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author Daniele Nicolodi <nicolodi@science.unitn.it>
date Wed, 23 Nov 2011 19:22:13 +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>Cross coherence 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_cpsd.html"><img src="b_prev.gif" border="0" align=
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30 "bottom" alt="Cross-spectral density estimates"></a>&nbsp;&nbsp;&nbsp;<a href=
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31 "sigproc_tfe.html"><img src="b_next.gif" border="0" align=
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32 "bottom" alt="Transfer function 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>Cross coherence 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/cohere')">ao/cohere</a> estimates the cross-coherence of time-series
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43 signals, included in the input <tt>ao</tt>s following the Welch's averaged, modified periodogram method <a href="#references">[1]</a>.
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44 Data are windowed prior to the estimation of the spectra, by multiplying
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45 it with a <a href="specwin.html">spectral window object</a>, and can be detrended by a polinomial of time in order to reduce the impact
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46 of the border discontinuities. The window length is adjustable to shorter lenghts to reduce the spectral
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47 density uncertainties, and the percentage of subsequent window overlap can be adjusted as well.
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48 <br>
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49 <br>
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50 <h2>Syntax</h2>
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51 </p>
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52 <div class="fragment"><pre>
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53 <br> b = cohere(a1,a2,pl)
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54 </pre>
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55 </div>
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56 <p>
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57 <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
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58 <tt>pl</tt> is an optional parameters list.
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59 <h2>Parameters</h2>
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60 The parameter list <tt>pl</tt> includes the following parameters:</p>
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61 <ul>
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62 <li> <tt>'Nfft'</tt> - number of samples in each fft [default: length of input data]
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63 Notice: analyzing a single segment produces as a result an object full of 1!
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64 A string value containing the variable 'fs' can
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65 also be used, e.g., plist('Nfft', '2*fs') </li>
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66 <li> <tt>'Win'</tt> - the window to be applied to the data to remove the
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67 discontinuities at edges of segments. [default: taken from user prefs].<br>
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68 The window is described by a string with its name and, only in the case of Kaiser window,
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69 the additional parameter <tt>'psll'</tt>. <br>For instance: plist('Win', 'Kaiser', 'psll', 200). </li>
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70 <li> <tt>'Olap'</tt> - segment percent overlap [default: -1, (taken from window function)] </li>
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71 <li> <tt>'Order'</tt> - order of segment detrending <ul>
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72 <li> -1 - no detrending </li>
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73 <li> 0 - subtract mean [default] </li>
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74 <li> 1 - subtract linear fit </li>
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75 <li> N - subtract fit of polynomial, order N </li> </ul> </li>
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76 <li><tt>'Navs'</tt> - number of averages. If set, and if Nfft was set to 0 or -1, the number of points for each window will be calculated to match the request. [default: -1, not set] </li>
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77 <li><tt>'Times'</tt> - interval of time to evaluate the calculation on. If empty [default], it will take the whole section.</li>
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78 <li><tt>'Type'</tt> - type of scaling of the coherence function. Choose between:</li>
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79 <ul>
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80 <li> <tt>'C'</tt> - Complex Coherence Sxy / sqrt(Sxx * Syy) [default]</li>
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81 <li> <tt>'MS'</tt> - Magnitude-Squared Coherence (abs(Sxy))^2 / (Sxx * Syy) </li>
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82 </ul>
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83 </ul>
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84 The length of the window is set by the value of the parameter <tt>'Nfft'</tt>, so that the window
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85 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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86 </p>
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87
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88 <p>As an alternative to setting the number of points <tt>'Nfft'</tt> in each window, it's possible to ask for a given number of coherence estimates by setting the <tt>'Navs'</tt> parameter, and the algorithm takes care of calculating the correct window length, according to the amount of overlap between subsequent segments.</p>
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89 <p>
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90 <table cellspacing="0" class="note" summary="Note" cellpadding="5" border="1">
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91 <tr width="90%">
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92 <td>
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93 If the user doesn't specify the value of a given parameter, the default value is used.
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94 </td>
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95 </tr>
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96 </table>
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97 </p>
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98
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99 <p>
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100 The function makes cross-coherence estimates between the 2 input <tt>ao</tt>s.
