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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 <!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"
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2 "http://www.w3.org/TR/1999/REC-html401-19991224/loose.dtd">
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3
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4 <html lang="en">
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5 <head>
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6 <meta name="generator" content=
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7 "HTML Tidy for Mac OS X (vers 1st December 2004), see www.w3.org">
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8 <meta http-equiv="Content-Type" content=
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9 "text/html; charset=us-ascii">
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10
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11 <title>Log-scale power spectral density estimates (LTPDA Toolbox)</title>
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12 <link rel="stylesheet" href="docstyle.css" type="text/css">
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13 <meta name="generator" content="DocBook XSL Stylesheets V1.52.2">
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14 <meta name="description" content=
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15 "Presents an overview of the features, system requirements, and starting the toolbox.">
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16 </head>
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17
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18 <body>
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19 <a name="top_of_page" id="top_of_page"></a>
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20
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21 <p style="font-size:1px;">&nbsp;</p>
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22
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23 <table class="nav" summary="Navigation aid" border="0" width=
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24 "100%" cellpadding="0" cellspacing="0">
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25 <tr>
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26 <td valign="baseline"><b>LTPDA Toolbox</b></td><td><a href="../helptoc.html">contents</a></td>
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27
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28 <td valign="baseline" align="right"><a href=
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29 "sigproc_tfe.html"><img src="b_prev.gif" border="0" align=
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30 "bottom" alt="Transfer function estimates"></a>&nbsp;&nbsp;&nbsp;<a href=
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31 "sigproc_lcpsd.html"><img src="b_next.gif" border="0" align=
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32 "bottom" alt="Log-scale cross-spectral density estimates"></a></td>
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33 </tr>
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34 </table>
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35
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36 <h1 class="title"><a name="f3-12899" id="f3-12899"></a>Log-scale power spectral density estimates</h1>
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37 <hr>
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38
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39 <p>
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40 <h2>Description</h2>
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41 <p>
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42 The LTPDA method <a href="matlab:doc('ao/lpsd')">ao/lpsd</a> estimates the power spectral density of time-series
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43 signals, included in the input <tt>ao</tt>s following the LPSD algorithm <a href="#references">[1]</a>. Spectral density estimates are not
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44 evaluated at frequencies which are linear multiples of the minimum frequency resolution <tt>1/T</tt>, where <tt>T</tt>
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45 is the window lenght, but on a logarithmic scale. The algorithm takes care of calculating the frequencies at which to evaluate
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46 the spectral estimate, aiming at minimizing the uncertainty in the estimate itself, and to recalculate a suitable
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47 window length for each frequency bin.
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48 </p>
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49 <p>
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50 Data are windowed prior to the estimation of the spectrum, by multiplying
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51 it with a <a href="specwin.html">spectral window object</a>, and can be detrended by polinomial of time in order to reduce the impact
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52 of the border discontinuities. Detrending is performed on each individual window.
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53 The user can choose the quantity being given in output among ASD (amplitude spectral density),
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54 PSD (power spectral density), AS (amplitude spectrum), and PS (power spectrum).
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55 </p>
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56 <br>
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57 <h2>Syntax</h2>
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58 </p>
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59 <div class="fragment"><pre>
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60 <br> bs = lpsd(a1,a2,a3,...,pl)
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61 bs = lpsd(as,pl)
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62 bs = as.lpsd(pl)
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63 </pre>
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64 </div>
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65 <p>
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66 <tt>a1</tt> and <tt>a2</tt> are the 2 <tt>ao</tt>s containing the input time series to be evaluated, <tt>b</tt> is the output object and <tt>pl</tt> is an optional parameter list.
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67
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68 <h2>Parameters</h2>
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69 <p>The parameter list <tt>pl</tt> includes the following parameters:</p>
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70 <ul>
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71 <li> <tt>'Kdes'</tt> - desired number of averages [default: 100]</li>
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72 <li> <tt>'Jdes'</tt> - number of spectral frequencies to compute [default: 1000]</li>
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73 <li> <tt>'Lmin'</tt> - minimum segment length [default: 0]</li>
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74 <li> <tt>'Win'</tt> - the window to be applied to the data to remove the
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75 discontinuities at edges of segments. [default: taken from user prefs].<br>
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76 The window is described by a string with its name and, only in the case of Kaiser window,
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77 the additional parameter <tt>'psll'</tt>. <br>For instance: plist('Win', 'Kaiser', 'psll', 200).
