line source
+ − <h2>Description</h2>
+ − <p>
+ − The LTPDA method <a href="matlab:doc('ao/tfe')">ao/tfe</a> estimates the transfer function 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 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 = tfe(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.
+ − </p>
+ − <h2>Parameters</h2>
+ − The parameter list <tt>pl</tt> includes the following parameters:
+ − <ul>
+ − <li> <tt>'Nfft'</tt> - number of samples in each fft [default: length of input data]
+ − 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>
+ − </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 TFE 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 transfer functions estimates between the 2 input <tt>ao</tt>s, and the output will contain the transfer function estimate from the first <tt>ao</tt> to the second.</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
+ − <div align="center">
+ − <img src="images/tfe_sigma1.png" >
+ − </div>
+ − where
+ − <div align="center">
+ − <img src="images/tfe_sigma2.png" >
+ − </div>
+ − is the coherence function.
+ − </p>
+ − <h2>Example</h2>
+ − <p>
+ − Evaluation of the transfer function between two time-series represented by:
+ − a low frequency sinewave signal superimposed to
+ − white noise, 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 = 1000;
+ − fs = 10;
+ −
+ − <span class="comment">% create first signal AO</span>
+ − 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)) + ...
+ − 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));
+ − x.setYunits(<span class="string">'m'</span>);
+ −
+ − <span class="comment">% create second signal AO</span>
+ − 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)) + ...
+ − 0.1*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));
+ − y.setYunits(<span class="string">'rad'</span>);
+ −
+ − <span class="comment">% compute transfer function</span>
+ − nfft = 1000;
+ − psll = 200;
+ − Txy = tfe(x,y,plist(<span class="string">'win'</span>,<span class="string">'Kaiser'</span>,<span class="string">'psll'</span>,psll,<span class="string">'nfft'</span>,nfft));
+ −
+ − <span class="comment">% plot</span>
+ − iplot(Txy)
+ − </pre>
+ − </div>
+ − <br>
+ − <img src="images/transfer_1.png" alt="" border="3">
+ −
+ − <h2><a name="references">References</a></h2>
+ −
+ − <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>
+ − </ol>
+ −
+ −