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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>Generation of noise with given PSD (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 "ltpda_training_topic_5_1.html"><img src="b_prev.gif" border="0" align=
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30 "bottom" alt="System identification in z-domain"></a>&nbsp;&nbsp;&nbsp;<a href=
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31 "ltpda_training_topic_5_3.html"><img src="b_next.gif" border="0" align=
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32 "bottom" alt="Fitting time series with polynomials"></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>Generation of noise with given PSD</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
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43 <p>
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44 Generation of model noise is performed with the function <tt>ao/noisegen1D</tt>.
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45 Details on the algorithm can be found in <a href="ng1D.html">noisegen1D help page</a>.
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46 </p>
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47
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48 <h2> Generation of noise with given PSD</h2>
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49
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50 <p>
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51 During this exercise we will:
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52 <ol>
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53 <li> Load from file an fsdata object with the model (obtained with a fit to the the PSD of test data with zDomainFit)
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54 <li> Genarate noise from this model
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55 <li> Compare PSD of the generated noise with original PSD
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56 </ol>
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57 </p>
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58
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59 <p>
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60 Let's open a new editor window and load the test data.
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61 </p>
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62
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63 <div class="fragment"><pre>
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64 tn = ao(plist(<span class="string">'filename'</span>, <span class="string">'topic5/T5_Ex03_TestNoise.xml'</span>));
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65 tn.setName;
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66 </pre></div>
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67
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68 <p>
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69 This command will load an Analysis Object containing a test time series
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70 10000 seconds long, sampled at 1 Hz. The command <tt>setName</tt> sets the name
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71 of the AO to be the same as the variable name, in this case <tt>tn</tt>.
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72 </p>
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73 <p>
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74 Now let's calculate the PSD of our data. We apply some averaging, in order to decrease the
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75 fluctuations in the data.
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76 </p>
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77
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78 <div class="fragment"><pre>
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79 tnxx = tn.psd(plist(<span class="string">'Nfft'</span>,2000));
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80 </pre></div>
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81
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82 <p>
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83 Additionally, we load a smooth model that represents well our data. It was obtained,
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84 as described <a href="ltpda_training_topic_5_2.html#brbme84">here</a>,
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85 by fitting the target PSD with z-domain fitting.
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86 We load the data from disk, and plot them against the target PSD. Please note that
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87 the colouring filters (whose response represents our model) have no units, so we force them to
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88 be the same as the PSD we compare with:
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89 <div class="fragment"><pre>
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90 fmod = ao(plist(<span class="string">'filename'</span>, <span class="string">'topic5/T5_Ex03_ModelNoise.xml'</span>));
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91 iplot(tnxx, fmod.setYunits(tnxx.yunits))
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92 </pre></div>
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93 </p>
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94
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95 <p>
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96 The comparison beyween the target PSD and the model should look like:
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97 <div align="center">
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98 <IMG src="images/ltpda_training_1/topic5/ltpda_training_5_2_4.png" align="center" border="0">
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99 </div>
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100 </p>
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101
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102 We can now start the noise generation process. The first step is to
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103 generate a white time series Analysis Object, with the desired duration and units:
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104 </p>
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105 <div class="fragment"><pre>
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106 a = ao(plist(<span class="string">'tsfcn'</span>,<span class="string">'randn(size(t))'</span>,<span class="string">'fs'</span>,1,<span class="string">'nsecs'</span>,10000,<span class="string">'yunits'</span>,<span class="string">'m'</span>));
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107 </pre></div>
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108 <p>
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109 Then we run the noise coloring process calling <tt>noisegen1D</tt>
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110 </p>
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111
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112 <div class="fragment"><pre>
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113 plng = plist(...
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114 <span class="string">'model'</span>, fmod, ... <span class="comment">% model for colored noise psd</span>
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115 <span class="string">'MaxIter'</span>, 50, ... <span class="comment">% maximum number of fit iteration per model order</span>
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116 <span class="string">'PoleType'</span>, 2, ... <span class="comment">% generates complex poles distributed in the unitary circle</span>
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117 <span class="string">'MinOrder'</span>, 20, ... <span class="comment">% minimum model order</span>
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118 <span class="string">'MaxOrder'</span>, 50, ... <span class="comment">% maximum model order</span>
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119 <span class="string">'Weights'</span>, 2, ... <span class="comment">% weight with 1/abs(model)</span>
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120 <span class="string">'Plot'</span>, false,... <span class="comment">% on to show the plot</span>
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121 <span class="string">'Disp'</span>, false,... <span class="comment">% on to display fit progress on the command window</span>
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122 <span class="string">'RMSEVar'</span>, 7,... <span class="comment">% Root Mean Squared Error Variation</span>
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123 <span class="string">'FitTolerance'</span>, 2); <span class="comment">% Residuals log difference</span>
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124
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125 ac = noisegen1D(a, plng);
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126 </pre></div>
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127
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128 <p>
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129 Let's check the result. We calculate the PSD of the generated noise and compare it
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130 with the PSD of the target data.
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131 </p>
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132
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133 <div class="fragment"><pre>
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134 acxx = ac.psd(plist(<span class="string">'Nfft'</span>,2000));
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135 iplot(tnxx,acxx)
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136 </pre></div>
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137
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138 <p>
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139 As can be seen, the result is in quite satisfactory agreement with the original data
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140 <div align="center">
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141 <IMG src="images/ltpda_training_1/topic5/ltpda_training_5_2_3.png" align="center" border="0">
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142 </div>
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143 </p>
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144
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145 <a name="brbme84"></a><h3 class="title" id="brbme84">Appendix: evaluation of the model for the noise PSD</h3>
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146 <p>
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147 The smooth model for the data, that we used to reproduce the synthesized noise, was actually obtained
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148 by applying the procedure of z-domain fitting that we discussed in <a href="ltpda_training_topic_5_1.html">the previous section</a>.
