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author Daniele Nicolodi <nicolodi@science.unitn.it>
date Wed, 23 Nov 2011 19:22:13 +0100
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1 <h2>Linear and Log-scale Methods</a></h2>
2
3 <p>
4 The LTPDA Toolbox offers two kind of spectral estimators. The first ones are based on <tt>pwelch</tt> from MATLAB, which is an
5 implementation of Welch's averaged, modified periodogram method <a href="#references"> [1]</a>. More details about spectral
6 estimation techniques can be found <a href="sigproc_intro.html" >here</a>.</p>
7
8 <p>
9 The following pages describe the different Welch-based spectral estimation <tt>ao</tt> methods
10 available in the LTPDA toolbox:
11 <ul>
12 <li><a href="sigproc_psd.html"> power spectral density estimates </a></li>
13 <li><a href="sigproc_cpsd.html"> cross-spectral density estimates </a></li>
14 <li><a href="sigproc_cohere.html"> cross-coherence estimates </a></li>
15 <li><a href="sigproc_tfe.html"> transfer function estimates </a></li>
16 </ul>
17 </p>
18
19 <p>
20 As an alternative, the LTPDA toolbox makes available the same set of estimators, based on an
21 implementation of the LPSD algorithm <a href="#references"> [2]</a>).
22 </p>
23 <p>
24 The following pages describe the different LPSD-based spectral estimation <tt>ao</tt> methods
25 available in the LTPDA toolbox:
26 <ul>
27 <li><a href="sigproc_lpsd.html"> log-scale power spectral density estimates </a></li>
28 <li><a href="sigproc_lcpsd.html"> log-scale cross-spectral density estimates </a></li>
29 <li><a href="sigproc_lcohere.html"> log-scale cross-coherence estimates </a></li>
30 <li><a href="sigproc_ltfe.html"> log-scale transfer function estimates</a></li>
31 </ul>
32 </p>
33
34 <p> More detailed help on spectral estimation can also be found in the help associated with
35 the <a href="matlab:doc('signal')" >Signal Processing Toolbox</a>.
36 </p>
37
38 <h2>Computing the sample variance</h2>
39 <p>
40 The spectral estimators previously described usually return the average of the spectral estimator applied
41 to different segments. This is a standard technique used in spectral analysis to reduce the variance of the
42 estimator.
43 </p>
44 <p>
45 When using one of the previous methods in the LTPDA Toolbox, the value of this average over different segments
46 is stored in the <tt>ao.y</tt> field of the output analysis object, but the user obtains also information about
47 the spectral estimator variance in the <tt>ao.dy</tt> field.
48 </p>
49 <p>
50 The methods listed above store in the <tt>ao.dy</tt> field the <b>standard deviation of the mean</b>, defined as
51 </p>
52 <div align="center">
53 <img src="images/mean_variance.png" >
54 </div>
55 <br>
56 <p>
57 For more details on how the variance of the mean is computed, please refer to the the help page of each method.
58 </p>
59 <p>
60 <table cellspacing="0" class="note" summary="Note" cellpadding="5" border="1">
61 <tr width="90%">
62 <td>
63 Note that when we only have one segment we can not evaluate the variance. This will happen in
64 <ul>
65 <li>linear estimators: when the number of averages is equal to one.</li>
66 <li>log-scale estimators: in the lowest frequency bins.</li>
67 </ul>
68 </td>
69 </tr>
70 </table>
71 </p>
72 <br>
73 <p>
74 The following example compares the sample variance computed by <tt>ao/psd</tt> with two different segment length.
75 </p>
76 <div class="fragment"><pre><br>
77 <span class="comment">% create white noise AO </span>
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>);
79 a = ao(pl);
80
81 <span class="comment">% compute psd with different Nfft</span>
82 b1 = psd(a, plist(<span class="string">'Nfft'</span>, 500));
83 b1.setName(<span class="string">'Nfft = 500'</span>);
84 b2 = psd(a, plist(<span class="string">'Nfft'</span>, 200));
85 b2.setName(<span class="string">'Nfft = 200'</span>);
86
87 <span class="comment">% plot with errorbars</span>
88 iplot(b1,b2,plist(<span class="string">'YErrU'</span>,{b1.dy,b2.dy}))
89 </pre></div>
90 <p>
91 <div align="center">
92 <p>
93 </p>
94 <IMG src="images/spectral_error.png" align="center" border="0">
95 </div>
96 </p>
97 <br>
98 <h2><a name="references">References</a></h2>
99
100 <ol>
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,
102 Modified Periodograms, <i>IEEE Trans. on Audio and Electroacoustics</i>, Vol. 15, No. 2 (1967), pp. 70 - 73</a></li>
103 <li> M. Troebs, G. Heinzel, Improved spectrum estimation from digitized time series
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>
105 </ol>