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author Daniele Nicolodi <nicolodi@science.unitn.it>
date Wed, 23 Nov 2011 19:22:13 +0100
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11 <title>Log-scale cross coherence density estimates (LTPDA Toolbox)</title>
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21 <p style="font-size:1px;">&nbsp;</p>
22
23 <table class="nav" summary="Navigation aid" border="0" width=
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25 <tr>
26 <td valign="baseline"><b>LTPDA Toolbox</b></td><td><a href="../helptoc.html">contents</a></td>
27
28 <td valign="baseline" align="right"><a href=
29 "sigproc_lcpsd.html"><img src="b_prev.gif" border="0" align=
30 "bottom" alt="Log-scale cross-spectral density estimates"></a>&nbsp;&nbsp;&nbsp;<a href=
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32 "bottom" alt="Log-scale transfer function estimates"></a></td>
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34 </table>
35
36 <h1 class="title"><a name="f3-12899" id="f3-12899"></a>Log-scale cross coherence density estimates</h1>
37 <hr>
38
39 <p>
40 <h2>Description</h2>
41 <p>
42 The LTPDA method <a href="matlab:doc('ao/lcohere')">ao/lcohere</a> estimates the coherence function of time-series
43 signals, included in the input <tt>ao</tt>s following the LPSD algorithm <a href="#references">[1]</a>. Spectral density estimates are not
44 evaluated at frequencies which are linear multiples of the minimum frequency resolution <tt>1/T</tt>, where <tt>T</tt>
45 is the window lenght, but on a logarithmic scale. The algorithm takes care of calculating the frequencies at which to evaluate
46 the spectral estimate, aiming at minimizing the uncertainty in the estimate itself, and to recalculate a suitable
47 window length for each frequency bin.
48 </p>
49 <p>
50 Data are windowed prior to the estimation of the spectrum, by multiplying
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
52 of the border discontinuities. Detrending is performed on each individual window.
53 The user can choose the quantity being given in output among ASD (amplitude spectral density),
54 PSD (power spectral density), AS (amplitude spectrum), and PS (power spectrum).
55 </p>
56 <br>
57 <h2>Syntax</h2>
58 </p>
59 <div class="fragment"><pre>
60 <br> b = lcohere(a1,a2,pl)
61 </pre>
62 </div>
63 <p> <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.
64
65 <h2>Parameters</h2>
66 <p>The parameter list <tt>pl</tt> includes the following parameters:</p>
67 <ul>
68 <li> <tt>'Kdes'</tt> - desired number of averages [default: 100]</li>
69 <li> <tt>'Jdes'</tt> - number of spectral frequencies to compute [default: 1000]</li>
70 <li> <tt>'Lmin'</tt> - minimum segment length [default: 0]</li>
71 <li> <tt>'Win'</tt> - the window to be applied to the data to remove the
72 discontinuities at edges of segments. [default: taken from user prefs].<br>
73 The window is described by a string with its name and, only in the case of Kaiser window,
74 the additional parameter <tt>'psll'</tt>. <br>For instance: plist('Win', 'Kaiser', 'psll', 200). </li>
75 <li> <tt>'Olap'</tt> - segment percent overlap [default: -1, (taken from window function)] </li>
76 <li> <tt>'Order'</tt> - order of segment detrending <ul>
77 <li> -1 - no detrending </li>
78 <li> 0 - subtract mean [default] </li>
79 <li> 1 - subtract linear fit </li>
80 <li> N - subtract fit of polynomial, order N </li> </ul> </li>
81 <li><tt>'Type'</tt> - type of scaling of the coherence function. Choose between:</li>
82 <ul>
83 <li> <tt>'C'</tt> - Complex Coherence Sxy / sqrt(Sxx * Syy) [default ]</li>
84 <li> <tt>'MS'</tt> - Magnitude-Squared Coherence (abs(Sxy))^2 / (Sxx * Syy) </li>
85 </ul>
86 </ul>
87 The length of the window is set by the value of the parameter <tt>'Nfft'</tt>, so that the window
88 is actually rebuilt using only the key features of the window, i.e. the name and, for Kaiser windows, the PSLL.
