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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>Cross-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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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_psd.html"><img src="b_prev.gif" border="0" align=
30 "bottom" alt="Power spectral density estimates"></a>&nbsp;&nbsp;&nbsp;<a href=
31 "sigproc_cohere.html"><img src="b_next.gif" border="0" align=
32 "bottom" alt="Cross coherence estimates"></a></td>
33 </tr>
34 </table>
35
36 <h1 class="title"><a name="f3-12899" id="f3-12899"></a>Cross-spectral density estimates</h1>
37 <hr>
38
39 <p>
40 <h2>Description</h2>
41 <p>
42 Cross-power spectral density is performed by the Welch's averaged, modified periodogram method.
43 The LTPDA method <a href="matlab:doc('ao/cpsd')">ao/cpsd</a> estimates the cross-spectral density of time-series
44 signals, included in the input <tt>ao</tt>s following the Welch's averaged, modified periodogram method <a href="#references">[1]</a>.
45 Data are windowed prior to the estimation of the spectra, by multiplying
46 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
47 of the border discontinuities. The window length is adjustable to shorter lenghts to reduce the spectral
48 density uncertainties, and the percentage of subsequent window overlap can be adjusted as well.
49 <br>
50 <br>
51 <h2>Syntax</h2>
52 </p>
53 <div class="fragment"><pre>
54 <br> b = cpsd(a1,a2,pl)
55 </pre>
56 </div>
57 <p>
58 <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,
59 and <tt>pl</tt> is an optional parameters list.
60 <h2>Parameters</h2>
61 The parameter list <tt>pl</tt> includes the following parameters:</p>
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 <p>
80 The length of the window is set by the value of the parameter <tt>'Nfft'</tt>, so that the window
81 is actually rebuilt using only the key features of the window, i.e. the name and, for Kaiser windows, the PSLL.
82 </p>
83
84 <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 CPSD 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>
85 <p>
86 <table cellspacing="0" class="note" summary="Note" cellpadding="5" border="1">
87 <tr width="90%">
88 <td>
89 If the user doesn't specify the value of a given parameter, the default value is used.
90 </td>
91 </tr>
92 </table>
93 </p>
94
95 <p>
96 The function makes CPSD estimates between the 2 input <tt>ao</tt>s. The input argument
97 list must contain 2 analysis objects, and the output will contain the CPSD estimate.
98 If passing two identical objects <tt>ai</tt>, the output will be equivalent to the output of <tt>psd(ai)</tt>.
99 </p>
100 </pre> </div>
101 </p>
102 <p>
103 <h2>Algorithm</h2>
104 <p>
105 The algorithm is based in standard MATLAB's tools, as the ones used by <a href="matlab:doc('pwelch')">pwelch</a>. However, in order to
106 compute the standard deviation of mean for each frequency bin, the averaging of the different segments is performed using Welford's
107 algorithm <a href="#references">[2]</a> which allows to compute mean and variance in one loop.
108 </p>
109 <h2>Example</h2>
110 </p>
111 <p>
112 Evaluation of the CPSD of two time-series represented by: a low frequency sinewave signal superimposed to
113 white noise, and a low frequency sinewave signal at the same frequency, phase shifted and with different
114 amplitude, superimposed to white noise.
115 </p>
116 <div class="fragment"><pre>
117 nsecs = 1000;
118 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>,10)) + ...
119 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>,10));
120 x.setYunits(<span class="string">'m'</span>);
121 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>,10,<span class="string">'phi'</span>,90)) + ...
122 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>,10));
123 y.setYunits(<span class="string">'V'</span>);
124 z = cpsd(x,y,plist(<span class="string">'nfft'</span>,1000));
125 iplot(z);
126 </pre>
127 </div>
128
129 <img src="images/cpsd_1.png" alt="" border="3">
130 <br>
131
132 <h2><a name="references">References</a></h2>
133
134 <ol>
135 <li> P.D. Welch, The Use of Fast Fourier Transform for the Estimation of Power Spectra: A Method Based on Time Averaging Over Short,
136 Modified Periodograms, <i>IEEE Trans. on Audio and Electroacoustics</i>, Vol. 15, No. 2 (1967), pp. 70 - 73</a></li>
137 <li> B. P. Weldford, Note on a Method for Calculating Corrected Sums of Squares and Products,
138 <i>Technometrics<i>, Vol. 4, No. 3 (1962), pp 419 - 420.</li>
139 </ol>
140
141
142
143
144
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_psd.html"><img src=
153 "b_prev.gif" border="0" align="bottom" alt=
154 "Power spectral density estimates"></a>&nbsp;</td>
155
156 <td align="left">Power spectral density estimates</td>
157
158 <td>&nbsp;</td>
159
160 <td align="right">Cross coherence estimates</td>
161
162 <td align="right" width="20"><a href=
163 "sigproc_cohere.html"><img src="b_next.gif" border="0" align=
164 "bottom" alt="Cross coherence estimates"></a></td>
165 </tr>
166 </table><br>
167
168 <p class="copy">&copy;LTP Team</p>
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