<h2>Linear and Log-scale Methods</a></h2>

<p>
  The LTPDA Toolbox offers two kind of spectral estimators. The first ones are based on <tt>pwelch</tt> from MATLAB, which is an
  implementation of Welch's averaged, modified periodogram method  <a href="#references"> [1]</a>. More details about spectral 
estimation techniques can be found <a href="sigproc_intro.html" >here</a>.</p>

<p>
  The following pages describe the different Welch-based spectral estimation <tt>ao</tt> methods
  available in the LTPDA toolbox:
  <ul>
    <li><a href="sigproc_psd.html"> power spectral density estimates </a></li>
    <li><a href="sigproc_cpsd.html"> cross-spectral density estimates </a></li>
    <li><a href="sigproc_cohere.html"> cross-coherence estimates </a></li>
    <li><a href="sigproc_tfe.html"> transfer function estimates </a></li>
  </ul>
</p>

<p>
  As an alternative, the LTPDA toolbox makes available the same set of estimators, based on an
  implementation of the LPSD algorithm  <a href="#references"> [2]</a>).
</p>
<p>
  The following pages describe the different LPSD-based spectral estimation <tt>ao</tt> methods
  available in the LTPDA toolbox:
  <ul>
    <li><a href="sigproc_lpsd.html"> log-scale power spectral density estimates </a></li>
    <li><a href="sigproc_lcpsd.html"> log-scale cross-spectral density estimates </a></li>
    <li><a href="sigproc_lcohere.html"> log-scale cross-coherence estimates </a></li>
    <li><a href="sigproc_ltfe.html"> log-scale transfer function estimates</a></li>
  </ul>
</p>

<p> More detailed help on spectral estimation can also be found in the help associated with
  the <a href="matlab:doc('signal')" >Signal Processing Toolbox</a>.
</p>

<h2>Computing the sample variance</h2>
<p>
  The spectral estimators previously described usually return the average of the spectral estimator applied 
  to different segments. This is a standard technique used in spectral analysis to reduce the variance of the 
  estimator.
</p>
<p>
  When using one of the previous methods in the LTPDA Toolbox, the value of this average over different segments 
  is stored in the <tt>ao.y</tt> field of the output analysis object, but the user obtains also information about
  the spectral estimator variance in the <tt>ao.dy</tt> field.
</p>
<p>
  The methods listed above store in the <tt>ao.dy</tt> field the <b>standard deviation of the mean</b>, defined as
</p>
<div align="center">
  <img src="images/mean_variance.png" >
</div>
<br>
<p>
  For more details on how the variance of the mean is computed, please refer to the the help page of each method. 
</p>
 <p>
  <table cellspacing="0" class="note" summary="Note" cellpadding="5" border="1">
    <tr width="90%">
      <td>
        Note that when we only have one segment we can not evaluate the variance. This will happen in 
        <ul>
          <li>linear estimators: when the number of averages is equal to one.</li>
          <li>log-scale estimators: in the lowest frequency bins.</li>
        </ul>
      </td>
    </tr>
  </table>
</p>
<br> 
<p>
  The following example compares the sample variance computed by <tt>ao/psd</tt> with two different segment length.
</p>
<div class="fragment"><pre><br>
<span class="comment">% create white noise AO </span>
pl = plist(<span class="string">'nsecs'</span>, 500, <span class="string">'fs'</span>, 5, <span class="string">'tsfcn'</span>, <span class="string">'randn(size(t))'</span>);
a = ao(pl);

<span class="comment">% compute psd with different Nfft</span>
b1 = psd(a, plist(<span class="string">'Nfft'</span>, 500));
b1.setName(<span class="string">'Nfft = 500'</span>);
b2 = psd(a, plist(<span class="string">'Nfft'</span>, 200));
b2.setName(<span class="string">'Nfft = 200'</span>);

<span class="comment">% plot with errorbars</span>
iplot(b1,b2,plist(<span class="string">'YErrU'</span>,{b1.dy,b2.dy}))   
</pre></div>
<p>
  <div align="center">
    <p>
    </p>
    <IMG src="images/spectral_error.png" align="center" border="0">
  </div>
</p>
<br>
<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>
 <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>  
</ol>