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author | Daniele Nicolodi <nicolodi@science.unitn.it> |
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date | Mon, 05 Dec 2011 16:20:06 +0100 |
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<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/1999/REC-html401-19991224/loose.dtd"> <html lang="en"> <head> <meta name="generator" content= "HTML Tidy for Mac OS X (vers 1st December 2004), see www.w3.org"> <meta http-equiv="Content-Type" content= "text/html; charset=us-ascii"> <title>Spectral Estimation Methods (LTPDA Toolbox)</title> <link rel="stylesheet" href="docstyle.css" type="text/css"> <meta name="generator" content="DocBook XSL Stylesheets V1.52.2"> <meta name="description" content= "Presents an overview of the features, system requirements, and starting the toolbox."> </head> <body> <a name="top_of_page" id="top_of_page"></a> <p style="font-size:1px;"> </p> <table class="nav" summary="Navigation aid" border="0" width= "100%" cellpadding="0" cellspacing="0"> <tr> <td valign="baseline"><b>LTPDA Toolbox</b></td><td><a href="../helptoc.html">contents</a></td> <td valign="baseline" align="right"><a href= "specwin_using.html"><img src="b_prev.gif" border="0" align= "bottom" alt="Using spectral windows"></a> <a href= "sigproc_psd.html"><img src="b_next.gif" border="0" align= "bottom" alt="Power spectral density estimates"></a></td> </tr> </table> <h1 class="title"><a name="f3-12899" id="f3-12899"></a>Spectral Estimation Methods</h1> <hr> <p> <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> </p> <br> <br> <table class="nav" summary="Navigation aid" border="0" width= "100%" cellpadding="0" cellspacing="0"> <tr valign="top"> <td align="left" width="20"><a href="specwin_using.html"><img src= "b_prev.gif" border="0" align="bottom" alt= "Using spectral windows"></a> </td> <td align="left">Using spectral windows</td> <td> </td> <td align="right">Power spectral density estimates</td> <td align="right" width="20"><a href= "sigproc_psd.html"><img src="b_next.gif" border="0" align= "bottom" alt="Power spectral density estimates"></a></td> </tr> </table><br> <p class="copy">©LTP Team</p> </body> </html>