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author Daniele Nicolodi <nicolodi@science.unitn.it>
date Mon, 05 Dec 2011 16:20:06 +0100
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1 <h2>Description</h2>
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2 <p>
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3 The LTPDA method <a href="matlab:doc('ao/psd')">ao/psd</a> estimates the power spectral density of time-series
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4 signals, included in the input <tt>ao</tt>s following the Welch's averaged, modified periodogram method <a href="#references">[1]</a>.
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5 Data are windowed prior to the estimation of the spectrum, by multiplying
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6 it with a <a href="specwin.html">spectral window object</a>, and can be detrended by a polinomial of time in order to reduce the impact
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7 of the border discontinuities. The window length is adjustable to shorter lenghts to reduce the spectral
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8 density uncertainties, and the percentage of subsequent window overlap can be adjusted as well. The detrending is
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9 performed on the individual windows. The user can choose the quantity being given in output among
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10 ASD (amplitude spectral density), PSD (power spectral density), AS (amplitude spectrum), and PS (power spectrum).
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11 <br>
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12 <br>
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13 <h2>Syntax</h2>
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14 </p>
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15 <div class="fragment"><pre>
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16 <br> bs = psd(a1, a2, a3, ..., pl)
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17 bs = psd(as, pl)
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18 bs = as.psd(pl)
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19 </pre> </div>
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20 <p>
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21 <tt>a1</tt>, <tt>a2</tt>, <tt>a3</tt>, ... are <tt>ao</tt>(s) containing the input time series to be evaluated. <tt>bs</tt> includes
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22 the output object(s) and <tt>pl</tt> is an optional parameter list.
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23 </p>
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24 <h2>Parameters</h2>
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25 <p>
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26 The parameter list <tt>pl</tt> includes the following parameters:
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27 </p>
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28 <ul>
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29 <li> <tt>'Nfft'</tt> - number of samples in each fft [default: length of input data]
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30 A string value containing the variable 'fs' can
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31 also be used, e.g., plist('Nfft', '2*fs') </li>
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32 <li> <tt>'Win'</tt> - the window to be applied to the data to remove the
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33 discontinuities at edges of segments. [default: taken from user prefs].<br>
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34 The window is described by a string with its name and, only in the case of Kaiser window,
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35 the additional parameter <tt>'psll'</tt>. <br>For instance: plist('Win', 'Kaiser', 'psll', 200). </li>
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36 </li>
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37 <li> <tt>'Olap'</tt> - segment percent overlap [default: -1, (taken from window function)] </li>
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38 <li> <tt>'Scale'</tt> - scaling of output. Choose from: <ul>
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39 <li> 'ASD' - amplitude spectral density </li>
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40 <li> 'PSD' - power spectral density [default] </li>
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41 <li> 'AS' - amplitude spectrum </li>
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42 <li> 'PS' - power spectrum </li> </ul> </li>
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43 <li> <tt>'Order'</tt> - order of segment detrending <ul>
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44 <li> -1 - no detrending </li>
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45 <li> 0 - subtract mean [default] </li>
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46 <li> 1 - subtract linear fit </li>
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47 <li> N - subtract fit of polynomial, order N </li> </ul> </li>
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48 <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>
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49 <li><tt>'Times'</tt> - interval of time to evaluate the calculation on. If empty [default], it will take the whole section.</li>
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50 </ul>
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51 <p>
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52 The length of the window is set by the value of the parameter <tt>'Nfft'</tt>, so that the window
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53 is actually built using only the key features of the window: the name and, for Kaiser windows, the psll.
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54 </p>
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55 <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 PSD 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>
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56 <p>
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57 <table cellspacing="0" class="note" summary="Note" cellpadding="5" border="1">
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58 <tr width="90%">
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59 <td>
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60 If the user doesn't specify the value of a given parameter, the default value is used.