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78 </li>
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79 <li> <tt>'Olap'</tt> - segment percent overlap [default: -1, (taken from window function)] </li>
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80 <li> <tt>'Scale'</tt> - scaling of output. Choose from: <ul>
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81 <li> 'ASD' - amplitude spectral density </li>
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82 <li> 'PSD' - power spectral density [default] </li>
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83 <li> 'AS' - amplitude spectrum </li>
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84 <li> 'PS' - power spectrum </li> </ul> </li>
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85 <li> <tt>'Order'</tt> - order of segment detrending <ul>
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86 <li> -1 - no detrending </li>
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87 <li> 0 - subtract mean [default] </li>
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88 <li> 1 - subtract linear fit </li>
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89 <li> N - subtract fit of polynomial, order N </li> </ul> </li>
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90 </ul>
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91 The length of the window is set by the value of the parameter <tt>'Nfft'</tt>, so that the window
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92 is actually rebuilt using only the key features of the window, i.e. the name and, for Kaiser windows, the PSLL.
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93 </p>
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94
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95 <p>
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96 <table cellspacing="0" class="note" summary="Note" cellpadding="5" border="1">
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97 <tr width="90%">
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98 <td>
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99 If the user doesn't specify the value of a given parameter, the default value is used.
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100 </td>
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101 </tr>
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102 </table>
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103 </p>
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104 <h2>Algorithm</h2>
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105 <p>
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106 The algorithm is implemented according to <a href="#references">[1]</a>. In order to
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107 compute the standard deviation of the mean for each frequency bin, the averaging of the different segments is performed using Welford's
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108 algorithm <a href="#references">[2]</a> which allows to compute mean and variance in one loop. <br>
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109 In the LPSD algorithm, the first frequencies bins are usually computed using a single segment containing all the data.
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110 For these bins, the sample variance is set to <tt>Inf</tt>.
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111 </p>
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112 <h2>Examples</h2>
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113 <p>
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114 1. Evaluation of the ASD of a time-series represented by a low frequency sinewave signal, superimposed to
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115 white noise. Comparison of the effect of using standard Pwelch and LPSD on the estimate
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116 of the white noise level and on resolving the signal.
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117 </p>
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118 <div class="fragment"><pre>
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119 <br> <span class="comment">% Create input AO</span>
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120 x1 = ao(plist(<span class="string">'waveform'</span>,<span class="string">'sine wave'</span>,<span class="string">'f'</span>,0.1,<span class="string">'A'</span>,1,<span class="string">'nsecs'</span>,1000,<span class="string">'fs'</span>,10,<span class="string">'yunits'</span>,<span class="string">'rad'</span>));
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121 x2 = ao(plist(<span class="string">'waveform'</span>,<span class="string">'noise'</span>,<span class="string">'type'</span>,<span class="string">'normal'</span>,<span class="string">'nsecs'</span>,1000,<span class="string">'fs'</span>,10,<span class="string">'yunits'</span>,<span class="string">'rad'</span>));
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122 x = x1 + x2;
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123
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124 <span class="comment">% Compute psd and lpsd </span>
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125 pl = plist(<span class="string">'scale'</span>,<span class="string">'ASD'</span>,<span class="string">'order'</span>,-1,<span class="string">'win'</span>,<span class="string">'Kaiser'</span>,<span class="string">'psll'</span>,200);
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126 y1 = psd(x, pl);
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127 y2 = lpsd(x, pl);
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128
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129 <span class="comment">% Compare</span>
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130 iplot(y1, y2)
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131 </pre>
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132 </div>
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133
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134 <img src="images/l_psd_1.png" border="3">
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135
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136
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137 <h2><a name="references">References</a></h2>
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138
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139 <ol>
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140 <li> M. Troebs, G. Heinzel, Improved spectrum estimation from digitized time series
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141 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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142 <li> B. P. Weldford, Note on a Method for Calculating Corrected Sums of Squares and Products,
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143 <i>Technometrics<i>, Vol. 4, No. 3 (1962), pp 419 - 420.</li>
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144 </ol>
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145 </p>
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146
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147 <br>
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148 <br>
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149 <table class="nav" summary="Navigation aid" border="0" width=
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150 "100%" cellpadding="0" cellspacing="0">
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151 <tr valign="top">
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152 <td align="left" width="20"><a href="sigproc_tfe.html"><img src=
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153 "b_prev.gif" border="0" align="bottom" alt=
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154 "Transfer function estimates"></a>&nbsp;</td>
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155
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156 <td align="left">Transfer function estimates</td>
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157
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158 <td>&nbsp;</td>
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159
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160 <td align="right">Log-scale cross-spectral density estimates</td>
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161
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162 <td align="right" width="20"><a href=
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163 "sigproc_lcpsd.html"><img src="b_next.gif" border="0" align=
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164 "bottom" alt="Log-scale cross-spectral density estimates"></a></td>
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165 </tr>
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166 </table><br>
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167
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168 <p class="copy">&copy;LTP Team</p>
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169 </body>
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170 </html>