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Import.
| author | Daniele Nicolodi <nicolodi@science.unitn.it> |
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| date | Wed, 23 Nov 2011 19:22:13 +0100 |
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| 1 <h2>Description</h2> | |
| 2 <p> | |
| 3 The LTPDA method <a href="matlab:doc('ao/lpsd')">ao/lpsd</a> estimates the power spectral density of time-series | |
| 4 signals, included in the input <tt>ao</tt>s following the LPSD algorithm <a href="#references">[1]</a>. Spectral density estimates are not | |
| 5 evaluated at frequencies which are linear multiples of the minimum frequency resolution <tt>1/T</tt>, where <tt>T</tt> | |
| 6 is the window lenght, but on a logarithmic scale. The algorithm takes care of calculating the frequencies at which to evaluate | |
| 7 the spectral estimate, aiming at minimizing the uncertainty in the estimate itself, and to recalculate a suitable | |
| 8 window length for each frequency bin. | |
| 9 </p> | |
| 10 <p> | |
| 11 Data are windowed prior to the estimation of the spectrum, by multiplying | |
| 12 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 | |
| 13 of the border discontinuities. Detrending is performed on each individual window. | |
| 14 The user can choose the quantity being given in output among ASD (amplitude spectral density), | |
| 15 PSD (power spectral density), AS (amplitude spectrum), and PS (power spectrum). | |
| 16 </p> | |
| 17 <br> | |
| 18 <h2>Syntax</h2> | |
| 19 </p> | |
| 20 <div class="fragment"><pre> | |
| 21 <br> bs = lpsd(a1,a2,a3,...,pl) | |
| 22 bs = lpsd(as,pl) | |
| 23 bs = as.lpsd(pl) | |
| 24 </pre> | |
| 25 </div> | |
| 26 <p> | |
| 27 <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 and <tt>pl</tt> is an optional parameter list. | |
| 28 | |
| 29 <h2>Parameters</h2> | |
| 30 <p>The parameter list <tt>pl</tt> includes the following parameters:</p> | |
| 31 <ul> | |
| 32 <li> <tt>'Kdes'</tt> - desired number of averages [default: 100]</li> | |
| 33 <li> <tt>'Jdes'</tt> - number of spectral frequencies to compute [default: 1000]</li> | |
| 34 <li> <tt>'Lmin'</tt> - minimum segment length [default: 0]</li> | |
| 35 <li> <tt>'Win'</tt> - the window to be applied to the data to remove the | |
| 36 discontinuities at edges of segments. [default: taken from user prefs].<br> | |
| 37 The window is described by a string with its name and, only in the case of Kaiser window, | |
| 38 the additional parameter <tt>'psll'</tt>. <br>For instance: plist('Win', 'Kaiser', 'psll', 200). | |
| 39 </li> | |
| 40 <li> <tt>'Olap'</tt> - segment percent overlap [default: -1, (taken from window function)] </li> | |
| 41 <li> <tt>'Scale'</tt> - scaling of output. Choose from: <ul> | |
| 42 <li> 'ASD' - amplitude spectral density </li> | |
| 43 <li> 'PSD' - power spectral density [default] </li> | |
| 44 <li> 'AS' - amplitude spectrum </li> | |
| 45 <li> 'PS' - power spectrum </li> </ul> </li> | |
| 46 <li> <tt>'Order'</tt> - order of segment detrending <ul> | |
| 47 <li> -1 - no detrending </li> | |
| 48 <li> 0 - subtract mean [default] </li> | |
| 49 <li> 1 - subtract linear fit </li> | |
| 50 <li> N - subtract fit of polynomial, order N </li> </ul> </li> | |
| 51 </ul> | |
| 52 The length of the window is set by the value of the parameter <tt>'Nfft'</tt>, so that the window | |
| 53 is actually rebuilt using only the key features of the window, i.e. the name and, for Kaiser windows, the PSLL. | |
| 54 </p> | |
| 55 | |
| 56 <p> | |
| 57 <table cellspacing="0" class="note" summary="Note" cellpadding="5" border="1"> | |
| 58 <tr width="90%"> | |
| 59 <td> | |
| 60 If the user doesn't specify the value of a given parameter, the default value is used. | |
| 61 </td> | |
| 62 </tr> | |
| 63 </table> | |
| 64 </p> | |
| 65 <h2>Algorithm</h2> | |
| 66 <p> | |
| 67 The algorithm is implemented according to <a href="#references">[1]</a>. In order to | |
| 68 compute the standard deviation of the mean for each frequency bin, the averaging of the different segments is performed using Welford's | |
| 69 algorithm <a href="#references">[2]</a> which allows to compute mean and variance in one loop. <br> | |
| 70 In the LPSD algorithm, the first frequencies bins are usually computed using a single segment containing all the data. | |
| 71 For these bins, the sample variance is set to <tt>Inf</tt>. | |
| 72 </p> | |
| 73 <h2>Examples</h2> | |
| 74 <p> | |
| 75 1. Evaluation of the ASD of a time-series represented by a low frequency sinewave signal, superimposed to | |
| 76 white noise. Comparison of the effect of using standard Pwelch and LPSD on the estimate | |
| 77 of the white noise level and on resolving the signal. | |
| 78 </p> | |
| 79 <div class="fragment"><pre> | |
| 80 <br> <span class="comment">% Create input AO</span> | |
| 81 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">'rad'</span>)); | |
| 82 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">'rad'</span>)); | |
| 83 x = x1 + x2; | |
| 84 | |
| 85 <span class="comment">% Compute psd and lpsd </span> | |
| 86 pl = 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">'Kaiser'</span>,<span class="string">'psll'</span>,200); | |
| 87 y1 = psd(x, pl); | |
| 88 y2 = lpsd(x, pl); | |
| 89 | |
| 90 <span class="comment">% Compare</span> | |
| 91 iplot(y1, y2) | |
| 92 </pre> | |
| 93 </div> | |
| 94 | |
| 95 <img src="images/l_psd_1.png" border="3"> | |
| 96 | |
| 97 | |
| 98 <h2><a name="references">References</a></h2> | |
| 99 | |
| 100 <ol> | |
| 101 <li> M. Troebs, G. Heinzel, Improved spectrum estimation from digitized time series | |
| 102 on a logarithmic frequency axis, <a href="http://dx.doi.org/10.1016/j.measurement.2005.10.010" ><i>Measurement</i>, Vol. 39 (2006), pp. 120 - 129</a>. See also the <a href="http://dx.doi.org/10.1016/j.measurement.2008.04.004" >Corrigendum</a>.</li> | |
| 103 <li> B. P. Weldford, Note on a Method for Calculating Corrected Sums of Squares and Products, | |
| 104 <i>Technometrics<i>, Vol. 4, No. 3 (1962), pp 419 - 420.</li> | |
| 105 </ol> |