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61 </td>
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62 </tr>
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63 </table>
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64 </p>
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65 <h2>Algorithm</h2>
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66 <p>
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67 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
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68 compute the standard deviation of the mean for each frequency bin, the averaging of the different segments is performed using Welford's
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69 algorithm <a href="#references">[2]</a> which allows to compute mean and variance in one loop.
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70 </p>
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71 <h2>Examples</h2>
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72 <p>
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73 1. Evaluation of the PSD of a time-series represented by a low frequency sinewave signal, superimposed to
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74 white noise. Comparison of the effect of windowing on the estimate of the white noise level and
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75 on resolving the signal.
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76 </p>
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77 <div class="fragment"><pre>
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78 <br> <span class="comment">% create two AOs</span>
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79 x1 = 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>,1000,<span class="string">'fs'</span>,10));
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80 x2 = 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>,1000,<span class="string">'fs'</span>,10));
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81 <span class="comment">% add both AOs</span>
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82 x = x1 + x2;
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83 <span class="comment">% compute the psd changing the 'nfft'</span>
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84 y_lf = psd(x);
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85 y_hf = psd(x,plist(<span class="string">'nfft'</span>,1000));
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86 <span class="comment">% compare </span>
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87 iplot(y_lf, y_hf)
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88 </pre></div>
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89
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90 <img src="images/psd_1.png" alt="" border="3">
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91
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92 <p>
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93 2. Evaluation of the PSD of a time-series represented by a low frequency sinewave signal, superimposed to
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94 white noise and to a low frequency linear drift. In the example, the same spectrum is computed with different
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95 spectral windows.
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96 </p>
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97 <div class="fragment"><pre>
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98 <br> <span class="comment">% create three AOs</span>
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99 x1 = 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>,1000,<span class="string">'fs'</span>,10,<span class="string">'yunits'</span>,<span class="string">'m'</span>));
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100 x2 = 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>,1000,<span class="string">'fs'</span>,10,<span class="string">'yunits'</span>,<span class="string">'m'</span>));
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101 x3 = ao(plist(<span class="string">'tsfcn'</span>, <span class="string">'t.^2 + t'</span>,<span class="string">'nsecs'</span>,1000,<span class="string">'fs'</span>,10,<span class="string">'yunits'</span>,<span class="string">'m'</span>));
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102 <span class="comment">% add them</span>
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103 x = x1 + x2 + x3;
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104 <span class="comment">% compute psd with different windows</span>
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105 y_1 = psd(x,plist(<span class="string">'scale'</span>,<span class="string">'ASD'</span>,<span class="string">'order'</span>,1,<span class="string">'win'</span>,<span class="string">'BH92'</span>));
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106 y_2 = psd(x,plist(<span class="string">'scale'</span>,<span class="string">'ASD'</span>,<span class="string">'order'</span>,2,<span class="string">'win'</span>,<span class="string">'Hamming'</span>));
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107 y_3 = psd(x,plist(<span class="string">'scale'</span>,<span class="string">'ASD'</span>,<span class="string">'order'</span>,2,<span class="string">'win'</span>,<span class="string">'Kaiser'</span>,<span class="string">'psll'</span>,200));
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108 <span class="comment">% compare</span>
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109 iplot(y_1, y_2, y_3);
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110 </pre></div>
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111 <p>
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112 <img src="images/psd_2.png" alt="" border="3">
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113 </p>
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114 <h2><a name="references">References</a></h2>
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115
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116 <ol>
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117 <li> P.D. Welch, The Use of Fast Fourier Transform for the Estimation of Power Spectra: A Method Based on Time Averaging Over Short,
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118 Modified Periodograms, <i>IEEE Trans. on Audio and Electroacoustics</i>, Vol. 15, No. 2 (1967), pp. 70 - 73.</a></li>
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119 <li> B. P. Weldford, Note on a Method for Calculating Corrected Sums of Squares and Products,
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120 <i>Technometrics<i>, Vol. 4, No. 3 (1962), pp 419 - 420.</li>
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121 </ol>
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122