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+ − 3
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+ − 4 <html lang="en">
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+ − 5 <head>
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+ − 10
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+ − 11 <title>Power spectral density estimates (LTPDA Toolbox)</title>
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+ − 12 <link rel="stylesheet" href="docstyle.css" type="text/css">
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+ − 13 <meta name="generator" content="DocBook XSL Stylesheets V1.52.2">
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+ − 14 <meta name="description" content=
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+ − 15 "Presents an overview of the features, system requirements, and starting the toolbox.">
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+ − 16 </head>
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+ − 17
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+ − 18 <body>
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+ − 19 <a name="top_of_page" id="top_of_page"></a>
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+ − 20
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+ − 21 <p style="font-size:1px;"> </p>
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+ − 22
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+ − 23 <table class="nav" summary="Navigation aid" border="0" width=
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+ − 24 "100%" cellpadding="0" cellspacing="0">
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+ − 25 <tr>
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+ − 26 <td valign="baseline"><b>LTPDA Toolbox</b></td><td><a href="../helptoc.html">contents</a></td>
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+ − 27
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+ − 28 <td valign="baseline" align="right"><a href=
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+ − 29 "sigproc_methods.html"><img src="b_prev.gif" border="0" align=
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+ − 30 "bottom" alt="Spectral Estimation Methods"></a> <a href=
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+ − 31 "sigproc_cpsd.html"><img src="b_next.gif" border="0" align=
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+ − 32 "bottom" alt="Cross-spectral density estimates"></a></td>
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+ − 33 </tr>
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+ − 34 </table>
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+ − 35
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+ − 36 <h1 class="title"><a name="f3-12899" id="f3-12899"></a>Power spectral density estimates</h1>
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+ − 37 <hr>
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+ − 38
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+ − 39 <p>
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+ − 40 <h2>Description</h2>
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+ − 41 <p>
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+ − 42 The LTPDA method <a href="matlab:doc('ao/psd')">ao/psd</a> estimates the power spectral density of time-series
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+ − 43 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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+ − 44 Data are windowed prior to the estimation of the spectrum, by multiplying
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+ − 45 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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+ − 46 of the border discontinuities. The window length is adjustable to shorter lenghts to reduce the spectral
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+ − 47 density uncertainties, and the percentage of subsequent window overlap can be adjusted as well. The detrending is
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+ − 48 performed on the individual windows. The user can choose the quantity being given in output among
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+ − 49 ASD (amplitude spectral density), PSD (power spectral density), AS (amplitude spectrum), and PS (power spectrum).
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+ − 50 <br>
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+ − 51 <br>
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+ − 52 <h2>Syntax</h2>
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+ − 53 </p>
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+ − 54 <div class="fragment"><pre>
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+ − 55 <br> bs = psd(a1, a2, a3, ..., pl)
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+ − 56 bs = psd(as, pl)
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+ − 57 bs = as.psd(pl)
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+ − 58 </pre> </div>
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+ − 59 <p>
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+ − 60 <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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+ − 61 the output object(s) and <tt>pl</tt> is an optional parameter list.
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+ − 62 </p>
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+ − 63 <h2>Parameters</h2>
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+ − 64 <p>
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+ − 65 The parameter list <tt>pl</tt> includes the following parameters:
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+ − 66 </p>
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+ − 67 <ul>
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+ − 68 <li> <tt>'Nfft'</tt> - number of samples in each fft [default: length of input data]
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+ − 69 A string value containing the variable 'fs' can
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+ − 70 also be used, e.g., plist('Nfft', '2*fs') </li>
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+ − 71 <li> <tt>'Win'</tt> - the window to be applied to the data to remove the
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+ − 72 discontinuities at edges of segments. [default: taken from user prefs].<br>
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+ − 73 The window is described by a string with its name and, only in the case of Kaiser window,
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+ − 74 the additional parameter <tt>'psll'</tt>. <br>For instance: plist('Win', 'Kaiser', 'psll', 200). </li>
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+ − 75 </li>
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+ − 76 <li> <tt>'Olap'</tt> - segment percent overlap [default: -1, (taken from window function)] </li>
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+ − 77 <li> <tt>'Scale'</tt> - scaling of output. Choose from: <ul>
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+ − 78 <li> 'ASD' - amplitude spectral density </li>
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+ − 79 <li> 'PSD' - power spectral density [default] </li>
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+ − 80 <li> 'AS' - amplitude spectrum </li>
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+ − 81 <li> 'PS' - power spectrum </li> </ul> </li>
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+ − 82 <li> <tt>'Order'</tt> - order of segment detrending <ul>
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+ − 83 <li> -1 - no detrending </li>
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+ − 84 <li> 0 - subtract mean [default] </li>
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+ − 85 <li> 1 - subtract linear fit </li>
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+ − 86 <li> N - subtract fit of polynomial, order N </li> </ul> </li>
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+ − 87 <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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+ − 88 <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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+ − 89 </ul>
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+ − 90 <p>
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+ − 91 The length of the window is set by the value of the parameter <tt>'Nfft'</tt>, so that the window
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+ − 92 is actually built using only the key features of the window: the name and, for Kaiser windows, the psll.
