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+ − 1 <h2>Linear and Log-scale Methods</a></h2>
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+ − 2
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+ − 3 <p>
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+ − 4 The LTPDA Toolbox offers two kind of spectral estimators. The first ones are based on <tt>pwelch</tt> from MATLAB, which is an
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+ − 5 implementation of Welch's averaged, modified periodogram method <a href="#references"> [1]</a>. More details about spectral
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+ − 6 estimation techniques can be found <a href="sigproc_intro.html" >here</a>.</p>
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+ − 7
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+ − 8 <p>
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+ − 9 The following pages describe the different Welch-based spectral estimation <tt>ao</tt> methods
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+ − 10 available in the LTPDA toolbox:
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+ − 11 <ul>
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+ − 12 <li><a href="sigproc_psd.html"> power spectral density estimates </a></li>
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+ − 13 <li><a href="sigproc_cpsd.html"> cross-spectral density estimates </a></li>
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+ − 14 <li><a href="sigproc_cohere.html"> cross-coherence estimates </a></li>
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+ − 15 <li><a href="sigproc_tfe.html"> transfer function estimates </a></li>
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+ − 16 </ul>
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+ − 17 </p>
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+ − 18
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+ − 19 <p>
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+ − 20 As an alternative, the LTPDA toolbox makes available the same set of estimators, based on an
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+ − 21 implementation of the LPSD algorithm <a href="#references"> [2]</a>).
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+ − 22 </p>
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+ − 23 <p>
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+ − 24 The following pages describe the different LPSD-based spectral estimation <tt>ao</tt> methods
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+ − 25 available in the LTPDA toolbox:
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+ − 26 <ul>
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+ − 27 <li><a href="sigproc_lpsd.html"> log-scale power spectral density estimates </a></li>
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+ − 28 <li><a href="sigproc_lcpsd.html"> log-scale cross-spectral density estimates </a></li>
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+ − 29 <li><a href="sigproc_lcohere.html"> log-scale cross-coherence estimates </a></li>
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+ − 30 <li><a href="sigproc_ltfe.html"> log-scale transfer function estimates</a></li>
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+ − 31 </ul>
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+ − 32 </p>
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+ − 33
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+ − 34 <p> More detailed help on spectral estimation can also be found in the help associated with
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+ − 35 the <a href="matlab:doc('signal')" >Signal Processing Toolbox</a>.
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+ − 36 </p>
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+ − 37
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+ − 38 <h2>Computing the sample variance</h2>
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+ − 39 <p>
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+ − 40 The spectral estimators previously described usually return the average of the spectral estimator applied
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+ − 41 to different segments. This is a standard technique used in spectral analysis to reduce the variance of the
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+ − 42 estimator.
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+ − 43 </p>
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+ − 44 <p>
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+ − 45 When using one of the previous methods in the LTPDA Toolbox, the value of this average over different segments
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+ − 46 is stored in the <tt>ao.y</tt> field of the output analysis object, but the user obtains also information about
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+ − 47 the spectral estimator variance in the <tt>ao.dy</tt> field.
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+ − 48 </p>
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+ − 49 <p>
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+ − 50 The methods listed above store in the <tt>ao.dy</tt> field the <b>standard deviation of the mean</b>, defined as
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+ − 51 </p>
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+ − 52 <div align="center">
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+ − 53 <img src="images/mean_variance.png" >
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+ − 54 </div>
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+ − 55 <br>
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+ − 56 <p>
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+ − 57 For more details on how the variance of the mean is computed, please refer to the the help page of each method.
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+ − 58 </p>
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+ − 59 <p>
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+ − 60 <table cellspacing="0" class="note" summary="Note" cellpadding="5" border="1">
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+ − 61 <tr width="90%">
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+ − 62 <td>
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+ − 63 Note that when we only have one segment we can not evaluate the variance. This will happen in
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+ − 64 <ul>
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+ − 65 <li>linear estimators: when the number of averages is equal to one.</li>
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+ − 66 <li>log-scale estimators: in the lowest frequency bins.</li>
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+ − 67 </ul>
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+ − 68 </td>
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+ − 69 </tr>
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+ − 70 </table>
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+ − 71 </p>
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+ − 72 <br>
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+ − 73 <p>
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+ − 74 The following example compares the sample variance computed by <tt>ao/psd</tt> with two different segment length.
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+ − 75 </p>
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+ − 76 <div class="fragment"><pre><br>
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+ − 77 <span class="comment">% create white noise AO </span>
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+ − 78 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>);
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+ − 79 a = ao(pl);
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+ − 80
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+ − 81 <span class="comment">% compute psd with different Nfft</span>
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+ − 82 b1 = psd(a, plist(<span class="string">'Nfft'</span>, 500));
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+ − 83 b1.setName(<span class="string">'Nfft = 500'</span>);
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+ − 84 b2 = psd(a, plist(<span class="string">'Nfft'</span>, 200));
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+ − 85 b2.setName(<span class="string">'Nfft = 200'</span>);
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+ − 86
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+ − 87 <span class="comment">% plot with errorbars</span>
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+ − 88 iplot(b1,b2,plist(<span class="string">'YErrU'</span>,{b1.dy,b2.dy}))
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+ − 89 </pre></div>
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+ − 90 <p>
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+ − 91 <div align="center">
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+ − 92 <p>
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+ − 93 </p>
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+ − 94 <IMG src="images/spectral_error.png" align="center" border="0">
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+ − 95 </div>
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+ − 96 </p>
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+ − 97 <br>
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+ − 98 <h2><a name="references">References</a></h2>
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+ − 99
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+ − 100 <ol>
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+ − 101 <li> P.D. Welch, The Use of Fast Fourier Transform for the Estimation of Power Spectra: A Method Based on Time Averaging Over Short,
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+ − 102 Modified Periodograms, <i>IEEE Trans. on Audio and Electroacoustics</i>, Vol. 15, No. 2 (1967), pp. 70 - 73</a></li>
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+ − 103 <li> M. Troebs, G. Heinzel, Improved spectrum estimation from digitized time series
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+ − 104 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>
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+ − 105 </ol>