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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 <!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 generation with given cross-spectral density (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 "franklin_ng.html"><img src="b_prev.gif" border="0" align=
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30 "bottom" alt="Franklin noise-generator"></a>&nbsp;&nbsp;&nbsp;<a href=
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31 "smodel.html"><img src="b_next.gif" border="0" align=
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32 "bottom" alt="Parameteric models"></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 generation with given cross-spectral density</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="#mchspectra">Multichannel Spectra</a>
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58 </td>
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59 <td>Theoretical background on multichannel spectra.</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="#NGTheory">Noise generation</a>
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64 </td>
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65 <td>Theoretical introduction to multichannel noise generation.</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="#ngMCH">Multichannel Noise Generation</a>
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70 </td>
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71 <td>Generation of multichannel noise with given cross-spectral density matrix.</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="#ng1D">Noisegen 1D</a>
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76 </td>
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77 <td>Generation of one-dimensional noise with given spectral density.</td>
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78 </tr>
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79 <tr valign="top">
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80 <td>
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81 <a href="#ng2D">Noisegen 2D</a>
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82 </td>
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83 <td>Generation of two-dimensional noise with given cross-spectral density.</td>
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84 </tr>
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85 </tbody>
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86 </table>
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87 </td>
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88 </tr>
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89 </table>
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90 </p>
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91 <!-- ===== link box: End ====== -->
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92
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93
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94
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95 <p>
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96 </p>
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97 <p>
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98 The following sections gives an introduction to the generation of model
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99 noise with a given cross spectral density. Further details can be found
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100 in ref. [1].
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101 </p>
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102
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103 <!-- ===== Multichannel Spectra Theory ====== -->
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104 <h2><a name="mchspectra">Theoretical background on multichannel spectra</a></h2>
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105 <p>
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106 We define the autocorrelation function (ACF) of a stationary multichannel process as:
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107 </p>
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108 <div>
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109 <IMG src="images/ngEqn1.gif" align="center" border="0">
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110 </div>
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111 <p>
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112 </p>
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113 <p>
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114 If the multichannel process is L dimensional then the kth element of the ACF is a LxL matrix:
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115 </p>
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116 <div>
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117 <IMG src="images/ngEqn2.gif" align="center" border="0">
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118 </div>
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119 <p>
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120 </p>
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121 <p>
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122 The ACF matrix is not hermitian but have the property that:
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123 </p>
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124 <div>
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125 <IMG src="images/ngEqn3.gif" align="center" border="0">
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126 </div>
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127 <p>
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128 </p>
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129 <p>
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130 The cross-spectral density matrix (CSD) is defined as the fourier transform of the ACF:
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131 </p>
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132 <div>
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133 <IMG src="images/ngEqn4.gif" align="center" border="0">
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134 </div>
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135 <p>
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136 </p>
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137 <p>
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138 the CSD matrix is hermitian.
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139 </p>
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140 <p>
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141 A multichannel white noise process is defined as the process whose ACF satisfies:
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142 </p>
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143 <div>
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144 <IMG src="images/ngEqn5.gif" align="center" border="0">
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145 </div>
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146 <p>
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147 </p>
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148 <p>
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149 therefore the cross-spectral matrix has constant terms as a function of the frequency:
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150 </p>
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151 <div>
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152 <IMG src="images/ngEqn6.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 The individual processes are each white noise processes with power spectral density (PSD) given by
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158 <IMG src="images/ngEqn7.gif" align="center" border="0">.
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159 The cross-correlation between the processes is zero except at the same time instant
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160 where they are correlated with a cross-correlation given by the off-diagonal elements of
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161 <IMG src="images/ngEqn8.gif" align="center" border="0">.
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162 A common assumption is
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163 <IMG src="images/ngEqn9.gif" align="center" border="0">
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164 (identity matrix) that is equivalent to assume the white processes having unitary variance
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165 and are completely uncorrelated being zero the off diagonal terms of the CSD matrix.
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166 Further details can be found in [1 - 3].
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167 </p>
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168
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169 <!-- ===== Multichannel Noise Generation Theory ====== -->
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170 <h2><a name="NGTheory">Theoretical introduction to multichannel noise generation</a></h2>
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171 <p>
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172 The problem of multichannel noise generation with a given cross-spectrum
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173 is formulated in frequency domain as follows:
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174 </p>
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175 <div>
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176 <IMG src="images/ngEqn10.gif" align="center" border="0">
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177 </div>
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178 <p>
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179 </p>
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180 <p>
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181 <IMG src="images/ngEqn11.gif" align="center" border="0"> is a
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182 multichannel digital filter that generating colored noise data with given cross-spectrum
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183 <IMG src="images/ngEqn12.gif" align="center" border="0">
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184 starting from a set of mutually independent unitary variance with noise processes.
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185 </p>
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186 <p>
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187 After some mathematics it can be showed that the desired multichannel coloring filter can be written as:
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188 </p>
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189 <div>
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190 <IMG src="images/ngEqn13.gif" align="center" border="0">
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191 </div>
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192 <p>
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193 </p>
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194 <p>
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195 where <IMG src="images/ngEqn14.gif" align="center" border="0">
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196 and <IMG src="images/ngEqn15.gif" align="center" border="0">
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197 are the eigenvectors and eigenvalues matrices of
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198 <IMG src="images/ngEqn12.gif" align="center" border="0">
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199 matrix.
