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