Mercurial > hg > ltpda
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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/lcpsd')">ao/lcpsd</a> estimates the cross-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> b = lcpsd(a1,a2,pl) | |
| 22 </pre> | |
| 23 </div> | |
| 24 <p> | |
| 25 <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. | |
| 26 | |
| 27 <h2>Parameters</h2> | |
| 28 <p>The parameter list <tt>pl</tt> includes the following parameters:</p> | |
| 29 <ul> | |
| 30 <li> <tt>'Kdes'</tt> - desired number of averages [default: 100]</li> | |
| 31 <li> <tt>'Jdes'</tt> - number of spectral frequencies to compute [default: 1000]</li> | |
| 32 <li> <tt>'Lmin'</tt> - minimum segment length [default: 0]</li> | |
| 33 <li> <tt>'Win'</tt> - the window to be applied to the data to remove the | |
| 34 discontinuities at edges of segments. [default: taken from user prefs].<br> | |
| 35 The window is described by a string with its name and, only in the case of Kaiser window, | |
| 36 the additional parameter <tt>'psll'</tt>. <br>For instance: plist('Win', 'Kaiser', 'psll', 200). </li> | |
| 37 <li> <tt>'Olap'</tt> - segment percent overlap [default: -1, (taken from window function)] </li> | |
| 38 <li> <tt>'Order'</tt> - order of segment detrending <ul> | |
| 39 <li> -1 - no detrending </li> | |
| 40 <li> 0 - subtract mean [default] </li> | |
| 41 <li> 1 - subtract linear fit </li> | |
| 42 <li> N - subtract fit of polynomial, order N </li> </ul> </li> | |
| 43 </ul> | |
| 44 The length of the window is set by the value of the parameter <tt>'Nfft'</tt>, so that the window | |
| 45 is actually rebuilt using only the key features of the window, i.e. the name and, for Kaiser windows, the PSLL. | |
| 46 </p> | |
| 47 <p> | |
| 48 <table cellspacing="0" class="note" summary="Note" cellpadding="5" border="1"> | |
| 49 <tr width="90%"> | |
| 50 <td> | |
| 51 If the user doesn't specify the value of a given parameter, the default value is used. | |
| 52 </td> | |
| 53 </tr> | |
| 54 </table> | |
| 55 </p> | |
| 56 | |
| 57 <p>The function makes log-scale CPSD estimates between the 2 input <tt>ao</tt>s. The input argument | |
| 58 list must contain 2 analysis objects, and the output will contain the LCPSD estimate. | |
| 59 If passing two identical objects <tt>ai</tt>, the output will be equivalent to the output of <tt>lpsd(ai)</tt>. | |
| 60 </p> | |
| 61 </pre> </div> | |
| 62 </p> | |
| 63 <h2>Algorithm</h2> | |
| 64 <p> | |
| 65 The algorithm is implemented according to <a href="#references">[1]</a>. In order to | |
| 66 compute the standard deviation of the mean for each frequency bin, the averaging of the different segments is performed using Welford's | |
| 67 algorithm <a href="#references">[2]</a> which allows to compute mean and variance in one loop. <br> | |
| 68 In the LPSD algorithm, the first frequencies bins are usually computed using a single segment containing all the data. | |
| 69 For these bins, the sample variance is set to <tt>Inf</tt>. | |
| 70 </p> | |
| 71 <b>Example</b> | |
| 72 <p> | |
| 73 Evaluation of the log-scale CPSD of two time-series represented by: a low frequency sinewave signal superimposed to | |
| 74 white noise, and a low frequency sinewave signal at the same frequency, phase shifted and with different | |
| 75 amplitude, superimposed to white noise. | |
| 76 </p> | |
| 77 <div class="fragment"><pre> | |
| 78 <br> <span class="comment">% Parameters</span> | |
| 79 nsecs = 1000; | |
| 80 fs = 10; | |
| 81 | |
| 82 <span class="comment">% Create input AOs</span> | |
| 83 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>,fs)) + ... | |
| 84 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>,fs)); | |
| 85 x.setYunits(<span class="string">'m'</span>); | |
| 86 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>,fs,<span class="string">'phi'</span>,90)) + ... | |
| 87 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>,fs)); | |
| 88 y.setYunits(<span class="string">'V'</span>); | |
| 89 | |
| 90 <span class="comment">% Compute log cpsd</span> | |
| 91 z = lcpsd(x,y,plist(<span class="string">'nfft'</span>,1000)); | |
| 92 | |
| 93 <span class="comment">% Plot</span> | |
| 94 iplot(z); | |
| 95 </pre> | |
| 96 </div> | |
| 97 | |
| 98 <img src="images/l_cpsd_1.png" alt="" border="3"> | |
| 99 <br> | |
| 100 <h2><a name="references">References</a></h2> | |
| 101 | |
| 102 <ol> | |
| 103 <li> M. Troebs, G. Heinzel, Improved spectrum estimation from digitized time series | |
| 104 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> | |
| 105 <li> B. P. Weldford, Note on a Method for Calculating Corrected Sums of Squares and Products, | |
| 106 <i>Technometrics<i>, Vol. 4, No. 3 (1962), pp 419 - 420.</li> | |
| 107 </ol> |