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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>Log-scale cross-spectral density estimates (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 "sigproc_lpsd.html"><img src="b_prev.gif" border="0" align=
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30 "bottom" alt="Log-scale power spectral density estimates"></a>&nbsp;&nbsp;&nbsp;<a href=
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31 "sigproc_lcohere.html"><img src="b_next.gif" border="0" align=
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32 "bottom" alt="Log-scale cross coherence 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>Log-scale cross-spectral density estimates</h1>
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37 <hr>
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38
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39 <p>
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40 <h2>Description</h2>
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41 <p>
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42 The LTPDA method <a href="matlab:doc('ao/lcpsd')">ao/lcpsd</a> estimates the cross-power spectral density of time-series
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43 signals, included in the input <tt>ao</tt>s following the LPSD algorithm <a href="#references">[1]</a>. Spectral density estimates are not
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44 evaluated at frequencies which are linear multiples of the minimum frequency resolution <tt>1/T</tt>, where <tt>T</tt>
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45 is the window lenght, but on a logarithmic scale. The algorithm takes care of calculating the frequencies at which to evaluate
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46 the spectral estimate, aiming at minimizing the uncertainty in the estimate itself, and to recalculate a suitable
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47 window length for each frequency bin.
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48 </p>
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49 <p>
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50 Data are windowed prior to the estimation of the spectrum, by multiplying
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51 it with a<a href="specwin.html">spectral window object</a>, and can be detrended by polinomial of time in order to reduce the impact
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52 of the border discontinuities. Detrending is performed on each individual window.
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53 The user can choose the quantity being given in output among ASD (amplitude spectral density),
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54 PSD (power spectral density), AS (amplitude spectrum), and PS (power spectrum).
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55 </p>
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56 <br>
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57 <h2>Syntax</h2>
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58 </p>
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59 <div class="fragment"><pre>
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60 <br> b = lcpsd(a1,a2,pl)
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61 </pre>
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62 </div>
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63 <p>
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64 <tt>a1</tt> and <tt>a2</tt> are the 2 <tt>ao</tt>s containing the input time series to be evaluated, <tt>b</tt> is the output object and <tt>pl</tt> is an optional parameter list.
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65
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66 <h2>Parameters</h2>
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67 <p>The parameter list <tt>pl</tt> includes the following parameters:</p>
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68 <ul>
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69 <li> <tt>'Kdes'</tt> - desired number of averages [default: 100]</li>
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70 <li> <tt>'Jdes'</tt> - number of spectral frequencies to compute [default: 1000]</li>
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71 <li> <tt>'Lmin'</tt> - minimum segment length [default: 0]</li>
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72 <li> <tt>'Win'</tt> - the window to be applied to the data to remove the
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73 discontinuities at edges of segments. [default: taken from user prefs].<br>
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74 The window is described by a string with its name and, only in the case of Kaiser window,
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75 the additional parameter <tt>'psll'</tt>. <br>For instance: plist('Win', 'Kaiser', 'psll', 200). </li>
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76 <li> <tt>'Olap'</tt> - segment percent overlap [default: -1, (taken from window function)] </li>
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77 <li> <tt>'Order'</tt> - order of segment detrending <ul>
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78 <li> -1 - no detrending </li>
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79 <li> 0 - subtract mean [default] </li>
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80 <li> 1 - subtract linear fit </li>
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81 <li> N - subtract fit of polynomial, order N </li> </ul> </li>
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82 </ul>
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83 The length of the window is set by the value of the parameter <tt>'Nfft'</tt>, so that the window
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84 is actually rebuilt using only the key features of the window, i.e. the name and, for Kaiser windows, the PSLL.
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85 </p>
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86 <p>
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87 <table cellspacing="0" class="note" summary="Note" cellpadding="5" border="1">
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88 <tr width="90%">
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89 <td>
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90 If the user doesn't specify the value of a given parameter, the default value is used.
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91 </td>
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92 </tr>
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93 </table>
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94 </p>
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95
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96 <p>The function makes log-scale CPSD estimates between the 2 input <tt>ao</tt>s. The input argument
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97 list must contain 2 analysis objects, and the output will contain the LCPSD estimate.
