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+ − 1 <h2>Description</h2>
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+ − 2 <p>
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+ − 3 The LTPDA method <a href="matlab:doc('ao/lcohere')">ao/lcohere</a> estimates the coherence function of time-series
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+ − 4 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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+ − 5 evaluated at frequencies which are linear multiples of the minimum frequency resolution <tt>1/T</tt>, where <tt>T</tt>
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+ − 6 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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+ − 7 the spectral estimate, aiming at minimizing the uncertainty in the estimate itself, and to recalculate a suitable
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+ − 8 window length for each frequency bin.
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+ − 9 </p>
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+ − 10 <p>
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+ − 11 Data are windowed prior to the estimation of the spectrum, by multiplying
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+ − 12 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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+ − 13 of the border discontinuities. Detrending is performed on each individual window.
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+ − 14 The user can choose the quantity being given in output among ASD (amplitude spectral density),
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+ − 15 PSD (power spectral density), AS (amplitude spectrum), and PS (power spectrum).
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+ − 16 </p>
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+ − 17 <br>
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+ − 18 <h2>Syntax</h2>
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+ − 19 </p>
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+ − 20 <div class="fragment"><pre>
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+ − 21 <br> b = lcohere(a1,a2,pl)
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+ − 22 </pre>
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+ − 23 </div>
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+ − 24 <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.
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+ − 25
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+ − 26 <h2>Parameters</h2>
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+ − 27 <p>The parameter list <tt>pl</tt> includes the following parameters:</p>
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+ − 28 <ul>
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+ − 29 <li> <tt>'Kdes'</tt> - desired number of averages [default: 100]</li>
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+ − 30 <li> <tt>'Jdes'</tt> - number of spectral frequencies to compute [default: 1000]</li>
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+ − 31 <li> <tt>'Lmin'</tt> - minimum segment length [default: 0]</li>
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+ − 32 <li> <tt>'Win'</tt> - the window to be applied to the data to remove the
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+ − 33 discontinuities at edges of segments. [default: taken from user prefs].<br>
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+ − 34 The window is described by a string with its name and, only in the case of Kaiser window,
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+ − 35 the additional parameter <tt>'psll'</tt>. <br>For instance: plist('Win', 'Kaiser', 'psll', 200). </li>
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+ − 36 <li> <tt>'Olap'</tt> - segment percent overlap [default: -1, (taken from window function)] </li>
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+ − 37 <li> <tt>'Order'</tt> - order of segment detrending <ul>
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+ − 38 <li> -1 - no detrending </li>
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+ − 39 <li> 0 - subtract mean [default] </li>
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+ − 40 <li> 1 - subtract linear fit </li>
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+ − 41 <li> N - subtract fit of polynomial, order N </li> </ul> </li>
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+ − 42 <li><tt>'Type'</tt> - type of scaling of the coherence function. Choose between:</li>
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+ − 43 <ul>
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+ − 44 <li> <tt>'C'</tt> - Complex Coherence Sxy / sqrt(Sxx * Syy) [default ]</li>
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+ − 45 <li> <tt>'MS'</tt> - Magnitude-Squared Coherence (abs(Sxy))^2 / (Sxx * Syy) </li>
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+ − 46 </ul>
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+ − 47 </ul>
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+ − 48 The length of the window is set by the value of the parameter <tt>'Nfft'</tt>, so that the window
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+ − 49 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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+ − 50 </p>
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+ − 51 <p>
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+ − 52 <table cellspacing="0" class="note" summary="Note" cellpadding="5" border="1">
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+ − 53 <tr width="90%">
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+ − 54 <td>
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+ − 55 If the user doesn't specify the value of a given parameter, the default value is used.
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+ − 56 </td>
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+ − 57 </tr>
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+ − 58 </table>
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+ − 59 </p>
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+ − 60 <p>
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+ − 61 The function makes magnitude-squadred coherence estimates between the 2 input <tt>ao</tt>s, on a logaritmic frequency scale.
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+ − 62 If passing two identical objects <tt>ai</tt> or linearly combined signals, the output will be 1 at all frequencies.</p>
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+ − 63 </pre> </div>
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+ − 64 </p>
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+ − 65 <h2>Algorithm</h2>
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+ − 66 <p>
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+ − 67 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>:
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+ − 68 </p>
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+ − 69 <div align="center">
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+ − 70 <img src="images/cohere_sigma1.png" >
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+ − 71 </div>
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+ − 72 where
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+ − 73 <div align="center">
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+ − 74 <img src="images/tfe_sigma2.png" >
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+ − 75 </div>
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+ − 76 <br>
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+ − 77 <p>
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+ − 78 is the coherence function.
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+ − 79 In the LPSD algorithm, the first frequencies bins are usually computed using a single segment containing all the data.
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+ − 80 For these bins, the sample variance is set to <tt>Inf</tt>.
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+ − 81 </p>
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+ − 82 <h2>Example</h2>
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+ − 83 <p>
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+ − 84 Evaluation of the coherence of two time-series represented by: a low frequency sinewave signal superimposed to
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+ − 85 white noise, and a low frequency sinewave signal at the same frequency, phase shifted and with different
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+ − 86 amplitude, superimposed to white noise.
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+ − 87 </p>
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+ − 88 <div class="fragment"><pre>
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+ − 89 <br> <span class="comment">% Parameters</span>
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+ − 90 nsecs = 1000;
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+ − 91 fs = 10;
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+ − 92 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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+ − 93 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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+ − 94 x.setYunits(<span class="string">'m'</span>);
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+ − 95 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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+ − 96 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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+ − 97 y.setYunits(<span class="string">'V'</span>);
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+ − 98
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+ − 99 <span class="comment">% Compute log coherence</span>
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+ − 100 Cxy = lcohere(x,y,plist(<span class="string">'win'</span>,<span class="string">'Kaiser'</span>,<span class="string">'psll'</span>,200));
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+ − 101
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+ − 102 <span class="comment">% Plot</span>
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+ − 103 iplot(Cxy);
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+ − 104 </pre>
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+ − 105 </div>
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+ − 106
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+ − 107 <img src="images/l_cohere_1.png" alt="" border="3">
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+ − 108 <br>
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+ − 109 <!-- <img src="images/l_cohere_2.png" alt="" border="3">
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+ − 110 <br> -->
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+ − 111
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+ − 112 <h2><a name="references">References</a></h2>
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+ − 113
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+ − 114 <ol>
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+ − 115 <li> M. Troebs, G. Heinzel, Improved spectrum estimation from digitized time series
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+ − 116 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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+ − 117 <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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+ − 118 , <i>IEEE Trans. on Audio and Electroacoustics</i>, Vol. 21, No. 4 (1973), pp. 337 - 344.</a></li>
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+ − 119 </ol>