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author Daniele Nicolodi <nicolodi@science.unitn.it>
date Wed, 23 Nov 2011 19:22:13 +0100
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1 <p>
2 Multivariate power spectral density is performed by the Welch's averaged, modified periodogram method.
3 <tt>ao/cpsd</tt> estimates the cross-spectral density of time-series
4 signals, included in the input <tt>ao</tt>s. Data are windowed prior to the estimation of the spectra, by multiplying
5 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
6 of the border discontinuities. The window length is adjustable to shorter lenghts to reduce the spectral
7 density uncertainties, and the percentage of subsequent window overlap can be adjusted as well.
8 <br>
9 <br>
10 <b>Syntaxis</b>
11 </p>
12 <div class="fragment"><pre>
13 b = cpsd(a1,a2,pl)
14 </pre>
15 </div>
16 <p>
17 <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.
18 The parameter list <tt>pl</tt> includes the following parameters:</p>
19 <ul>
20 <li> <tt>'Nfft'</tt> - number of samples in each fft [default: length of input data]
21 A string value containing the variable 'fs' can
22 also be used, e.g., plist('Nfft', '2*fs') </li>
23 <li> <tt>'Win'</tt> - the window to be applied to the data to remove the
24 discontinuities at edges of segments. [default: taken from user prefs]
25 Only the design parameters of the window object are used. Enter either:
26 <ul>
27 <li>a specwin window object OR</li>
28 <li>a string value containing the window name e.g., plist('Win', 'Kaiser', 'psll', 200)</li></ul>
29 </li>
30 <li> <tt>'Olap'</tt> - segment percent overlap [default: -1, (taken from window function)] </li>
31 <li> <tt>'Order'</tt> - order of segment detrending <ul>
32 <li> -1 - no detrending </li>
33 <li> 0 - subtract mean [default] </li>
34 <li> 1 - subtract linear fit </li>
35 <li> N - subtract fit of polynomial, order N </li> </ul> </li>
36 <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>
37 </ul>
38 <p>
39 The length of the window is set by the value of the parameter <tt>'Nfft'</tt>, so that the window
40 is actually rebuilt using only the key features of the window, i.e. the name and, for Keiser windows, the PSLL.
41 </p>
42 <p>As an alternative, the user can input, as a value for the <tt>'Win'</tt> key, a string corresponding to the name of the window. In the case of Kaiser window, it's necessary to specify the additional parameter <tt>'psll'</tt>.</p>
43 <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>
44 <p>
45 <table cellspacing="0" class="note" summary="Note" cellpadding="5" border="1">
46 <tr width="90%">
47 <td>
48 If the user doesn't specify the value of a given parameter, the default value is used.
49 </td>
50 </tr>
51 </table>
52 </p>
53
54 <p>
55 The function makes CPSD estimates between the 2 input <tt>ao</tt>s. The input argument
56 list must contain 2 analysis objects, and the output will contain the CPSD estimate.
57 If passing two identical objects <tt>ai</tt>, the output will be equivalent to the output of <tt>psd(ai)</tt>.
58 </p>
59 </pre> </div>
60 </p>
61 <p>
62 <b>Example</b>
63 </p>
64 <p>
65 Evaluation of the CPSD of two time-series represented by: a low frequency sinewave signal superimposed to
66 white noise, and a low frequency sinewave signal at the same frequency, phase shifted and with different
67 amplitude, superimposed to white noise.
68 </p>
69 <div class="fragment"><pre>
70 nsecs = 1000;
71 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)) + ...
72 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));
73 x.setYunits(<span class="string">'m'</span>);
74 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)) + ...
75 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));
76 y.setYunits(<span class="string">'V'</span>);
77 z = cpsd(x,y,plist(<span class="string">'nfft'</span>,1000));
78 iplot(z);
79 </pre>
80 </div>
81
82 <img src="images/cpsd_2.png" alt="" border="3">
83 <br>