+
−<h2>Description</h2>
+
−<p>
+
−  The LTPDA method <a href="matlab:doc('ao/lpsd')">ao/lpsd</a> estimates the 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>    bs = lpsd(a1,a2,a3,...,pl)
+
−    bs = lpsd(as,pl)
+
−    bs = as.lpsd(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>'Scale'</tt> - scaling of output. Choose from: <ul>
+
−      <li>  'ASD' - amplitude spectral density </li>
+
−      <li>  'PSD' - power spectral density [default] </li>
+
−      <li>  'AS'  - amplitude spectrum </li>
+
−  <li>  'PS'  - power spectrum </li> </ul> </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>
+
−<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>
+
−<h2>Examples</h2>
+
−<p>
+
−  1. Evaluation of the ASD of a time-series represented by a low frequency sinewave signal, superimposed to
+
−  white noise. Comparison of the effect of using standard Pwelch and LPSD on the estimate
+
−  of the white noise level and on resolving the signal.
+
−</p>
+
−<div class="fragment"><pre>
+
−    <br>    <span class="comment">% Create input AO</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>,1000,<span class="string">'fs'</span>,10,<span class="string">'yunits'</span>,<span class="string">'rad'</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>,1000,<span class="string">'fs'</span>,10,<span class="string">'yunits'</span>,<span class="string">'rad'</span>));
+
−    x  = x1 + x2;
+
−    
+
−     <span class="comment">% Compute psd and lpsd </span> 
+
−    pl = plist(<span class="string">'scale'</span>,<span class="string">'ASD'</span>,<span class="string">'order'</span>,-1,<span class="string">'win'</span>,<span class="string">'Kaiser'</span>,<span class="string">'psll'</span>,200);
+
−    y1 = psd(x, pl);
+
−    y2 = lpsd(x, pl);
+
−    
+
−     <span class="comment">% Compare</span> 
+
−    iplot(y1, y2)
+
−  </pre>
+
−</div>
+
−
+
−<img src="images/l_psd_1.png"  border="3">
+
−
+
−
+
−<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>