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