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author Daniele Nicolodi <nicolodi@science.unitn.it>
date Mon, 05 Dec 2011 16:20:06 +0100
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1 <!-- $Id: ndim_ng_content.html,v 1.6 2011/05/02 19:08:05 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="#mchspectra">Multichannel Spectra</a>
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20 </td>
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21 <td>Theoretical background on multichannel spectra.</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="#NGTheory">Noise generation</a>
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26 </td>
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27 <td>Theoretical introduction to multichannel noise generation.</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="#ngMCH">Multichannel Noise Generation</a>
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32 </td>
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33 <td>Generation of multichannel noise with given cross-spectral density matrix.</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="#ng1D">Noisegen 1D</a>
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38 </td>
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39 <td>Generation of one-dimensional noise with given spectral density.</td>
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40 </tr>
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41 <tr valign="top">
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42 <td>
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43 <a href="#ng2D">Noisegen 2D</a>
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44 </td>
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45 <td>Generation of two-dimensional noise with given cross-spectral density.</td>
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46 </tr>
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47 </tbody>
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48 </table>
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49 </td>
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50 </tr>
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51 </table>
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52 </p>
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53 <!-- ===== link box: End ====== -->
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54
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55
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56
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57 <p>
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58 </p>
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59 <p>
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60 The following sections gives an introduction to the generation of model
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61 noise with a given cross spectral density. Further details can be found
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62 in ref. [1].
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63 </p>
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64
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65 <!-- ===== Multichannel Spectra Theory ====== -->
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66 <h2><a name="mchspectra">Theoretical background on multichannel spectra</a></h2>
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67 <p>
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68 We define the autocorrelation function (ACF) of a stationary multichannel process as:
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69 </p>
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70 <div>
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71 <IMG src="images/ngEqn1.gif" align="center" border="0">
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72 </div>
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73 <p>
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74 </p>
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75 <p>
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76 If the multichannel process is L dimensional then the kth element of the ACF is a LxL matrix:
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77 </p>
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78 <div>
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79 <IMG src="images/ngEqn2.gif" align="center" border="0">
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80 </div>
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81 <p>
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82 </p>
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83 <p>
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84 The ACF matrix is not hermitian but have the property that:
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85 </p>
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86 <div>
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87 <IMG src="images/ngEqn3.gif" align="center" border="0">
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88 </div>
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89 <p>
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90 </p>
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91 <p>
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92 The cross-spectral density matrix (CSD) is defined as the fourier transform of the ACF:
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93 </p>
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94 <div>
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95 <IMG src="images/ngEqn4.gif" align="center" border="0">
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96 </div>
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97 <p>
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98 </p>
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99 <p>
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100 the CSD matrix is hermitian.
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101 </p>
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102 <p>
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103 A multichannel white noise process is defined as the process whose ACF satisfies:
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104 </p>
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105 <div>
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106 <IMG src="images/ngEqn5.gif" align="center" border="0">
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107 </div>
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108 <p>
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109 </p>
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110 <p>
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111 therefore the cross-spectral matrix has constant terms as a function of the frequency:
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112 </p>
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113 <div>
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114 <IMG src="images/ngEqn6.gif" align="center" border="0">
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115 </div>
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116 <p>
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117 </p>
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118 <p>
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119 The individual processes are each white noise processes with power spectral density (PSD) given by
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120 <IMG src="images/ngEqn7.gif" align="center" border="0">.
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121 The cross-correlation between the processes is zero except at the same time instant
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122 where they are correlated with a cross-correlation given by the off-diagonal elements of
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123 <IMG src="images/ngEqn8.gif" align="center" border="0">.
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124 A common assumption is
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125 <IMG src="images/ngEqn9.gif" align="center" border="0">
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126 (identity matrix) that is equivalent to assume the white processes having unitary variance
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127 and are completely uncorrelated being zero the off diagonal terms of the CSD matrix.
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128 Further details can be found in [1 - 3].
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129 </p>
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130
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131 <!-- ===== Multichannel Noise Generation Theory ====== -->
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132 <h2><a name="NGTheory">Theoretical introduction to multichannel noise generation</a></h2>
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133 <p>
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134 The problem of multichannel noise generation with a given cross-spectrum
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135 is formulated in frequency domain as follows:
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136 </p>
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137 <div>
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138 <IMG src="images/ngEqn10.gif" align="center" border="0">
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139 </div>
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140 <p>
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141 </p>
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142 <p>
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143 <IMG src="images/ngEqn11.gif" align="center" border="0"> is a
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144 multichannel digital filter that generating colored noise data with given cross-spectrum
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145 <IMG src="images/ngEqn12.gif" align="center" border="0">
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146 starting from a set of mutually independent unitary variance with noise processes.
