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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>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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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_cohere.html"><img src="b_prev.gif" border="0" align=
30 "bottom" alt="Cross coherence estimates"></a>&nbsp;&nbsp;&nbsp;<a href=
31 "sigproc_lpsd.html"><img src="b_next.gif" border="0" align=
32 "bottom" alt="Log-scale power spectral density estimates"></a></td>
33 </tr>
34 </table>
35
36 <h1 class="title"><a name="f3-12899" id="f3-12899"></a>Transfer function estimates</h1>
37 <hr>
38
39 <p>
40 <h2>Description</h2>
41 <p>
42 The LTPDA method <a href="matlab:doc('ao/tfe')">ao/tfe</a> estimates the transfer function of time-series
43 signals, included in the input <tt>ao</tt>s following the Welch's averaged, modified periodogram method <a href="#references">[1]</a>.
44 Data are windowed prior to the estimation of the spectra, by multiplying
45 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
46 of the border discontinuities. The window length is adjustable to shorter lenghts to reduce the spectral
47 density uncertainties, and the percentage of subsequent window overlap can be adjusted as well.
48 <br>
49 <br>
50 <h2>Syntax</h2>
51 </p>
52 <div class="fragment"><pre>
53 <br> b = tfe(a1,a2,pl)
54 </pre>
55 </div>
56 <p>
57 <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
58 <tt>pl</tt> is an optional parameters list.
59 </p>
60 <h2>Parameters</h2>
61 The parameter list <tt>pl</tt> includes the following parameters:
62 <ul>
63 <li> <tt>'Nfft'</tt> - number of samples in each fft [default: length of input data]
64 A string value containing the variable 'fs' can
65 also be used, e.g., plist('Nfft', '2*fs') </li>
66 <li> <tt>'Win'</tt> - the window to be applied to the data to remove the
67 discontinuities at edges of segments. [default: taken from user prefs].<br>
68 The window is described by a string with its name and, only in the case of Kaiser window,
69 the additional parameter <tt>'psll'</tt>. <br>For instance: plist('Win', 'Kaiser', 'psll', 200). </li>
70 <li> <tt>'Olap'</tt> - segment percent overlap [default: -1, (taken from window function)] </li>
71 <li> <tt>'Order'</tt> - order of segment detrending <ul>
72 <li> -1 - no detrending </li>
73 <li> 0 - subtract mean [default] </li>
74 <li> 1 - subtract linear fit </li>
75 <li> N - subtract fit of polynomial, order N </li> </ul> </li>
76 <li><tt>'Navs'</tt> - number of averages. If set, and if Nfft was set to 0 or -1, the number of points for each window will be calculated to match the request. [default: -1, not set] </li>
77 <li><tt>'Times'</tt> - interval of time to evaluate the calculation on. If empty [default], it will take the whole section.</li>
78 </ul>
79 The length of the window is set by the value of the parameter <tt>'Nfft'</tt>, so that the window
80 is actually rebuilt using only the key features of the window, i.e. the name and, for Kaiser windows, the PSLL.
81 </p>
82
83 <p>As an alternative to setting the number of points <tt>'Nfft'</tt> in each window, it's possible to ask for a given number of TFE estimates by setting the <tt>'Navs'</tt> parameter, and the algorithm takes care of calculating the correct window length, according to the amount of overlap between subsequent segments.</p>
84 <p>
85 <table cellspacing="0" class="note" summary="Note" cellpadding="5" border="1">
86 <tr width="90%">
87 <td>
88 If the user doesn't specify the value of a given parameter, the default value is used.
89 </td>
90 </tr>
91 </table>
92 </p>
93
94 <p>The function makes 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>
95 <h2>Algorithm</h2>
96 <p>
97 The algorithm is based in standard MATLAB's tools, as the ones used by <a href="matlab:doc('pwelch')">pwelch</a>. The standard deviation of the mean
98 is computed as
99 <div align="center">
100 <img src="images/tfe_sigma1.png" >
101 </div>
102 where
103 <div align="center">
104 <img src="images/tfe_sigma2.png" >
105 </div>
106 is the coherence function.
107 </p>
108 <h2>Example</h2>
109 <p>
110 Evaluation of the transfer function between two time-series represented by:
111 a low frequency sinewave signal superimposed to
112 white noise, and a low frequency sinewave signal at the same frequency, phase shifted and with different
113 amplitude, superimposed to white noise.
114 </p>
115 <div class="fragment"><pre>
116 <br> <span class="comment">% parameters</span>
117 nsecs = 1000;
118 fs = 10;
119
120 <span class="comment">% create first signal AO</span>
121 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)) + ...
122 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));
123 x.setYunits(<span class="string">'m'</span>);
124
125 <span class="comment">% create second signal AO</span>
126 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)) + ...
127 0.1*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));
128 y.setYunits(<span class="string">'rad'</span>);
129
130 <span class="comment">% compute transfer function</span>
131 nfft = 1000;
132 psll = 200;
133 Txy = tfe(x,y,plist(<span class="string">'win'</span>,<span class="string">'Kaiser'</span>,<span class="string">'psll'</span>,psll,<span class="string">'nfft'</span>,nfft));
134
135 <span class="comment">% plot</span>
136 iplot(Txy)
137 </pre>
138 </div>
139 <br>
140 <img src="images/transfer_1.png" alt="" border="3">
141
142 <h2><a name="references">References</a></h2>
143
144 <ol>
145 <li> P.D. Welch, The Use of Fast Fourier Transform for the Estimation of Power Spectra: A Method Based on Time Averaging Over Short,
146 Modified Periodograms, <i>IEEE Trans. on Audio and Electroacoustics</i>, Vol. 15, No. 2 (1967), pp. 70 - 73.</a></li>
147 </ol>
148
149
150
151 </p>
152
153 <br>
154 <br>
155 <table class="nav" summary="Navigation aid" border="0" width=
156 "100%" cellpadding="0" cellspacing="0">
157 <tr valign="top">
158 <td align="left" width="20"><a href="sigproc_cohere.html"><img src=
159 "b_prev.gif" border="0" align="bottom" alt=
160 "Cross coherence estimates"></a>&nbsp;</td>
161
162 <td align="left">Cross coherence estimates</td>
163
164 <td>&nbsp;</td>
165
166 <td align="right">Log-scale power spectral density estimates</td>
167
168 <td align="right" width="20"><a href=
169 "sigproc_lpsd.html"><img src="b_next.gif" border="0" align=
170 "bottom" alt="Log-scale power spectral density estimates"></a></td>
171 </tr>
172 </table><br>
173
174 <p class="copy">&copy;LTP Team</p>
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