147 <li> Call <a href="matlab:doc('mfir')">mfir()</a> on noise-floor estimate to produce whitening filter.
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148 <li> Filter data.
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149 </ol>
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150 </p>
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151
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152 <h3>Whiten2D</h3>
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153 <p>
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154 <tt>whiten2D</tt> whitens cross-correlated time-series. Whitening
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155 filters are constructed by a fitting procedure to the models
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156 for the corss-spectral matrix provided.
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157 In order to work with <tt>whiten2D</tt> you must provide
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158 a model (frequency series analysis objects) for the cross-spectral density
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159 matrix of the process.
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160 <ol>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
161 <li> Whitening filters frequency response is calculated by the
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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162 eigendecomposition of the cross-spectral matrix.
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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163 <li> Calculated responses are fit in z-domain in order to identify
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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164 corresponding autoregressive moving average filters.
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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165 <li> Input time-series is filtered. The filtering process corresponds to:<br>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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166 w(1) = Filt11(a(1)) + Filt12(a(2))<br>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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167 w(2) = Filt21(a(1)) + Filt22(a(2))
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
168 </ol>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
169 </p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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170
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Daniele Nicolodi <nicolodi@science.unitn.it>
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171
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Daniele Nicolodi <nicolodi@science.unitn.it>
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172 <!-- ===== 1D Examples ====== -->
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Daniele Nicolodi <nicolodi@science.unitn.it>
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173 <h2><a name="buildWhitener1D">Whitening noise in one-dimensional data</a></h2>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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174 <p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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175 We can now test an example of the one-dimensinal whitening filters capabilities.
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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176 With the following commands we can generate a colored noise data series
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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177 for parameters description please refer to the
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Daniele Nicolodi <nicolodi@science.unitn.it>
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178 <a href="matlab:doc('ao')">ao</a>,
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Daniele Nicolodi <nicolodi@science.unitn.it>
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179 <a href="matlab:doc('miir')">miir</a> and
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Daniele Nicolodi <nicolodi@science.unitn.it>
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180 <a href="matlab:doc('ao/filter')">filter</a>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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181 documentation pages.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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182 </p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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183 <div class="fragment"><pre>
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184
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Daniele Nicolodi <nicolodi@science.unitn.it>
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185 fs = 1; <span class="comment">% sampling frequency</span>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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186
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Daniele Nicolodi <nicolodi@science.unitn.it>
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187 <span class="comment">% Generate gaussian white noise</span>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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188 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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189 <span class="string">'fs'</span>, fs, ...
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Daniele Nicolodi <nicolodi@science.unitn.it>
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190 <span class="string">'nsecs'</span>, 1e5, ...
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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191 <span class="string">'yunits'</span>, <span class="string">'m'</span>);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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192 a = ao(pl);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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193
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Daniele Nicolodi <nicolodi@science.unitn.it>
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194 <span class="comment">% Get a coloring filter</span>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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195 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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196 <span class="string">'fs'</span>, fs, ...
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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197 <span class="string">'order'</span>, 3, ...
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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198 <span class="string">'gain'</span>, 1, ...
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Daniele Nicolodi <nicolodi@science.unitn.it>
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199 <span class="string">'fc'</span>, [0.03 0.1]);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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200 ft = miir(pl);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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201
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Daniele Nicolodi <nicolodi@science.unitn.it>
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202 <span class="comment">% Coloring noise</span>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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203 af = filter(a, ft);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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204
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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205 </pre></div>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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206
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Daniele Nicolodi <nicolodi@science.unitn.it>
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207 <p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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208 Now we can try to white colored noise.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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209 </p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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210
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Daniele Nicolodi <nicolodi@science.unitn.it>
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211
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Daniele Nicolodi <nicolodi@science.unitn.it>
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212 <h3>buildWhitener1D</h3>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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213 <p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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214 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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215 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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216 <a href="matlab:doc('ao/buildWhitener1D')">buildWhitener1D</a> documentation page
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Daniele Nicolodi <nicolodi@science.unitn.it>
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217 for the meaning of any parameter. The result of the whitening procedure
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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218 is reported in figure 2.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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219 </p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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220
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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>
parents:
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222
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Daniele Nicolodi <nicolodi@science.unitn.it>
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223 pl = plist(...
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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224 <span class="string">'MaxIter'</span>, 30, ...
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
225 <span class="string">'MinOrder'</span>, 9, ...
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
226 <span class="string">'MaxOrder'</span>, 15, ...
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
227 <span class="string">'FITTOL'</span>, 5e-2);
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
228
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Daniele Nicolodi <nicolodi@science.unitn.it>
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229 wfil = buildWhitener1D(af,pl);
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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230
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Daniele Nicolodi <nicolodi@science.unitn.it>
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diff changeset
231 aw = filter(af,wfil);
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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232
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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233 </pre></div>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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234
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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235 <div align="center">
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
236 <table border="0">
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
237 <caption align="bottom">
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
238 <b> Figure 2:</b> Power spectral density (estimated with the welch method)
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
239 of colored and whitened processes.
