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author Daniele Nicolodi <nicolodi@science.unitn.it>
date Wed, 23 Nov 2011 19:22:13 +0100
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11 <title>Log-scale power spectral density estimates (LTPDA Toolbox)</title>
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15 "Presents an overview of the features, system requirements, and starting the toolbox.">
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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 "sigproc_tfe.html"><img src="b_prev.gif" border="0" align=
30 "bottom" alt="Transfer function estimates"></a>&nbsp;&nbsp;&nbsp;<a href=
31 "sigproc_lcpsd.html"><img src="b_next.gif" border="0" align=
32 "bottom" alt="Log-scale cross-spectral density estimates"></a></td>
33 </tr>
34 </table>
35
36 <h1 class="title"><a name="f3-12899" id="f3-12899"></a>Log-scale power spectral density estimates</h1>
37 <hr>
38
39 <p>
40 <h2>Description</h2>
41 <p>
42 The LTPDA method <a href="matlab:doc('ao/lpsd')">ao/lpsd</a> estimates the power spectral density of time-series
43 signals, included in the input <tt>ao</tt>s following the LPSD algorithm <a href="#references">[1]</a>. Spectral density estimates are not
44 evaluated at frequencies which are linear multiples of the minimum frequency resolution <tt>1/T</tt>, where <tt>T</tt>
45 is the window lenght, but on a logarithmic scale. The algorithm takes care of calculating the frequencies at which to evaluate
46 the spectral estimate, aiming at minimizing the uncertainty in the estimate itself, and to recalculate a suitable
47 window length for each frequency bin.
48 </p>
49 <p>
50 Data are windowed prior to the estimation of the spectrum, by multiplying
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
52 of the border discontinuities. Detrending is performed on each individual window.
53 The user can choose the quantity being given in output among ASD (amplitude spectral density),
54 PSD (power spectral density), AS (amplitude spectrum), and PS (power spectrum).
55 </p>
56 <br>
57 <h2>Syntax</h2>
58 </p>
59 <div class="fragment"><pre>
60 <br> bs = lpsd(a1,a2,a3,...,pl)
61 bs = lpsd(as,pl)
62 bs = as.lpsd(pl)
63 </pre>
64 </div>
65 <p>
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.
67
68 <h2>Parameters</h2>
69 <p>The parameter list <tt>pl</tt> includes the following parameters:</p>
70 <ul>
71 <li> <tt>'Kdes'</tt> - desired number of averages [default: 100]</li>
72 <li> <tt>'Jdes'</tt> - number of spectral frequencies to compute [default: 1000]</li>
73 <li> <tt>'Lmin'</tt> - minimum segment length [default: 0]</li>
74 <li> <tt>'Win'</tt> - the window to be applied to the data to remove the
75 discontinuities at edges of segments. [default: taken from user prefs].<br>
76 The window is described by a string with its name and, only in the case of Kaiser window,
77 the additional parameter <tt>'psll'</tt>. <br>For instance: plist('Win', 'Kaiser', 'psll', 200).
78 </li>
79 <li> <tt>'Olap'</tt> - segment percent overlap [default: -1, (taken from window function)] </li>
80 <li> <tt>'Scale'</tt> - scaling of output. Choose from: <ul>
81 <li> 'ASD' - amplitude spectral density </li>
82 <li> 'PSD' - power spectral density [default] </li>
83 <li> 'AS' - amplitude spectrum </li>
84 <li> 'PS' - power spectrum </li> </ul> </li>
85 <li> <tt>'Order'</tt> - order of segment detrending <ul>
86 <li> -1 - no detrending </li>
87 <li> 0 - subtract mean [default] </li>
88 <li> 1 - subtract linear fit </li>
89 <li> N - subtract fit of polynomial, order N </li> </ul> </li>
90 </ul>
91 The length of the window is set by the value of the parameter <tt>'Nfft'</tt>, so that the window
92 is actually rebuilt using only the key features of the window, i.e. the name and, for Kaiser windows, the PSLL.
93 </p>
94
95 <p>
96 <table cellspacing="0" class="note" summary="Note" cellpadding="5" border="1">
97 <tr width="90%">
98 <td>
99 If the user doesn't specify the value of a given parameter, the default value is used.
100 </td>
101 </tr>
102 </table>
103 </p>
104 <h2>Algorithm</h2>
105 <p>
106 The algorithm is implemented according to <a href="#references">[1]</a>. In order to
107 compute the standard deviation of the mean for each frequency bin, the averaging of the different segments is performed using Welford's
108 algorithm <a href="#references">[2]</a> which allows to compute mean and variance in one loop. <br>
109 In the LPSD algorithm, the first frequencies bins are usually computed using a single segment containing all the data.
110 For these bins, the sample variance is set to <tt>Inf</tt>.
111 </p>
112 <h2>Examples</h2>
113 <p>
114 1. Evaluation of the ASD of a time-series represented by a low frequency sinewave signal, superimposed to
115 white noise. Comparison of the effect of using standard Pwelch and LPSD on the estimate
116 of the white noise level and on resolving the signal.
117 </p>
118 <div class="fragment"><pre>
119 <br> <span class="comment">% Create input AO</span>
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>));
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>));
122 x = x1 + x2;
123
124 <span class="comment">% Compute psd and lpsd </span>
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);
126 y1 = psd(x, pl);
127 y2 = lpsd(x, pl);
128
129 <span class="comment">% Compare</span>
130 iplot(y1, y2)
131 </pre>
132 </div>
133
134 <img src="images/l_psd_1.png" border="3">
135
136
137 <h2><a name="references">References</a></h2>
138
139 <ol>
140 <li> M. Troebs, G. Heinzel, Improved spectrum estimation from digitized time series
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>
142 <li> B. P. Weldford, Note on a Method for Calculating Corrected Sums of Squares and Products,
143 <i>Technometrics<i>, Vol. 4, No. 3 (1962), pp 419 - 420.</li>
144 </ol>
145 </p>
146
147 <br>
148 <br>
149 <table class="nav" summary="Navigation aid" border="0" width=
150 "100%" cellpadding="0" cellspacing="0">
151 <tr valign="top">
152 <td align="left" width="20"><a href="sigproc_tfe.html"><img src=
153 "b_prev.gif" border="0" align="bottom" alt=
154 "Transfer function estimates"></a>&nbsp;</td>
155
156 <td align="left">Transfer function estimates</td>
157
158 <td>&nbsp;</td>
159
160 <td align="right">Log-scale cross-spectral density estimates</td>
161
162 <td align="right" width="20"><a href=
163 "sigproc_lcpsd.html"><img src="b_next.gif" border="0" align=
164 "bottom" alt="Log-scale cross-spectral density estimates"></a></td>
165 </tr>
166 </table><br>
167
168 <p class="copy">&copy;LTP Team</p>
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