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101 If passing two identical objects or linearly combined signals, the output will be 1 at all frequencies. The same will happen if analyzing only a single window.</p>
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102 <h2>Algorithm</h2>
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103 <p>
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104 The algorithm is based in standard MATLAB's tools, as the ones used by <a href="matlab:doc('pwelch')">pwelch</a>. The standard deviation of the mean is computed as <a href="#references">[2]</a>
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105 <div align="center">
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106 <img src="images/cohere_sigma1.png" >
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107 </div>
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108 where
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109 <div align="center">
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110 <img src="images/tfe_sigma2.png" >
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111 </div>
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112 is the coherence function.
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113 <p>
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114 <h2>Example</h2>
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115 </p>
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116 <p>
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117 Evaluation of the cross-coherence of two time-series represented by: a low frequency sinewave signal superimposed to
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118 white noise and a linear drift, and a low frequency sinewave signal at the same frequency, phase shifted and with different
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119 amplitude, superimposed to white noise.
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120 </p>
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121 <div class="fragment"><pre>
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122 <br> <span class="comment">% parameters</span>
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123 nsecs = 5000;
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124 fs = 10;
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125 nfft = 1000;
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126
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127 <span class="comment">% build first signal components</span>
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128 x1 = 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,<span class="string">'yunits'</span>,<span class="string">'m'</span>))
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129 x2 = 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,<span class="string">'yunits'</span>,<span class="string">'m'</span>))
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130 x3 = ao(plist(<span class="string">'tsfcn'</span>, <span class="string">'t'</span>,<span class="string">'nsecs'</span>,nsecs,<span class="string">'fs'</span>,fs,<span class="string">'yunits'</span>,<span class="string">'m'</span>));
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131
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132 <span class="comment">% add components</span>
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133 x = x1 + x2 + x3;
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134
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135 <span class="comment">% build second signal components</span>
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136 y1 = 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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137 y2 = 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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138
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139 <span class="comment">% add components and set units</span>
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140 y = y1 + y2;
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141 y.setYunits(<span class="string">'V'</span>);
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142
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143 <span class="comment">% compute coherence</span>
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144 pl = plist(<span class="string">'win'</span>,<span class="string">'BH92'</span>,<span class="string">'nfft'</span>,nfft, <span class="string">'order'</span>,1);
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145 Cxy = cohere(x,y,pl);
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146
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147 <span class="comment">%plot</span>
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148 iplot(Cxy);
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149 </pre>
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150 </div>
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151 <br>
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152
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153 <img src="images/cohere_1.png" border="3">
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154
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155 <h2><a name="references">References</a></h2>
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156 <br>
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157 <ol>
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158 <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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159 Modified Periodograms, <i>IEEE Trans. on Audio and Electroacoustics</i>, Vol. 15, No. 2 (1967), pp. 70 - 73.</a></li>
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160 <li> G.C. Carter, C.H. Knapp, A.H. Nuttall, Estimation of the Magnitude-Squared Coherence Function Via Overlapped Fast Fourier Transform Processing
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161 , <i>IEEE Trans. on Audio and Electroacoustics</i>, Vol. 21, No. 4 (1973), pp. 337 - 344.</a></li>
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162 </ol>
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163
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164 </p>
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165
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166 <br>
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167 <br>
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168 <table class="nav" summary="Navigation aid" border="0" width=
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169 "100%" cellpadding="0" cellspacing="0">
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170 <tr valign="top">
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171 <td align="left" width="20"><a href="sigproc_cpsd.html"><img src=
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172 "b_prev.gif" border="0" align="bottom" alt=
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173 "Cross-spectral density estimates"></a>&nbsp;</td>
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174
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175 <td align="left">Cross-spectral density estimates</td>
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176
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177 <td>&nbsp;</td>
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178
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179 <td align="right">Transfer function estimates</td>
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180
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181 <td align="right" width="20"><a href=
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182 "sigproc_tfe.html"><img src="b_next.gif" border="0" align=
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183 "bottom" alt="Transfer function estimates"></a></td>
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184 </tr>
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185 </table><br>
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186
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187 <p class="copy">&copy;LTP Team</p>
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188 </body>
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189 </html>