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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240 </caption>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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241 <tr>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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242 <td>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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243 <IMG src="images/whitening04.png" align="center" border="0">
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
244 </td>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
245 </tr>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
246 </table>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
247 </div>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
248
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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249
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
250 <h3>Firwhiten</h3>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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251 <p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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252 As an alternative you can try <tt>firwhiten</tt> to whiten the present
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
253 colored noise. Please refer to the
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
254 <a href="matlab:doc('ao/firwhiten')">firwhiten</a> documentation page
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
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 3.
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
257 </p>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
258
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
259 <div class="fragment"><pre>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
260
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
261 pl = plist(...
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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262 <span class="string">'Ntaps'</span>, 5000, ...
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
263 <span class="string">'Nfft'</span>, 1e5, ...
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
264 <span class="string">'BW'</span>, 5);
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
265
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
266 aw = firwhiten(af, pl);
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
267
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
268 </pre></div>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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269
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
270 <div align="center">
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
271 <table border="0">
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
272 <caption align="bottom">
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
273 <b> Figure 3:</b> Power spectral density (estimated with the welch method)
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
274 of colored and whitened processes.
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
275 </caption>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
276 <tr>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
277 <td>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
278 <IMG src="images/whitening05.png" align="center" border="0">
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
279 </td>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
280 </tr>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
281 </table>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
282 </div>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
283
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
284
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
285 <!-- ===== 2D Examples ====== -->
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
286 <h2><a name="Whiten2D">Whitening noise in two-dimensional data</a></h2>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
287
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
288 <p>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
289 We consider now the problem of whitening cross correlated data series.
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
290 As a example we consider a typical couple of x-dynamics LTP data series.
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
291 <tt>a1</tt> and <tt>a2</tt> are interferometer output noise data series.
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
292 In oreder to whiten data we must input a frequency response model of the
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
293 cross spectral matrix of the cross-correlated process.
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
294 </p>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
295 <p>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
296 <IMG src="images/whitening10.gif" align="center" border="0">
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
297 </p>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
298 <p>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
299 Refer to <a href="matlab:doc('ao/firwhiten')">whiten2D</a> documentation page
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
300 for the meaning of any parameter.
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
301 </p>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
302
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
303 <div class="fragment"><pre>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
304
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
305 pl = plist(...
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
306 <span class="string">'csd11'</span>, mod11, ...
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
307 <span class="string">'csd12'</span>, mod12, ...
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
308 <span class="string">'csd21'</span>, mod21, ...
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
309 <span class="string">'csd22'</span>, mod22, ...
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
310 <span class="string">'MaxIter'</span>, 75, ...
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
311 <span class="string">'PoleType'</span>, 3, ...
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
312 <span class="string">'MinOrder'</span>, 20, ...
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
313 <span class="string">'MaxOrder'</span>, 40, ...
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
314 <span class="string">'Weights'</span>, 2, ...
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
315 <span class="string">'Plot'</span>, false,...
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
316 <span class="string">'Disp'</span>, false,...
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
317 <span class="string">'MSEVARTOL'</span>, 1e-2,...
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
318 <span class="string">'FITTOL'</span>, 1e-3);
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
319
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
320 [aw1,aw2] = whiten2D(a1,a2,pl);
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
321
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322 </pre></div>
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323
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Daniele Nicolodi <nicolodi@science.unitn.it>
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324 <div align="center">
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Daniele Nicolodi <nicolodi@science.unitn.it>
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325 <table border="0">
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Daniele Nicolodi <nicolodi@science.unitn.it>
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326 <caption align="bottom">
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327 <b> Figure 4:</b> Power spectral density of the noisy data series
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328 before (left) and after (right) the whitening.
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329 </caption>
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330 <tr>
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331 <td>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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332 <IMG src="images/whitening06.png" align="center" border="0">
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Daniele Nicolodi <nicolodi@science.unitn.it>
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333 </td>
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334 <td>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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335 <IMG src="images/whitening07.png" align="center" border="0">
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Daniele Nicolodi <nicolodi@science.unitn.it>
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336 </td>
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337 </tr>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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338 </table>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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339 </div>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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340
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341 <div align="center">
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Daniele Nicolodi <nicolodi@science.unitn.it>
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342 <table border="0">
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Daniele Nicolodi <nicolodi@science.unitn.it>
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343 <caption align="bottom">
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Daniele Nicolodi <nicolodi@science.unitn.it>
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344 <b> Figure 5:</b> Real (left) and Imaginary (right) part of the
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Daniele Nicolodi <nicolodi@science.unitn.it>
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345 <a href="matlab:doc('ao/cohere')">coherence</a> function.
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346 Blue line refers to theoretical expectation for colored noise data.
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347 Red line refers to calculated values for colored noise data.
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348 Green line refers to calculated values for whitened noise data.
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349
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350 </caption>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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351 <tr>
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352 <td>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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353 <IMG src="images/whitening08.png" align="center" border="0">
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Daniele Nicolodi <nicolodi@science.unitn.it>
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354 </td>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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355 <td>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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356 <IMG src="images/whitening09.png" align="center" border="0">
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Daniele Nicolodi <nicolodi@science.unitn.it>
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357 </td>
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358 </tr>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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359 </table>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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360 </div>
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361
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362
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363
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364
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365
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366
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367
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368