0
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 1 <!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 2 "http://www.w3.org/TR/1999/REC-html401-19991224/loose.dtd">
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 3
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 4 <html lang="en">
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 5 <head>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 6 <meta name="generator" content=
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 7 "HTML Tidy for Mac OS X (vers 1st December 2004), see www.w3.org">
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 8 <meta http-equiv="Content-Type" content=
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 9 "text/html; charset=us-ascii">
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 10
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 11 <title>Spectral Estimation Methods (LTPDA Toolbox)</title>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 12 <link rel="stylesheet" href="docstyle.css" type="text/css">
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 13 <meta name="generator" content="DocBook XSL Stylesheets V1.52.2">
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 14 <meta name="description" content=
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 15 "Presents an overview of the features, system requirements, and starting the toolbox.">
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 16 </head>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 17
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 18 <body>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 19 <a name="top_of_page" id="top_of_page"></a>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 20
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 21 <p style="font-size:1px;"> </p>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 22
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 23 <table class="nav" summary="Navigation aid" border="0" width=
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 24 "100%" cellpadding="0" cellspacing="0">
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 25 <tr>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 26 <td valign="baseline"><b>LTPDA Toolbox</b></td><td><a href="../helptoc.html">contents</a></td>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 27
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 28 <td valign="baseline" align="right"><a href=
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 29 "specwin_using.html"><img src="b_prev.gif" border="0" align=
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 30 "bottom" alt="Using spectral windows"></a> <a href=
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 31 "sigproc_psd.html"><img src="b_next.gif" border="0" align=
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 32 "bottom" alt="Power spectral density estimates"></a></td>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 33 </tr>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 34 </table>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 35
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 36 <h1 class="title"><a name="f3-12899" id="f3-12899"></a>Spectral Estimation Methods</h1>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 37 <hr>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 38
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 39 <p>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 40 <h2>Linear and Log-scale Methods</a></h2>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 41
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 42 <p>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 43 The LTPDA Toolbox offers two kind of spectral estimators. The first ones are based on <tt>pwelch</tt> from MATLAB, which is an
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 44 implementation of Welch's averaged, modified periodogram method <a href="#references"> [1]</a>. More details about spectral
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 45 estimation techniques can be found <a href="sigproc_intro.html" >here</a>.</p>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 46
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 47 <p>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 48 The following pages describe the different Welch-based spectral estimation <tt>ao</tt> methods
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 49 available in the LTPDA toolbox:
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 50 <ul>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 51 <li><a href="sigproc_psd.html"> power spectral density estimates </a></li>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 52 <li><a href="sigproc_cpsd.html"> cross-spectral density estimates </a></li>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 53 <li><a href="sigproc_cohere.html"> cross-coherence estimates </a></li>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 54 <li><a href="sigproc_tfe.html"> transfer function estimates </a></li>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 55 </ul>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 56 </p>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 57
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 58 <p>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 59 As an alternative, the LTPDA toolbox makes available the same set of estimators, based on an
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 60 implementation of the LPSD algorithm <a href="#references"> [2]</a>).
