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author Daniele Nicolodi <nicolodi@science.unitn.it>
date Wed, 07 Dec 2011 17:29:47 +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>Spectral Estimation Methods (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 "specwin_using.html"><img src="b_prev.gif" border="0" align=
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30 "bottom" alt="Using spectral windows"></a>&nbsp;&nbsp;&nbsp;<a href=
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31 "sigproc_psd.html"><img src="b_next.gif" border="0" align=
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32 "bottom" alt="Power spectral density estimates"></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>Spectral Estimation Methods</h1>
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37 <hr>
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38
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39 <p>
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40 <h2>Linear and Log-scale Methods</a></h2>
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41
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42 <p>
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43 The LTPDA Toolbox offers two kind of spectral estimators. The first ones are based on <tt>pwelch</tt> from MATLAB, which is an
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44 implementation of Welch's averaged, modified periodogram method <a href="#references"> [1]</a>. More details about spectral
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45 estimation techniques can be found <a href="sigproc_intro.html" >here</a>.</p>
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46
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47 <p>
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48 The following pages describe the different Welch-based spectral estimation <tt>ao</tt> methods
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49 available in the LTPDA toolbox:
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50 <ul>
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51 <li><a href="sigproc_psd.html"> power spectral density estimates </a></li>
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52 <li><a href="sigproc_cpsd.html"> cross-spectral density estimates </a></li>
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53 <li><a href="sigproc_cohere.html"> cross-coherence estimates </a></li>
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54 <li><a href="sigproc_tfe.html"> transfer function estimates </a></li>
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55 </ul>
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56 </p>
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57
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58 <p>
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59 As an alternative, the LTPDA toolbox makes available the same set of estimators, based on an
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60 implementation of the LPSD algorithm <a href="#references"> [2]</a>).
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61 </p>
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62 <p>
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63 The following pages describe the different LPSD-based spectral estimation <tt>ao</tt> methods
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64 available in the LTPDA toolbox:
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65 <ul>
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66 <li><a href="sigproc_lpsd.html"> log-scale power spectral density estimates </a></li>
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67 <li><a href="sigproc_lcpsd.html"> log-scale cross-spectral density estimates </a></li>
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68 <li><a href="sigproc_lcohere.html"> log-scale cross-coherence estimates </a></li>
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69 <li><a href="sigproc_ltfe.html"> log-scale transfer function estimates</a></li>
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70 </ul>
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71 </p>
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72
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73 <p> More detailed help on spectral estimation can also be found in the help associated with
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74 the <a href="matlab:doc('signal')" >Signal Processing Toolbox</a>.
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75 </p>
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76
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77 <h2>Computing the sample variance</h2>
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78 <p>
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79 The spectral estimators previously described usually return the average of the spectral estimator applied
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80 to different segments. This is a standard technique used in spectral analysis to reduce the variance of the
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81 estimator.
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82 </p>
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83 <p>
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84 When using one of the previous methods in the LTPDA Toolbox, the value of this average over different segments
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85 is stored in the <tt>ao.y</tt> field of the output analysis object, but the user obtains also information about
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86 the spectral estimator variance in the <tt>ao.dy</tt> field.
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87 </p>
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88 <p>
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89 The methods listed above store in the <tt>ao.dy</tt> field the <b>standard deviation of the mean</b>, defined as
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90 </p>
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91 <div align="center">
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92 <img src="images/mean_variance.png" >
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93 </div>
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94 <br>
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95 <p>
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96 For more details on how the variance of the mean is computed, please refer to the the help page of each method.
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97 </p>
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98 <p>
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99 <table cellspacing="0" class="note" summary="Note" cellpadding="5" border="1">
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100 <tr width="90%">
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101 <td>
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102 Note that when we only have one segment we can not evaluate the variance. This will happen in
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103 <ul>
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104 <li>linear estimators: when the number of averages is equal to one.</li>
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105 <li>log-scale estimators: in the lowest frequency bins.</li>
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106 </ul>
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107 </td>
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108 </tr>
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109 </table>
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110 </p>
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111 <br>
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112 <p>
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113 The following example compares the sample variance computed by <tt>ao/psd</tt> with two different segment length.
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114 </p>
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115 <div class="fragment"><pre><br>
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116 <span class="comment">% create white noise AO </span>
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117 pl = plist(<span class="string">'nsecs'</span>, 500, <span class="string">'fs'</span>, 5, <span class="string">'tsfcn'</span>, <span class="string">'randn(size(t))'</span>);
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118 a = ao(pl);
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119
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120 <span class="comment">% compute psd with different Nfft</span>
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121 b1 = psd(a, plist(<span class="string">'Nfft'</span>, 500));
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122 b1.setName(<span class="string">'Nfft = 500'</span>);
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123 b2 = psd(a, plist(<span class="string">'Nfft'</span>, 200));
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124 b2.setName(<span class="string">'Nfft = 200'</span>);
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125
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126 <span class="comment">% plot with errorbars</span>
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127 iplot(b1,b2,plist(<span class="string">'YErrU'</span>,{b1.dy,b2.dy}))
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128 </pre></div>
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129 <p>
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130 <div align="center">
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131 <p>
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132 </p>
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133 <IMG src="images/spectral_error.png" align="center" border="0">
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134 </div>
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135 </p>
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136 <br>
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137 <h2><a name="references">References</a></h2>
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138
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139 <ol>
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140 <li> P.D. Welch, The Use of Fast Fourier Transform for the Estimation of Power Spectra: A Method Based on Time Averaging Over Short,
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141 Modified Periodograms, <i>IEEE Trans. on Audio and Electroacoustics</i>, Vol. 15, No. 2 (1967), pp. 70 - 73</a></li>
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142 <li> M. Troebs, G. Heinzel, Improved spectrum estimation from digitized time series
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143 on a logarithmic frequency axis, <a href="http://dx.doi.org/10.1016/j.measurement.2005.10.010" ><i>Measurement</i>, Vol. 39 (2006), pp. 120 - 129</a>. See also the <a href="http://dx.doi.org/10.1016/j.measurement.2008.04.004" >Corrigendum</a>. </li>
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144 </ol>
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145
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146 </p>
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147
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148 <br>
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149 <br>
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150 <table class="nav" summary="Navigation aid" border="0" width=
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151 "100%" cellpadding="0" cellspacing="0">
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152 <tr valign="top">
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153 <td align="left" width="20"><a href="specwin_using.html"><img src=
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154 "b_prev.gif" border="0" align="bottom" alt=
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155 "Using spectral windows"></a>&nbsp;</td>
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156
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157 <td align="left">Using spectral windows</td>
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158
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159 <td>&nbsp;</td>
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160
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161 <td align="right">Power spectral density estimates</td>
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162
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163 <td align="right" width="20"><a href=
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164 "sigproc_psd.html"><img src="b_next.gif" border="0" align=
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165 "bottom" alt="Power spectral density estimates"></a></td>
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166 </tr>
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167 </table><br>
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168
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169 <p class="copy">&copy;LTP Team</p>
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170 </body>
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171 </html>