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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 transfer function 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_lcohere.html"><img src="b_prev.gif" border="0" align=
30 "bottom" alt="Log-scale cross coherence density estimates"></a>&nbsp;&nbsp;&nbsp;<a href=
31 "sigproc_fit.html"><img src="b_next.gif" border="0" align=
32 "bottom" alt="Fitting Algorithms"></a></td>
33 </tr>
34 </table>
35
36 <h1 class="title"><a name="f3-12899" id="f3-12899"></a>Log-scale transfer function estimates</h1>
37 <hr>
38
39 <p>
40 <h2>Description</h2>
41 <p>
42 The LTPDA method <a href="matlab:doc('ao/ltfe')">ao/ltfe</a> estimates the transfer function 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> b = ltfe(a1,a2,pl)
61 </pre>
62 </div>
63 <p>
64 <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.
65
66 <h2>Parameters</h2>
67 <p>The parameter list <tt>pl</tt> includes the following parameters:</p>
68 <ul>
69 <li> <tt>'Kdes'</tt> - desired number of averages [default: 100]</li>
70 <li> <tt>'Jdes'</tt> - number of spectral frequencies to compute [default: 1000]</li>
71 <li> <tt>'Lmin'</tt> - minimum segment length [default: 0]</li>
72 <li> <tt>'Win'</tt> - the window to be applied to the data to remove the
73 discontinuities at edges of segments. [default: taken from user prefs].<br>
74 The window is described by a string with its name and, only in the case of Kaiser window,
75 the additional parameter <tt>'psll'</tt>. <br>For instance: plist('Win', 'Kaiser', 'psll', 200). </li>
76 <li> <tt>'Olap'</tt> - segment percent overlap [default: -1, (taken from window function)] </li>
77 <li> <tt>'Order'</tt> - order of segment detrending <ul>
78 <li> -1 - no detrending </li>
79 <li> 0 - subtract mean [default] </li>
80 <li> 1 - subtract linear fit </li>
81 <li> N - subtract fit of polynomial, order N </li> </ul> </li>
82 </ul>
83 The length of the window is set by the value of the parameter <tt>'Nfft'</tt>, so that the window
84 is actually rebuilt using only the key features of the window, i.e. the name and, for Kaiser windows, the PSLL.
85 </p>
86
87 <p>
88 <table cellspacing="0" class="note" summary="Note" cellpadding="5" border="1">
89 <tr width="90%">
90 <td>
91 If the user doesn't specify the value of a given parameter, the default value is used.
92 </td>
93 </tr>
94 </table>
95 </p>
96
97 </p>
98 <p>The function makes logaritmic frequencyscale transfer functions estimates between the 2 input <tt>ao</tt>s, and the output will contain the transfer function estimate from the first <tt>ao</tt> to the second.</p>
99
100 </pre> </div>
101 </p>
102 <h2>Algorithm</h2>
103 <p>
104 The algorithm is implemented according to <a href="#references">[1]</a>. The sample variance is computed according to:
105 </p>
106 <div align="center">
107 <img src="images/tfe_sigma1.png" >
108 </div>
109 where
110 <div align="center">
111 <img src="images/tfe_sigma2.png" >
112 </div>
113 <br>
114 <p>
115 is the coherence function.
116 In the LPSD algorithm, the first frequencies bins are usually computed using a single segment containing all the data.
117 For these bins, the standard deviation of the mean is set to <tt>Inf</tt>.
118 </p>
119 <h2>Example</h2>
120 <p>
121 Evaluation of the transfer function between two time-series represented by:
122 a low frequency sinewave signal superimposed to
123 white noise, and a low frequency sinewave signal at the same frequency, phase shifted and with different
124 amplitude, superimposed to white noise.
125 </p>
126 <div class="fragment"><pre>
127 <br> <span class="comment">% Parameters</span>
128 nsecs = 1000;
129 fs = 10;
130 nfft = 1000;
131
132 <span class="comment">% Create input AOs</span>
133 x = 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>,nsecs,<span class="string">'fs'</span>,fs)) + ...
134 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>,nsecs,<span class="string">'fs'</span>,fs));
135 x.setYunits(<span class="string">'m'</span>);
136 y = 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>,2,<span class="string">'nsecs'</span>,nsecs,<span class="string">'fs'</span>,fs,<span class="string">'phi'</span>,90)) + ...
137 4*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>,nsecs,<span class="string">'fs'</span>,fs));
138 y.setYunits(<span class="string">'V'</span>);
139
140 <span class="comment">% Compute transfer function</span>
141 Txy = ltfe(x,y,plist(<span class="string">'win'</span>,specwin(<span class="string">'Kaiser'</span>,1,200),<span class="string">'nfft'</span>,nfft));
142
143 <span class="comment">% Plot</span>
144 iplot(Txy);
145 </pre>
146 </div>
147
148 <img src="images/l_transfer_1.png" alt="" border="3">
149 <br>
150 <!-- <img src="images/l_transfer_2.png" alt="" border="3">
151 <br> -->
152
153 <h2><a name="references">References</a></h2>
154
155 <ol>
156 <li> M. Troebs, G. Heinzel, Improved spectrum estimation from digitized time series
157 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>
158 </ol>
159 </p>
160
161 <br>
162 <br>
163 <table class="nav" summary="Navigation aid" border="0" width=
164 "100%" cellpadding="0" cellspacing="0">
165 <tr valign="top">
166 <td align="left" width="20"><a href="sigproc_lcohere.html"><img src=
167 "b_prev.gif" border="0" align="bottom" alt=
168 "Log-scale cross coherence density estimates"></a>&nbsp;</td>
169
170 <td align="left">Log-scale cross coherence density estimates</td>
171
172 <td>&nbsp;</td>
173
174 <td align="right">Fitting Algorithms</td>
175
176 <td align="right" width="20"><a href=
177 "sigproc_fit.html"><img src="b_next.gif" border="0" align=
178 "bottom" alt="Fitting Algorithms"></a></td>
179 </tr>
180 </table><br>
181
182 <p class="copy">&copy;LTP Team</p>
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184 </html>