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+ − 93 </p>
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+ − 94 <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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+ − 95 <p>
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+ − 96 <table cellspacing="0" class="note" summary="Note" cellpadding="5" border="1">
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+ − 97 <tr width="90%">
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+ − 98 <td>
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+ − 99 If the user doesn't specify the value of a given parameter, the default value is used.
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+ − 100 </td>
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+ − 101 </tr>
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+ − 102 </table>
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+ − 103 </p>
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+ − 104 <h2>Algorithm</h2>
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+ − 105 <p>
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+ − 106 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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+ − 107 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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+ − 108 algorithm <a href="#references">[2]</a> which allows to compute mean and variance in one loop.
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+ − 109 </p>
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+ − 110 <h2>Examples</h2>
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+ − 111 <p>
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+ − 112 1. Evaluation of the PSD of a time-series represented by a low frequency sinewave signal, superimposed to
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+ − 113 white noise. Comparison of the effect of windowing on the estimate of the white noise level and
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+ − 114 on resolving the signal.
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+ − 115 </p>
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+ − 116 <div class="fragment"><pre>
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+ − 117 <br> <span class="comment">% create two AOs</span>
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+ − 118 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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+ − 119 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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+ − 120 <span class="comment">% add both AOs</span>
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+ − 121 x = x1 + x2;
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+ − 122 <span class="comment">% compute the psd changing the 'nfft'</span>
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+ − 123 y_lf = psd(x);
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+ − 124 y_hf = psd(x,plist(<span class="string">'nfft'</span>,1000));
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+ − 125 <span class="comment">% compare </span>
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+ − 126 iplot(y_lf, y_hf)
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+ − 127 </pre></div>
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+ − 128
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+ − 129 <img src="images/psd_1.png" alt="" border="3">
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+ − 130
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+ − 131 <p>
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+ − 132 2. Evaluation of the PSD of a time-series represented by a low frequency sinewave signal, superimposed to
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+ − 133 white noise and to a low frequency linear drift. In the example, the same spectrum is computed with different
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+ − 134 spectral windows.
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+ − 135 </p>
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+ − 136 <div class="fragment"><pre>
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+ − 137 <br> <span class="comment">% create three AOs</span>
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+ − 138 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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+ − 139 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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+ − 140 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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+ − 141 <span class="comment">% add them</span>
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+ − 142 x = x1 + x2 + x3;
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+ − 143 <span class="comment">% compute psd with different windows</span>
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+ − 144 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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+ − 145 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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+ − 146 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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+ − 147 <span class="comment">% compare</span>
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+ − 148 iplot(y_1, y_2, y_3);
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+ − 149 </pre></div>
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+ − 150 <p>
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+ − 151 <img src="images/psd_2.png" alt="" border="3">
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+ − 152 </p>
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+ − 153 <h2><a name="references">References</a></h2>
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+ − 154
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+ − 155 <ol>
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+ − 156 <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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+ − 157 Modified Periodograms, <i>IEEE Trans. on Audio and Electroacoustics</i>, Vol. 15, No. 2 (1967), pp. 70 - 73.</a></li>
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