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149 If you want to practise more with this fitting technique, we repost here the steps.
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150 </p>
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151 <p>
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152 In order to extract a reliable model from PSD data we need to discard the first frequency bins;
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153 we do that by means of the <tt>split</tt> method.
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154 </p>
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155
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156 <div class="fragment"><pre>
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157 tnxxr = split(tnxx,plist(<span class="string">'frequencies'</span>, [2e-3 +inf]));
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158 iplot(tnxx,tnxxr)
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159 </pre></div>
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160
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161 <p>
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162 The result should look like:
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163 <div align="center">
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164 <IMG src="images/ltpda_training_1/topic5/ltpda_training_5_2_1.png"align="center" border="0">
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165 </div>
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166 </p>
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167 <p>
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168 Now it's the moment to fit our PSD to extract a smooth model to pass to the noise generator.
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169
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170 First of all we should define a set of proper weights for our fit process.
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171 We smooth our PSD data and then define the weights as the inverse of the absolute value
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172 of the smoothed PSD. This should help the fit function to do a good job with noisy data.
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173 It is worth noting here that weights are not univocally defined and there could be better
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174 ways to define them.
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175 </p>
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176 <div class="fragment"><pre>
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177 stnxx = smoother(tnxxr);
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178 iplot(tnxxr, stnxx)
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179 wgh = 1./abs(stnxx);
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180 </pre></div>
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181 <p>
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182 The result of the <tt>smoother</tt> method is shown in the plot below:
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183 </p>
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184 <div align="center">
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185 <IMG src="images/ltpda_training_1/topic5/ltpda_training_5_2_smoother.png" align="center" border="0">
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186 </div>
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187 <p>
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188 Now let's run an automatic search for the proper model and pass the set
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189 of externally defined weights. The first output of <tt>zDomainFit</tt> is a <tt>miir</tt> filter model;
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190 the second output is the model response. Note that we are setting
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191 <span class="string">'ResFlat'</span> parameter to define the exit condition.
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192 <span class="string">'ResFlat'</span> check the spectral flatness of the
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193 absolute value of the fit residuals.
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194 </p>
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195 <div class="fragment"><pre>
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196 plfit = plist(<span class="string">'fs'</span>,1,...
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197 <span class="string">'AutoSearch'</span>,<span class="string">'on'</span>,...
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198 <span class="string">'StartPolesOpt'</span>,<span class="string">'clog'</span>,...
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199 <span class="string">'maxiter'</span>,50,...
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200 <span class="string">'minorder'</span>,30,...
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201 <span class="string">'maxorder'</span>,45,...
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202 <span class="string">'weights'</span>,wgh,... <span class="comment">% assign externally calculated weights</span>
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203 <span class="string">'rmse'</span>,5,...
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204 <span class="string">'condtype'</span>,<span class="string">'MSE'</span>,...
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205 <span class="string">'msevartol'</span>,0.1,...
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206 <span class="string">'fittol'</span>,0.01,...
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207 <span class="string">'Plot'</span>,<span class="string">'on'</span>,...
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208 <span class="string">'ForceStability'</span>,<span class="string">'off'</span>,...
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209 <span class="string">'CheckProgress'</span>,<span class="string">'off'</span>);
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210
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211 <span class="comment">% Do the fit</span>
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212 fit_results = zDomainFit(tnxxr,plfit);
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213 </pre></div>
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214
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215 <p>
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216 Fit result should look like:
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217 <div align="center">
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218 <IMG src="images/ltpda_training_1/topic5/ltpda_training_5_2_2.png" align="center" border="0">
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219 </div>
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220 </p>
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221
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222 <p>
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223 The fit results consist in a <tt>filterbank</tt> object; we can evaluate the absolute values of
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224 the response of these filters at the frequencies defined by the <tt>x</tt> field of the PSD we want to match.
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225
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226 <div class="fragment"><pre>
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227 <span class="comment">% Evaluate the absolute value of the response of the colouring filter</span>
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228 b = resp(fit_results,plist(<span class="string">'f'</span>,tnxxr.x));
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229 b.abs;
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230
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231 <span class="comment">% Save the model on disk</span>
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232 b.save(plist(<span class="string">'filename'</span>, <span class="string">'topic5/T5_Ex03_ModelNoise.xml'</span>));
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233 </pre></div>
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234
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235
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236
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237
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238
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239
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240
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241
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242
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243
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244
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245
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246 </p>
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247
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248 <br>
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249 <br>
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250 <table class="nav" summary="Navigation aid" border="0" width=
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251 "100%" cellpadding="0" cellspacing="0">
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252 <tr valign="top">
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253 <td align="left" width="20"><a href="ltpda_training_topic_5_1.html"><img src=
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254 "b_prev.gif" border="0" align="bottom" alt=
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255 "System identification in z-domain"></a>&nbsp;</td>
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256
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257 <td align="left">System identification in z-domain</td>
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258
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259 <td>&nbsp;</td>
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260
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261 <td align="right">Fitting time series with polynomials</td>
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262
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263 <td align="right" width="20"><a href=
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264 "ltpda_training_topic_5_3.html"><img src="b_next.gif" border="0" align=
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265 "bottom" alt="Fitting time series with polynomials"></a></td>
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266 </tr>
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267 </table><br>
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268
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269 <p class="copy">&copy;LTP Team</p>
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270 </body>
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271 </html>