line source
+ − <h2>Description</h2>
+ − <p>
+ − The LTPDA method <a href="matlab:doc('ao/cohere')">ao/cohere</a> estimates the cross-coherence of time-series
+ − signals, included in the input <tt>ao</tt>s following the Welch's averaged, modified periodogram method <a href="#references">[1]</a>.
+ − Data are windowed prior to the estimation of the spectra, by multiplying
+ − 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
+ − of the border discontinuities. The window length is adjustable to shorter lenghts to reduce the spectral
+ − density uncertainties, and the percentage of subsequent window overlap can be adjusted as well.
+ − <br>
+ − <br>
+ − <h2>Syntax</h2>
+ − </p>
+ − <div class="fragment"><pre>
+ − <br> b = cohere(a1,a2,pl)
+ − </pre>
+ − </div>
+ − <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 parameters list.
+ − <h2>Parameters</h2>
+ − The parameter list <tt>pl</tt> includes the following parameters:</p>
+ − <ul>
+ − <li> <tt>'Nfft'</tt> - number of samples in each fft [default: length of input data]
+ − Notice: analyzing a single segment produces as a result an object full of 1!
+ − A string value containing the variable 'fs' can
+ − also be used, e.g., plist('Nfft', '2*fs') </li>
+ − <li> <tt>'Win'</tt> - the window to be applied to the data to remove the
+ − discontinuities at edges of segments. [default: taken from user prefs].<br>
+ − The window is described by a string with its name and, only in the case of Kaiser window,
+ − the additional parameter <tt>'psll'</tt>. <br>For instance: plist('Win', 'Kaiser', 'psll', 200). </li>
+ − <li> <tt>'Olap'</tt> - segment percent overlap [default: -1, (taken from window function)] </li>
+ − <li> <tt>'Order'</tt> - order of segment detrending <ul>
+ − <li> -1 - no detrending </li>
+ − <li> 0 - subtract mean [default] </li>
+ − <li> 1 - subtract linear fit </li>
+ − <li> N - subtract fit of polynomial, order N </li> </ul> </li>
+ − <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>
+ − <li><tt>'Times'</tt> - interval of time to evaluate the calculation on. If empty [default], it will take the whole section.</li>
+ − <li><tt>'Type'</tt> - type of scaling of the coherence function. Choose between:</li>
+ − <ul>
+ − <li> <tt>'C'</tt> - Complex Coherence Sxy / sqrt(Sxx * Syy) [default]</li>
+ − <li> <tt>'MS'</tt> - Magnitude-Squared Coherence (abs(Sxy))^2 / (Sxx * Syy) </li>
+ − </ul>
+ − </ul>
+ − The length of the window is set by the value of the parameter <tt>'Nfft'</tt>, so that the window
+ − is actually rebuilt using only the key features of the window, i.e. the name and, for Kaiser windows, the PSLL.
+ − </p>
+ −
+ − <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>
+ − <p>
+ − <table cellspacing="0" class="note" summary="Note" cellpadding="5" border="1">
+ − <tr width="90%">
+ − <td>
+ − If the user doesn't specify the value of a given parameter, the default value is used.
+ − </td>
+ − </tr>
+ − </table>
+ − </p>
+ −
+ − <p>
+ − The function makes cross-coherence estimates between the 2 input <tt>ao</tt>s.
+ − 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>
+ − <h2>Algorithm</h2>
+ − <p>
+ − 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>
+ − <div align="center">
+ − <img src="images/cohere_sigma1.png" >
+ − </div>
+ − where
+ − <div align="center">
+ − <img src="images/tfe_sigma2.png" >
+ − </div>
+ − is the coherence function.
+ − <p>
+ − <h2>Example</h2>
+ − </p>
+ − <p>
+ − Evaluation of the cross-coherence of two time-series represented by: a low frequency sinewave signal superimposed to
+ − white noise and a linear drift, and a low frequency sinewave signal at the same frequency, phase shifted and with different
+ − amplitude, superimposed to white noise.
+ − </p>
+ − <div class="fragment"><pre>
+ − <br> <span class="comment">% parameters</span>
+ − nsecs = 5000;
+ − fs = 10;
+ − nfft = 1000;
+ −
+ − <span class="comment">% build first signal components</span>
+ − 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>))
+ − 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>))
+ − 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>));
+ −
+ − <span class="comment">% add components</span>
+ − x = x1 + x2 + x3;
+ −
+ − <span class="comment">% build second signal components</span>
+ − 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));
+ − 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));
+ −
+ − <span class="comment">% add components and set units</span>
+ − y = y1 + y2;
+ − y.setYunits(<span class="string">'V'</span>);
+ −
+ − <span class="comment">% compute coherence</span>
+ − pl = plist(<span class="string">'win'</span>,<span class="string">'BH92'</span>,<span class="string">'nfft'</span>,nfft, <span class="string">'order'</span>,1);
+ − Cxy = cohere(x,y,pl);
+ −
+ − <span class="comment">%plot</span>
+ − iplot(Cxy);
+ − </pre>
+ − </div>
+ − <br>
+ −
+ − <img src="images/cohere_1.png" border="3">
+ −
+ − <h2><a name="references">References</a></h2>
+ − <br>
+ − <ol>
+ − <li> P.D. Welch, The Use of Fast Fourier Transform for the Estimation of Power Spectra: A Method Based on Time Averaging Over Short,
+ − Modified Periodograms, <i>IEEE Trans. on Audio and Electroacoustics</i>, Vol. 15, No. 2 (1967), pp. 70 - 73.</a></li>
+ − <li> G.C. Carter, C.H. Knapp, A.H. Nuttall, Estimation of the Magnitude-Squared Coherence Function Via Overlapped Fast Fourier Transform Processing
+ − , <i>IEEE Trans. on Audio and Electroacoustics</i>, Vol. 21, No. 4 (1973), pp. 337 - 344.</a></li>
+ − </ol>