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200 </p>
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201
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202 <!-- ===== Multichannel Noise Generator ====== -->
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203 <h2><a name="ngMCH">Generation of multichannel noise with given cross-spectral density matrix</a></h2>
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204 <p>
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205 <tt>LTPDA Toolbox</tt> provides two methods (<a href="matlab:doc('matrix/mchNoisegenFilter')">mchNoisegenFilter</a> and
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206 <a href="matlab:doc('matrix/mchNoisegen')">mchNoisegen</a>) of the class <tt>matrix</tt> for the production
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207 of multichannel noise coloring filter and multichannel colored noise data series.
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208 Noise data are colored Gaussian distributed time series with given cross-spectral density matrix.
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209 Noise generation process is properly initialized in order to avoid starting transients on the data series.
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210 Details on frequency domain identification of noisegen filters and on the noise generation process
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211 can be found in ref. [1].
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212 <a href="matlab:doc('matrix/mchNoisegenFilter')">mchNoisegenFilter</a> needs a model for the one-sided
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213 cross-spectral density or power spectral density if we are considering one-dimensional problems.
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214 <a href="matlab:doc('matrix/mchNoisegen')">mchNoisegen</a> instead accepts as input the noise generating filter
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215 produced by <a href="matlab:doc('matrix/mchNoisegenFilter')">mchNoisegenFilter</a>.
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216 Details on accepted parameters can be found on the documentation pages of the two methods:
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217 <ul>
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218 <li> <a href="matlab:doc('matrix/mchNoisegenFilter')">mchNoisegenFilter</a>
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219 <li> <a href="matlab:doc('matrix/mchNoisegen')">mchNoisegen</a>
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220 </ul>
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221 </p>
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222
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223
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224 <!-- ===== Noisegen 1D ====== -->
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225 <h2><a name="ng1D">Generation of one-dimensional noise with given spectral density</a></h2>
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226 <p>
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227 <tt>noisegen1D</tt> is a coloring tool allowing the generation of colored noise from withe noise with a given spectrum.
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228 The function constructs a coloring filter through a fitting procedure to the model provided.
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229 If no model is provided an error is prompted. The colored noise provided has one-sided psd
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230 corresponding to the input model.
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231 The function needs a model for the one-sided power spectral density of
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232 the given process. Details on accepted parameters can be found on
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233 the <a href="matlab:doc('ao/noisegen1D')">noisegen1D</a> documentation page. <br>
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234 <ol>
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235 <li> The square root of the model for the power spectral
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236 density is fit in z-domain in order to determine a coloring
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237 filter.
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238 <li> Unstable poles are removed by an all-pass stabilization procedure.
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239 <li> White input data are filtered with the identified filter in order to be colored.
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240 </ol>
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241 </p>
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242
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243
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244 <!-- ===== Noisegen 2D ====== -->
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245 <h2><a name="ng2D">Generation of two-dimensional noise with given cross-spectral density</a></h2>
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246 <p>
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247 <tt>noisegen2D</tt> is a nose coloring tool allowing the generation
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248 two data series with the given cross-spectral density from two starting
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249 white and mutually uncorrelated data series.
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250 Coloring filters are constructed by a fitting procedure to a model
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251 for the corss-spectral density matrix provided.
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252 In order to work with <tt>noisegen2D</tt> you must provide
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253 a model (frequency series analysis objects) for the cross-spectral density
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254 matrix of the process.
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255 Details on accepted parameters can be found on
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256 the <a href="matlab:doc('ao/noisegen2D')">noisegen2D</a> documentation page. <br>
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257 <ol>
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258 <li> Coloring filters frequency response is calculated by the
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259 eigendecomposition of the model cross-spectral matrix.
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260 <li> Calculated responses are fit in z-domain in order to identify
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261 corresponding autoregressive moving average filters.
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262 <li> Input time-series are filtered. The filtering process corresponds to:<br>
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263 o(1) = Filt11(a(1)) + Filt12(a(2))<br>
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264 o(2) = Filt21(a(1)) + Filt22(a(2))
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265 </ol>
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266 </p>
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267
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268
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269 <h2>References</h2>
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270 <p>
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271 <ol>
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272 <li> L. Ferraioli et. al., Calibrating spectral estimation for the LISA
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273 Technology Package with multichannel synthetic noise generation, Phys. Rev. D 82, 042001 (2010).
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274 <li> S. M. Kay, Modern Spectral Estimation, Prentice-Hall, 1999 </li>
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275 <li> G. M. Jenkins and D. G. Watts, Spectral Analysis and Its Applications, Holden-Day 1968. </li>
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276 </ol>
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277 </p>
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278
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279 </p>
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280
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281 <br>
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282 <br>
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283 <table class="nav" summary="Navigation aid" border="0" width=
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284 "100%" cellpadding="0" cellspacing="0">
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285 <tr valign="top">
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286 <td align="left" width="20"><a href="franklin_ng.html"><img src=
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287 "b_prev.gif" border="0" align="bottom" alt=
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288 "Franklin noise-generator"></a>&nbsp;</td>
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289
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290 <td align="left">Franklin noise-generator</td>
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291
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292 <td>&nbsp;</td>
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293
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294 <td align="right">Parameteric models</td>
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295
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296 <td align="right" width="20"><a href=
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297 "smodel.html"><img src="b_next.gif" border="0" align=
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298 "bottom" alt="Parameteric models"></a></td>
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299 </tr>
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300 </table><br>
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301
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302 <p class="copy">&copy;LTP Team</p>
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303 </body>
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304 </html>