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98 If passing two identical objects <tt>ai</tt>, the output will be equivalent to the output of <tt>lpsd(ai)</tt>.
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99 </p>
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100 </pre> </div>
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101 </p>
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102 <h2>Algorithm</h2>
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103 <p>
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104 The algorithm is implemented according to <a href="#references">[1]</a>. In order to
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105 compute the standard deviation of the mean for each frequency bin, the averaging of the different segments is performed using Welford's
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106 algorithm <a href="#references">[2]</a> which allows to compute mean and variance in one loop. <br>
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107 In the LPSD algorithm, the first frequencies bins are usually computed using a single segment containing all the data.
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108 For these bins, the sample variance is set to <tt>Inf</tt>.
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109 </p>
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110 <b>Example</b>
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111 <p>
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112 Evaluation of the log-scale CPSD of two time-series represented by: a low frequency sinewave signal superimposed to
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113 white noise, and a low frequency sinewave signal at the same frequency, phase shifted and with different
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114 amplitude, superimposed to white noise.
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115 </p>
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116 <div class="fragment"><pre>
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117 <br> <span class="comment">% Parameters</span>
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118 nsecs = 1000;
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119 fs = 10;
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120
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121 <span class="comment">% Create input AOs</span>
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122 x = ao(plist(<span class="string">'waveform'</span>,<span class="string">'sine wave'</span>,<span class="string">'f'</span>,0.1,<span class="string">'A'</span>,1,<span class="string">'nsecs'</span>,nsecs,<span class="string">'fs'</span>,fs)) + ...
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123 ao(plist(<span class="string">'waveform'</span>,<span class="string">'noise'</span>,<span class="string">'type'</span>,<span class="string">'normal'</span>,<span class="string">'nsecs'</span>,nsecs,<span class="string">'fs'</span>,fs));
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124 x.setYunits(<span class="string">'m'</span>);
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125 y = ao(plist(<span class="string">'waveform'</span>,<span class="string">'sine wave'</span>,<span class="string">'f'</span>,0.1,<span class="string">'A'</span>,2,<span class="string">'nsecs'</span>,nsecs,<span class="string">'fs'</span>,fs,<span class="string">'phi'</span>,90)) + ...
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126 4*ao(plist(<span class="string">'waveform'</span>,<span class="string">'noise'</span>,<span class="string">'type'</span>,<span class="string">'normal'</span>,<span class="string">'nsecs'</span>,nsecs,<span class="string">'fs'</span>,fs));
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127 y.setYunits(<span class="string">'V'</span>);
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128
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129 <span class="comment">% Compute log cpsd</span>
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130 z = lcpsd(x,y,plist(<span class="string">'nfft'</span>,1000));
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131
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132 <span class="comment">% Plot</span>
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133 iplot(z);
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134 </pre>
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135 </div>
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136
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137 <img src="images/l_cpsd_1.png" alt="" border="3">
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138 <br>
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139 <h2><a name="references">References</a></h2>
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140
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141 <ol>
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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 <li> B. P. Weldford, Note on a Method for Calculating Corrected Sums of Squares and Products,
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145 <i>Technometrics<i>, Vol. 4, No. 3 (1962), pp 419 - 420.</li>
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146 </ol>
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147 </p>
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149 <br>
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150 <br>
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151 <table class="nav" summary="Navigation aid" border="0" width=
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152 "100%" cellpadding="0" cellspacing="0">
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153 <tr valign="top">
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154 <td align="left" width="20"><a href="sigproc_lpsd.html"><img src=
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155 "b_prev.gif" border="0" align="bottom" alt=
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156 "Log-scale power spectral density estimates"></a>&nbsp;</td>
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157
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158 <td align="left">Log-scale power spectral density estimates</td>
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159
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160 <td>&nbsp;</td>
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161
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162 <td align="right">Log-scale cross coherence density estimates</td>
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163
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164 <td align="right" width="20"><a href=
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165 "sigproc_lcohere.html"><img src="b_next.gif" border="0" align=
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166 "bottom" alt="Log-scale cross coherence density estimates"></a></td>
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167 </tr>
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168 </table><br>
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169
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170 <p class="copy">&copy;LTP Team</p>
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171 </body>
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172 </html>