line source
+ − <h2>Description</h2>
+ − <p>
+ − The LTPDA method <a href="matlab:doc('ao/lcpsd')">ao/lcpsd</a> estimates the cross-power spectral density of time-series
+ − signals, included in the input <tt>ao</tt>s following the LPSD algorithm <a href="#references">[1]</a>. Spectral density estimates are not
+ − evaluated at frequencies which are linear multiples of the minimum frequency resolution <tt>1/T</tt>, where <tt>T</tt>
+ − is the window lenght, but on a logarithmic scale. The algorithm takes care of calculating the frequencies at which to evaluate
+ − the spectral estimate, aiming at minimizing the uncertainty in the estimate itself, and to recalculate a suitable
+ − window length for each frequency bin.
+ − </p>
+ − <p>
+ − Data are windowed prior to the estimation of the spectrum, 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. Detrending is performed on each individual window.
+ − The user can choose the quantity being given in output among ASD (amplitude spectral density),
+ − PSD (power spectral density), AS (amplitude spectrum), and PS (power spectrum).
+ − </p>
+ − <br>
+ − <h2>Syntax</h2>
+ − </p>
+ − <div class="fragment"><pre>
+ − <br> b = lcpsd(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 parameter list.
+ −
+ − <h2>Parameters</h2>
+ − <p>The parameter list <tt>pl</tt> includes the following parameters:</p>
+ − <ul>
+ − <li> <tt>'Kdes'</tt> - desired number of averages [default: 100]</li>
+ − <li> <tt>'Jdes'</tt> - number of spectral frequencies to compute [default: 1000]</li>
+ − <li> <tt>'Lmin'</tt> - minimum segment length [default: 0]</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>
+ − </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>
+ − <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 log-scale CPSD estimates between the 2 input <tt>ao</tt>s. The input argument
+ − list must contain 2 analysis objects, and the output will contain the LCPSD estimate.
+ − If passing two identical objects <tt>ai</tt>, the output will be equivalent to the output of <tt>lpsd(ai)</tt>.
+ − </p>
+ − </pre> </div>
+ − </p>
+ − <h2>Algorithm</h2>
+ − <p>
+ − The algorithm is implemented according to <a href="#references">[1]</a>. In order to
+ − compute the standard deviation of the mean for each frequency bin, the averaging of the different segments is performed using Welford's
+ − algorithm <a href="#references">[2]</a> which allows to compute mean and variance in one loop. <br>
+ − In the LPSD algorithm, the first frequencies bins are usually computed using a single segment containing all the data.
+ − For these bins, the sample variance is set to <tt>Inf</tt>.
+ − </p>
+ − <b>Example</b>
+ − <p>
+ − Evaluation of the log-scale CPSD of 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 input AOs</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>);
+ − 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)) + ...
+ − 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));
+ − y.setYunits(<span class="string">'V'</span>);
+ −
+ − <span class="comment">% Compute log cpsd</span>
+ − z = lcpsd(x,y,plist(<span class="string">'nfft'</span>,1000));
+ −
+ − <span class="comment">% Plot</span>
+ − iplot(z);
+ − </pre>
+ − </div>
+ −
+ − <img src="images/l_cpsd_1.png" alt="" border="3">
+ − <br>
+ − <h2><a name="references">References</a></h2>
+ −
+ − <ol>
+ − <li> M. Troebs, G. Heinzel, Improved spectrum estimation from digitized time series
+ − 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>
+ − <li> B. P. Weldford, Note on a Method for Calculating Corrected Sums of Squares and Products,
+ − <i>Technometrics<i>, Vol. 4, No. 3 (1962), pp 419 - 420.</li>
+ − </ol>