89 </p>
90 <p>
91 <table cellspacing="0" class="note" summary="Note" cellpadding="5" border="1">
92 <tr width="90%">
93 <td>
94 If the user doesn't specify the value of a given parameter, the default value is used.
95 </td>
96 </tr>
97 </table>
98 </p>
99 <p>
100 The function makes magnitude-squadred coherence estimates between the 2 input <tt>ao</tt>s, on a logaritmic frequency scale.
101 If passing two identical objects <tt>ai</tt> or linearly combined signals, the output will be 1 at all frequencies.</p>
102 </pre> </div>
103 </p>
104 <h2>Algorithm</h2>
105 <p>
106 The algorithm is implemented according to <a href="#references">[1]</a>. The standard deviation of the mean is computed according to <a href="#references">[2]</a>:
107 </p>
108 <div align="center">
109 <img src="images/cohere_sigma1.png" >
110 </div>
111 where
112 <div align="center">
113 <img src="images/tfe_sigma2.png" >
114 </div>
115 <br>
116 <p>
117 is the coherence function.
118 In the LPSD algorithm, the first frequencies bins are usually computed using a single segment containing all the data.
119 For these bins, the sample variance is set to <tt>Inf</tt>.
120 </p>
121 <h2>Example</h2>
122 <p>
123 Evaluation of the coherence of two time-series represented by: a low frequency sinewave signal superimposed to
124 white noise, and a low frequency sinewave signal at the same frequency, phase shifted and with different
125 amplitude, superimposed to white noise.
126 </p>
127 <div class="fragment"><pre>
128 <br> <span class="comment">% Parameters</span>
129 nsecs = 1000;
130 fs = 10;
131 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)) + ...
132 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));
133 x.setYunits(<span class="string">'m'</span>);
134 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)) + ...
135 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));
136 y.setYunits(<span class="string">'V'</span>);
137
138 <span class="comment">% Compute log coherence</span>
139 Cxy = lcohere(x,y,plist(<span class="string">'win'</span>,<span class="string">'Kaiser'</span>,<span class="string">'psll'</span>,200));
140
141 <span class="comment">% Plot</span>
142 iplot(Cxy);
143 </pre>
144 </div>
145
146 <img src="images/l_cohere_1.png" alt="" border="3">
147 <br>
148 <!-- <img src="images/l_cohere_2.png" alt="" border="3">
149 <br> -->
150
151 <h2><a name="references">References</a></h2>
152
153 <ol>
154 <li> M. Troebs, G. Heinzel, Improved spectrum estimation from digitized time series
155 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>
156 <li> G.C. Carter, C.H. Knapp, A.H. Nuttall, Estimation of the Magnitude-Squared Coherence Function Via Overlapped Fast Fourier Transform Processing
157 , <i>IEEE Trans. on Audio and Electroacoustics</i>, Vol. 21, No. 4 (1973), pp. 337 - 344.</a></li>
158 </ol>
159 </p>
160
161 <br>
162 <br>
163 <table class="nav" summary="Navigation aid" border="0" width=
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165 <tr valign="top">
166 <td align="left" width="20"><a href="sigproc_lcpsd.html"><img src=
167 "b_prev.gif" border="0" align="bottom" alt=
168 "Log-scale cross-spectral density estimates"></a>&nbsp;</td>
169
170 <td align="left">Log-scale cross-spectral density estimates</td>
171
172 <td>&nbsp;</td>
173
174 <td align="right">Log-scale transfer function estimates</td>
175
176 <td align="right" width="20"><a href=
177 "sigproc_ltfe.html"><img src="b_next.gif" border="0" align=
178 "bottom" alt="Log-scale transfer function estimates"></a></td>
179 </tr>
180 </table><br>
181
182 <p class="copy">&copy;LTP Team</p>
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