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+ − 5 <head>
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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;"> </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> <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> </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> </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">©LTP Team</p>
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+ − 169 </body>
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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