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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 <h2>Description</h2>
2 <p>
3 The LTPDA method <a href="matlab:doc('ao/cohere')">ao/cohere</a> estimates the cross-coherence of time-series
4 signals, included in the input <tt>ao</tt>s following the Welch's averaged, modified periodogram method <a href="#references">[1]</a>.
5 Data are windowed prior to the estimation of the spectra, by multiplying
6 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
7 of the border discontinuities. The window length is adjustable to shorter lenghts to reduce the spectral
8 density uncertainties, and the percentage of subsequent window overlap can be adjusted as well.
9 <br>
10 <br>
11 <h2>Syntax</h2>
12 </p>
13 <div class="fragment"><pre>
14 <br> b = cohere(a1,a2,pl)
15 </pre>
16 </div>
17 <p>
18 <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
19 <tt>pl</tt> is an optional parameters list.
20 <h2>Parameters</h2>
21 The parameter list <tt>pl</tt> includes the following parameters:</p>
22 <ul>
23 <li> <tt>'Nfft'</tt> - number of samples in each fft [default: length of input data]
24 Notice: analyzing a single segment produces as a result an object full of 1!
25 A string value containing the variable 'fs' can
26 also be used, e.g., plist('Nfft', '2*fs') </li>
27 <li> <tt>'Win'</tt> - the window to be applied to the data to remove the
28 discontinuities at edges of segments. [default: taken from user prefs].<br>
29 The window is described by a string with its name and, only in the case of Kaiser window,
30 the additional parameter <tt>'psll'</tt>. <br>For instance: plist('Win', 'Kaiser', 'psll', 200). </li>
31 <li> <tt>'Olap'</tt> - segment percent overlap [default: -1, (taken from window function)] </li>
32 <li> <tt>'Order'</tt> - order of segment detrending <ul>
33 <li> -1 - no detrending </li>
34 <li> 0 - subtract mean [default] </li>
35 <li> 1 - subtract linear fit </li>
36 <li> N - subtract fit of polynomial, order N </li> </ul> </li>
37 <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>
38 <li><tt>'Times'</tt> - interval of time to evaluate the calculation on. If empty [default], it will take the whole section.</li>
39 <li><tt>'Type'</tt> - type of scaling of the coherence function. Choose between:</li>
40 <ul>
41 <li> <tt>'C'</tt> - Complex Coherence Sxy / sqrt(Sxx * Syy) [default]</li>
42 <li> <tt>'MS'</tt> - Magnitude-Squared Coherence (abs(Sxy))^2 / (Sxx * Syy) </li>
43 </ul>
44 </ul>
45 The length of the window is set by the value of the parameter <tt>'Nfft'</tt>, so that the window
46 is actually rebuilt using only the key features of the window, i.e. the name and, for Kaiser windows, the PSLL.
47 </p>
48
49 <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>
50 <p>
51 <table cellspacing="0" class="note" summary="Note" cellpadding="5" border="1">
52 <tr width="90%">
53 <td>
54 If the user doesn't specify the value of a given parameter, the default value is used.
55 </td>
56 </tr>
57 </table>
58 </p>
59
60 <p>
61 The function makes cross-coherence estimates between the 2 input <tt>ao</tt>s.
62 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>
63 <h2>Algorithm</h2>
64 <p>
65 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>
66 <div align="center">
67 <img src="images/cohere_sigma1.png" >
68 </div>
69 where
70 <div align="center">
71 <img src="images/tfe_sigma2.png" >
72 </div>
73 is the coherence function.
74 <p>
75 <h2>Example</h2>
76 </p>
77 <p>
78 Evaluation of the cross-coherence of two time-series represented by: a low frequency sinewave signal superimposed to
79 white noise and a linear drift, and a low frequency sinewave signal at the same frequency, phase shifted and with different
80 amplitude, superimposed to white noise.
81 </p>
82 <div class="fragment"><pre>
83 <br> <span class="comment">% parameters</span>
84 nsecs = 5000;
85 fs = 10;
86 nfft = 1000;
87
88 <span class="comment">% build first signal components</span>
89 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>))
90 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>))
91 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>));
92
93 <span class="comment">% add components</span>
94 x = x1 + x2 + x3;
95
96 <span class="comment">% build second signal components</span>
97 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));
98 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));
99
100 <span class="comment">% add components and set units</span>
101 y = y1 + y2;
102 y.setYunits(<span class="string">'V'</span>);
103
104 <span class="comment">% compute coherence</span>
105 pl = plist(<span class="string">'win'</span>,<span class="string">'BH92'</span>,<span class="string">'nfft'</span>,nfft, <span class="string">'order'</span>,1);
106 Cxy = cohere(x,y,pl);
107
108 <span class="comment">%plot</span>
109 iplot(Cxy);
110 </pre>
111 </div>
112 <br>
113
114 <img src="images/cohere_1.png" border="3">
115
116 <h2><a name="references">References</a></h2>
117 <br>
118 <ol>
119 <li> P.D. Welch, The Use of Fast Fourier Transform for the Estimation of Power Spectra: A Method Based on Time Averaging Over Short,
120 Modified Periodograms, <i>IEEE Trans. on Audio and Electroacoustics</i>, Vol. 15, No. 2 (1967), pp. 70 - 73.</a></li>
121 <li> G.C. Carter, C.H. Knapp, A.H. Nuttall, Estimation of the Magnitude-Squared Coherence Function Via Overlapped Fast Fourier Transform Processing
122 , <i>IEEE Trans. on Audio and Electroacoustics</i>, Vol. 21, No. 4 (1973), pp. 337 - 344.</a></li>
123 </ol>