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author Daniele Nicolodi <nicolodi@science.unitn.it>
date Mon, 05 Dec 2011 18:04:34 +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>Franklin noise-generator (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 "noisegen.html"><img src="b_prev.gif" border="0" align=
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30 "bottom" alt="Generating model noise"></a>&nbsp;&nbsp;&nbsp;<a href=
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31 "ndim_ng.html"><img src="b_next.gif" border="0" align=
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32 "bottom" alt="Noise generation with given cross-spectral density"></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>Franklin noise-generator</h1>
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37 <hr>
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38
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39 <p>
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40 The following sections gives an introduction to the <a href="noisegen.html">generation of model noise</a> using the noise generator implemented in LTPDA.
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41 <ul>
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42 <li><a href="#franklin">Franklin's noise generator</a></li>
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43 <li><a href="#description">Description</a></li>
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44 <li><a href="#inputs">Inputs</a></li>
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45 <li><a href="#outputs">Outputs</a></li>
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46 <li><a href="#usage">Usage</a></li>
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47 </ul>
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48 <h2><a name="franklin">Franklin's noise generator</a></h2>
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49 Franklin's noise generator is a method to generate arbitrarily long time series with a prescribed spectral density.
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50 The algorithm is based on the following paper:
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51 </p>
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52 <p>Franklin, Joel N.:
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53 <i> Numerical simulation of stationary and non-stationary gaussian
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54 random processes </i>, SIAM review, Volume {<b> 7</b>}, Issue 1, page 68--80, 1965.
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55 </p>
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56 <p>
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57 The Document <i> Generation of Random time series with prescribed spectra </i> by Gerhard Heinzel (S2-AEI-TN-3034) <br> corrects a mistake in the aforesaid paper and describes the practical implementation.
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58 </p>
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59 <p>
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60 See <a href="noisegen.html">Generating model noise</a> for more general information on this.
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61 </p>
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62 <p>
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63 Franklin's method does not require any 'warm up' period. It starts with a transfer function given as ratio of two polynomials.<br/>
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64 The generator operates on a real state vector y of length n which is
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65 maintained between invocations. It produces samples of the time series in equidistant steps <tt>T = 1/fs</tt>, where <tt>fs</tt> is the sampling frequency.
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66 </p>
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67 <p>
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68 <ul>
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69 <li> y0 = Tinit * r, on initialization
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70 <li> yi = E * yi-1 + Tprop * r, to propagate
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71 <li> xi = a * yi , the sampled time series.
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72 </ul>
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73 r is a vector of independent normal Gaussian random numbers
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74 Tinit, E, Tprop which are real matrices and a which is a real vector are determined once by the algorithm.
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75 </p>
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76
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77 <h2><a name="description">Description</a></h2>
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78 <p>
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79 When an analysis object is constructed from a pole zero model Franklin's noise generator is called (compare <a href="ao_create.html#pzmodel">Creating AOs from pole zero models</a>).
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80 </p>
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81
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82
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83 <h2><a name="inputs">Inputs</a></h2>
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84 for the function call the parameter list has to contain at least:
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85 <ul>
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86 <li> nsecs - number of seconds (length of time series)
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87 <li> fs - sampling frequency
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88 <li> pzmodel with gain
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89 </ul>
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90
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91 <h2><a name="outputs">Outputs</a></h2>
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92 <ul>
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93 <li> b - analysis object containing the resulting time series
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94 </ul>
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95 </p>
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96 <h2><a name="usage">Usage</a></h2>
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97 The analysis object constructor <a href="ao_create.html">ao</a> calls the following four functions when the input is a pzmodel.
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98 <ul>
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99 <li> ngconv
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100 <li> ngsetup
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101 <li> nginit
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102 <li> ngprop
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103 </ul>
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104 <p>
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105 First a parameter list of the input parameters is to be done. For further information on this look at <a href="plist_create.html#params">Creating parameter lists from parameters</a>.<br/>
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106 </p>
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107 <h2><a name="starting">Starting from a given pole/zero model</a></h2>
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108 <p>
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109 The parameter list should contain the number of seconds the resulting time series should have <tt>nsecs</tt> and the sampling frequency <tt>fs</tt>. <br/>
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110 The constructor call should look like this:
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111 </p>
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112 <div class="fragment"><pre>
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113 f1 = 5;
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114 f2 = 10;
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115 f3 = 1;
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116 gain = 1;
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117 fs = 10; <span class="comment">% sampling frequency</span>
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118 nsecs = 100; <span class="comment">% number of seconds to be generated</span>
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119 p = [pz(f1) pz(f2)];
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120 z = [pz(f3)];
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121 pzm = pzmodel(gain, p, z);
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122 a = ao(pzm, plist(<span class="string">'nsecs'</span>, nsecs, <span class="string">'fs'</span>,fs))
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123
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124 </pre></div>
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125 The output will be an analysis object <tt>a</tt> containing the time series with the spectrum described by the input pole-zero model.
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126 </p>
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127
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128
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129
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130
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131 </p>
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132
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133 <br>
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134 <br>
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135 <table class="nav" summary="Navigation aid" border="0" width=
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136 "100%" cellpadding="0" cellspacing="0">
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137 <tr valign="top">
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138 <td align="left" width="20"><a href="noisegen.html"><img src=
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139 "b_prev.gif" border="0" align="bottom" alt=
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140 "Generating model noise"></a>&nbsp;</td>
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141
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142 <td align="left">Generating model noise</td>
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143
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144 <td>&nbsp;</td>
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145
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146 <td align="right">Noise generation with given cross-spectral density</td>
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147
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148 <td align="right" width="20"><a href=
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149 "ndim_ng.html"><img src="b_next.gif" border="0" align=
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150 "bottom" alt="Noise generation with given cross-spectral density"></a></td>
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151 </tr>
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152 </table><br>
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153
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154 <p class="copy">&copy;LTP Team</p>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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155 </body>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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156 </html>