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147 </p>
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148 <p>
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149 After some mathematics it can be showed that the desired multichannel coloring filter can be written as:
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150 </p>
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151 <div>
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152 <IMG src="images/ngEqn13.gif" align="center" border="0">
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153 </div>
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154 <p>
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155 </p>
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156 <p>
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157 where <IMG src="images/ngEqn14.gif" align="center" border="0">
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158 and <IMG src="images/ngEqn15.gif" align="center" border="0">
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159 are the eigenvectors and eigenvalues matrices of
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160 <IMG src="images/ngEqn12.gif" align="center" border="0">
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161 matrix.
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162 </p>
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163
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164 <!-- ===== Multichannel Noise Generator ====== -->
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165 <h2><a name="ngMCH">Generation of multichannel noise with given cross-spectral density matrix</a></h2>
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166 <p>
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167 <tt>LTPDA Toolbox</tt> provides two methods (<a href="matlab:doc('matrix/mchNoisegenFilter')">mchNoisegenFilter</a> and
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168 <a href="matlab:doc('matrix/mchNoisegen')">mchNoisegen</a>) of the class <tt>matrix</tt> for the production
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169 of multichannel noise coloring filter and multichannel colored noise data series.
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170 Noise data are colored Gaussian distributed time series with given cross-spectral density matrix.
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171 Noise generation process is properly initialized in order to avoid starting transients on the data series.
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172 Details on frequency domain identification of noisegen filters and on the noise generation process
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173 can be found in ref. [1].
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174 <a href="matlab:doc('matrix/mchNoisegenFilter')">mchNoisegenFilter</a> needs a model for the one-sided
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175 cross-spectral density or power spectral density if we are considering one-dimensional problems.
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176 <a href="matlab:doc('matrix/mchNoisegen')">mchNoisegen</a> instead accepts as input the noise generating filter
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177 produced by <a href="matlab:doc('matrix/mchNoisegenFilter')">mchNoisegenFilter</a>.
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178 Details on accepted parameters can be found on the documentation pages of the two methods:
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179 <ul>
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180 <li> <a href="matlab:doc('matrix/mchNoisegenFilter')">mchNoisegenFilter</a>
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181 <li> <a href="matlab:doc('matrix/mchNoisegen')">mchNoisegen</a>
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182 </ul>
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183 </p>
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184
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185
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186 <!-- ===== Noisegen 1D ====== -->
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187 <h2><a name="ng1D">Generation of one-dimensional noise with given spectral density</a></h2>
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188 <p>
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189 <tt>noisegen1D</tt> is a coloring tool allowing the generation of colored noise from withe noise with a given spectrum.
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190 The function constructs a coloring filter through a fitting procedure to the model provided.
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191 If no model is provided an error is prompted. The colored noise provided has one-sided psd
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192 corresponding to the input model.
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193 The function needs a model for the one-sided power spectral density of
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194 the given process. Details on accepted parameters can be found on
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195 the <a href="matlab:doc('ao/noisegen1D')">noisegen1D</a> documentation page. <br>
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196 <ol>
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197 <li> The square root of the model for the power spectral
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198 density is fit in z-domain in order to determine a coloring
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199 filter.
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200 <li> Unstable poles are removed by an all-pass stabilization procedure.
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201 <li> White input data are filtered with the identified filter in order to be colored.
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202 </ol>
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203 </p>
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204
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205
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206 <!-- ===== Noisegen 2D ====== -->
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207 <h2><a name="ng2D">Generation of two-dimensional noise with given cross-spectral density</a></h2>
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208 <p>
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209 <tt>noisegen2D</tt> is a nose coloring tool allowing the generation
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210 two data series with the given cross-spectral density from two starting
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211 white and mutually uncorrelated data series.
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212 Coloring filters are constructed by a fitting procedure to a model
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213 for the corss-spectral density matrix provided.
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214 In order to work with <tt>noisegen2D</tt> you must provide
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215 a model (frequency series analysis objects) for the cross-spectral density
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216 matrix of the process.
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217 Details on accepted parameters can be found on
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218 the <a href="matlab:doc('ao/noisegen2D')">noisegen2D</a> documentation page. <br>
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219 <ol>
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220 <li> Coloring filters frequency response is calculated by the
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221 eigendecomposition of the model cross-spectral matrix.
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222 <li> Calculated responses are fit in z-domain in order to identify
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223 corresponding autoregressive moving average filters.
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224 <li> Input time-series are filtered. The filtering process corresponds to:<br>
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225 o(1) = Filt11(a(1)) + Filt12(a(2))<br>
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226 o(2) = Filt21(a(1)) + Filt22(a(2))
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227 </ol>
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228 </p>
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229
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230
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231 <h2>References</h2>
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232 <p>
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233 <ol>
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234 <li> L. Ferraioli et. al., Calibrating spectral estimation for the LISA
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235 Technology Package with multichannel synthetic noise generation, Phys. Rev. D 82, 042001 (2010).
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236 <li> S. M. Kay, Modern Spectral Estimation, Prentice-Hall, 1999 </li>
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237 <li> G. M. Jenkins and D. G. Watts, Spectral Analysis and Its Applications, Holden-Day 1968. </li>
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238 </ol>
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239 </p>