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 61 </p>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 62 <p>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 63 The following pages describe the different LPSD-based spectral estimation <tt>ao</tt> methods
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 64 available in the LTPDA toolbox:
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 65 <ul>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 66 <li><a href="sigproc_lpsd.html"> log-scale power spectral density estimates </a></li>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 67 <li><a href="sigproc_lcpsd.html"> log-scale cross-spectral density estimates </a></li>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 68 <li><a href="sigproc_lcohere.html"> log-scale cross-coherence estimates </a></li>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 69 <li><a href="sigproc_ltfe.html"> log-scale transfer function estimates</a></li>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 70 </ul>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 71 </p>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 72
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 73 <p> More detailed help on spectral estimation can also be found in the help associated with
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 74 the <a href="matlab:doc('signal')" >Signal Processing Toolbox</a>.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 75 </p>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 76
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 77 <h2>Computing the sample variance</h2>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 78 <p>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 79 The spectral estimators previously described usually return the average of the spectral estimator applied
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 80 to different segments. This is a standard technique used in spectral analysis to reduce the variance of the
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 81 estimator.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 82 </p>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 83 <p>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 84 When using one of the previous methods in the LTPDA Toolbox, the value of this average over different segments
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 85 is stored in the <tt>ao.y</tt> field of the output analysis object, but the user obtains also information about
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 86 the spectral estimator variance in the <tt>ao.dy</tt> field.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 87 </p>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 88 <p>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 89 The methods listed above store in the <tt>ao.dy</tt> field the <b>standard deviation of the mean</b>, defined as
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 90 </p>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 91 <div align="center">
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 92 <img src="images/mean_variance.png" >
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 93 </div>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 94 <br>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 95 <p>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 96 For more details on how the variance of the mean is computed, please refer to the the help page of each method.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 97 </p>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 98 <p>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 99 <table cellspacing="0" class="note" summary="Note" cellpadding="5" border="1">
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 100 <tr width="90%">
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 101 <td>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 102 Note that when we only have one segment we can not evaluate the variance. This will happen in
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 103 <ul>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 104 <li>linear estimators: when the number of averages is equal to one.</li>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 105 <li>log-scale estimators: in the lowest frequency bins.</li>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 106 </ul>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 107 </td>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 108 </tr>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 109 </table>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 110 </p>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 111 <br>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 112 <p>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 113 The following example compares the sample variance computed by <tt>ao/psd</tt> with two different segment length.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 114 </p>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 115 <div class="fragment"><pre><br>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 116 <span class="comment">% create white noise AO </span>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 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>);
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 118 a = ao(pl);
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 119
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 120 <span class="comment">% compute psd with different Nfft</span>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 121 b1 = psd(a, plist(<span class="string">'Nfft'</span>, 500));
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 122 b1.setName(<span class="string">'Nfft = 500'</span>);
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 123 b2 = psd(a, plist(<span class="string">'Nfft'</span>, 200));
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 124 b2.setName(<span class="string">'Nfft = 200'</span>);
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 125
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 126 <span class="comment">% plot with errorbars</span>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 127 iplot(b1,b2,plist(<span class="string">'YErrU'</span>,{b1.dy,b2.dy}))
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 128 </pre></div>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 129 <p>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 130 <div align="center">
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 131 <p>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 132 </p>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 133 <IMG src="images/spectral_error.png" align="center" border="0">
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 134 </div>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 135 </p>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 136 <br>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 137 <h2><a name="references">References</a></h2>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 138
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 139 <ol>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 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,
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 141 Modified Periodograms, <i>IEEE Trans. on Audio and Electroacoustics</i>, Vol. 15, No. 2 (1967), pp. 70 - 73</a></li>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 142 <li> M. Troebs, G. Heinzel, Improved spectrum estimation from digitized time series
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 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>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 144 </ol>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 145
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 146 </p>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 147
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 148 <br>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 149 <br>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 150 <table class="nav" summary="Navigation aid" border="0" width=
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 151 "100%" cellpadding="0" cellspacing="0">
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 152 <tr valign="top">
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 153 <td align="left" width="20"><a href="specwin_using.html"><img src=
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 154 "b_prev.gif" border="0" align="bottom" alt=
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 155 "Using spectral windows"></a> </td>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 156
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 157 <td align="left">Using spectral windows</td>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 158
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 159 <td> </td>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 160
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 161 <td align="right">Power spectral density estimates</td>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 162
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 163 <td align="right" width="20"><a href=
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 164 "sigproc_psd.html"><img src="b_next.gif" border="0" align=
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 165 "bottom" alt="Power spectral density estimates"></a></td>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 166 </tr>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 167 </table><br>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 168
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 169 <p class="copy">©LTP Team</p>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 170 </body>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff
changeset
+ − 171 </html>