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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 <!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"
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10
11 <title>Spectral Estimation Methods (LTPDA Toolbox)</title>
12 <link rel="stylesheet" href="docstyle.css" type="text/css">
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14 <meta name="description" content=
15 "Presents an overview of the features, system requirements, and starting the toolbox.">
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17
18 <body>
19 <a name="top_of_page" id="top_of_page"></a>
20
21 <p style="font-size:1px;">&nbsp;</p>
22
23 <table class="nav" summary="Navigation aid" border="0" width=
24 "100%" cellpadding="0" cellspacing="0">
25 <tr>
26 <td valign="baseline"><b>LTPDA Toolbox</b></td><td><a href="../helptoc.html">contents</a></td>
27
28 <td valign="baseline" align="right"><a href=
29 "specwin_using.html"><img src="b_prev.gif" border="0" align=
30 "bottom" alt="Using spectral windows"></a>&nbsp;&nbsp;&nbsp;<a href=
31 "sigproc_psd.html"><img src="b_next.gif" border="0" align=
32 "bottom" alt="Power spectral density estimates"></a></td>
33 </tr>
34 </table>
35
36 <h1 class="title"><a name="f3-12899" id="f3-12899"></a>Spectral Estimation Methods</h1>
37 <hr>
38
39 <p>
40 <h2>Linear and Log-scale Methods</a></h2>
41
42 <p>
43 The LTPDA Toolbox offers two kind of spectral estimators. The first ones are based on <tt>pwelch</tt> from MATLAB, which is an
44 implementation of Welch's averaged, modified periodogram method <a href="#references"> [1]</a>. More details about spectral
45 estimation techniques can be found <a href="sigproc_intro.html" >here</a>.</p>
46
47 <p>
48 The following pages describe the different Welch-based spectral estimation <tt>ao</tt> methods
49 available in the LTPDA toolbox:
50 <ul>
51 <li><a href="sigproc_psd.html"> power spectral density estimates </a></li>
52 <li><a href="sigproc_cpsd.html"> cross-spectral density estimates </a></li>
53 <li><a href="sigproc_cohere.html"> cross-coherence estimates </a></li>
54 <li><a href="sigproc_tfe.html"> transfer function estimates </a></li>
55 </ul>
56 </p>
57
58 <p>
59 As an alternative, the LTPDA toolbox makes available the same set of estimators, based on an
60 implementation of the LPSD algorithm <a href="#references"> [2]</a>).
61 </p>
62 <p>
63 The following pages describe the different LPSD-based spectral estimation <tt>ao</tt> methods
64 available in the LTPDA toolbox:
65 <ul>
66 <li><a href="sigproc_lpsd.html"> log-scale power spectral density estimates </a></li>
67 <li><a href="sigproc_lcpsd.html"> log-scale cross-spectral density estimates </a></li>
68 <li><a href="sigproc_lcohere.html"> log-scale cross-coherence estimates </a></li>
69 <li><a href="sigproc_ltfe.html"> log-scale transfer function estimates</a></li>
70 </ul>
71 </p>
72
73 <p> More detailed help on spectral estimation can also be found in the help associated with
74 the <a href="matlab:doc('signal')" >Signal Processing Toolbox</a>.
75 </p>
76
77 <h2>Computing the sample variance</h2>
78 <p>
79 The spectral estimators previously described usually return the average of the spectral estimator applied
80 to different segments. This is a standard technique used in spectral analysis to reduce the variance of the
81 estimator.
82 </p>
83 <p>
84 When using one of the previous methods in the LTPDA Toolbox, the value of this average over different segments
85 is stored in the <tt>ao.y</tt> field of the output analysis object, but the user obtains also information about
86 the spectral estimator variance in the <tt>ao.dy</tt> field.
87 </p>
88 <p>
89 The methods listed above store in the <tt>ao.dy</tt> field the <b>standard deviation of the mean</b>, defined as
90 </p>
91 <div align="center">
92 <img src="images/mean_variance.png" >
93 </div>
94 <br>
95 <p>
96 For more details on how the variance of the mean is computed, please refer to the the help page of each method.
97 </p>
98 <p>
99 <table cellspacing="0" class="note" summary="Note" cellpadding="5" border="1">
100 <tr width="90%">
101 <td>
102 Note that when we only have one segment we can not evaluate the variance. This will happen in
103 <ul>
104 <li>linear estimators: when the number of averages is equal to one.</li>
105 <li>log-scale estimators: in the lowest frequency bins.</li>
106 </ul>
107 </td>
108 </tr>
109 </table>
110 </p>
111 <br>
112 <p>
113 The following example compares the sample variance computed by <tt>ao/psd</tt> with two different segment length.
114 </p>
115 <div class="fragment"><pre><br>
116 <span class="comment">% create white noise AO </span>
117 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>);
118 a = ao(pl);
119
120 <span class="comment">% compute psd with different Nfft</span>
121 b1 = psd(a, plist(<span class="string">'Nfft'</span>, 500));
122 b1.setName(<span class="string">'Nfft = 500'</span>);
123 b2 = psd(a, plist(<span class="string">'Nfft'</span>, 200));
124 b2.setName(<span class="string">'Nfft = 200'</span>);
125
126 <span class="comment">% plot with errorbars</span>
127 iplot(b1,b2,plist(<span class="string">'YErrU'</span>,{b1.dy,b2.dy}))
128 </pre></div>
129 <p>
130 <div align="center">
131 <p>
132 </p>
133 <IMG src="images/spectral_error.png" align="center" border="0">
134 </div>
135 </p>
136 <br>
137 <h2><a name="references">References</a></h2>
138
139 <ol>
140 <li> P.D. Welch, The Use of Fast Fourier Transform for the Estimation of Power Spectra: A Method Based on Time Averaging Over Short,
141 Modified Periodograms, <i>IEEE Trans. on Audio and Electroacoustics</i>, Vol. 15, No. 2 (1967), pp. 70 - 73</a></li>
142 <li> M. Troebs, G. Heinzel, Improved spectrum estimation from digitized time series
143 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>
144 </ol>
145
146 </p>
147
148 <br>
149 <br>
150 <table class="nav" summary="Navigation aid" border="0" width=
151 "100%" cellpadding="0" cellspacing="0">
152 <tr valign="top">
153 <td align="left" width="20"><a href="specwin_using.html"><img src=
154 "b_prev.gif" border="0" align="bottom" alt=
155 "Using spectral windows"></a>&nbsp;</td>
156
157 <td align="left">Using spectral windows</td>
158
159 <td>&nbsp;</td>
160
161 <td align="right">Power spectral density estimates</td>
162
163 <td align="right" width="20"><a href=
164 "sigproc_psd.html"><img src="b_next.gif" border="0" align=
165 "bottom" alt="Power spectral density estimates"></a></td>
166 </tr>
167 </table><br>
168
169 <p class="copy">&copy;LTP Team</p>
170 </body>
171 </html>