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author Daniele Nicolodi <nicolodi@science.unitn.it>
date Mon, 05 Dec 2011 16:20:06 +0100
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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>