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| author | Daniele Nicolodi <nicolodi@science.unitn.it> |
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| date | Wed, 23 Nov 2011 19:22:13 +0100 |
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/m-toolbox/html_help/help/ug/sigproc_lcpsd.html Wed Nov 23 19:22:13 2011 +0100 @@ -0,0 +1,172 @@ +<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" + "http://www.w3.org/TR/1999/REC-html401-19991224/loose.dtd"> + +<html lang="en"> +<head> + <meta name="generator" content= + "HTML Tidy for Mac OS X (vers 1st December 2004), see www.w3.org"> + <meta http-equiv="Content-Type" content= + "text/html; charset=us-ascii"> + + <title>Log-scale cross-spectral density estimates (LTPDA Toolbox)</title> + <link rel="stylesheet" href="docstyle.css" type="text/css"> + <meta name="generator" content="DocBook XSL Stylesheets V1.52.2"> + <meta name="description" content= + "Presents an overview of the features, system requirements, and starting the toolbox."> + </head> + +<body> + <a name="top_of_page" id="top_of_page"></a> + + <p style="font-size:1px;"> </p> + + <table class="nav" summary="Navigation aid" border="0" width= + "100%" cellpadding="0" cellspacing="0"> + <tr> + <td valign="baseline"><b>LTPDA Toolbox</b></td><td><a href="../helptoc.html">contents</a></td> + + <td valign="baseline" align="right"><a href= + "sigproc_lpsd.html"><img src="b_prev.gif" border="0" align= + "bottom" alt="Log-scale power spectral density estimates"></a> <a href= + "sigproc_lcohere.html"><img src="b_next.gif" border="0" align= + "bottom" alt="Log-scale cross coherence density estimates"></a></td> + </tr> + </table> + + <h1 class="title"><a name="f3-12899" id="f3-12899"></a>Log-scale cross-spectral density estimates</h1> + <hr> + + <p> + <h2>Description</h2> +<p> + The LTPDA method <a href="matlab:doc('ao/lcpsd')">ao/lcpsd</a> estimates the cross-power spectral density of time-series + signals, included in the input <tt>ao</tt>s following the LPSD algorithm <a href="#references">[1]</a>. Spectral density estimates are not + evaluated at frequencies which are linear multiples of the minimum frequency resolution <tt>1/T</tt>, where <tt>T</tt> + is the window lenght, but on a logarithmic scale. The algorithm takes care of calculating the frequencies at which to evaluate + the spectral estimate, aiming at minimizing the uncertainty in the estimate itself, and to recalculate a suitable + window length for each frequency bin. + </p> + <p> + Data are windowed prior to the estimation of the spectrum, by multiplying + 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 + of the border discontinuities. Detrending is performed on each individual window. + The user can choose the quantity being given in output among ASD (amplitude spectral density), + PSD (power spectral density), AS (amplitude spectrum), and PS (power spectrum). + </p> + <br> +<h2>Syntax</h2> +</p> +<div class="fragment"><pre> + <br> b = lcpsd(a1,a2,pl) + </pre> +</div> +<p> + <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. + + <h2>Parameters</h2> + <p>The parameter list <tt>pl</tt> includes the following parameters:</p> + <ul> + <li> <tt>'Kdes'</tt> - desired number of averages [default: 100]</li> + <li> <tt>'Jdes'</tt> - number of spectral frequencies to compute [default: 1000]</li> + <li> <tt>'Lmin'</tt> - minimum segment length [default: 0]</li> + <li> <tt>'Win'</tt> - the window to be applied to the data to remove the + discontinuities at edges of segments. [default: taken from user prefs].<br> + The window is described by a string with its name and, only in the case of Kaiser window, + the additional parameter <tt>'psll'</tt>. <br>For instance: plist('Win', 'Kaiser', 'psll', 200). </li> + <li> <tt>'Olap'</tt> - segment percent overlap [default: -1, (taken from window function)] </li> + <li> <tt>'Order'</tt> - order of segment detrending <ul> + <li> -1 - no detrending </li> + <li> 0 - subtract mean [default] </li> + <li> 1 - subtract linear fit </li> + <li> N - subtract fit of polynomial, order N </li> </ul> </li> +</ul> + The length of the window is set by the value of the parameter <tt>'Nfft'</tt>, so that the window + is actually rebuilt using only the key features of the window, i.e. the name and, for Kaiser windows, the PSLL. +</p> +<p> + <table cellspacing="0" class="note" summary="Note" cellpadding="5" border="1"> + <tr width="90%"> + <td> + If the user doesn't specify the value of a given parameter, the default value is used. + </td> + </tr> + </table> +</p> + +<p>The function makes log-scale CPSD estimates between the 2 input <tt>ao</tt>s. The input argument + list must contain 2 analysis objects, and the output will contain the LCPSD estimate. + If passing two identical objects <tt>ai</tt>, the output will be equivalent to the output of <tt>lpsd(ai)</tt>. +</p> +</pre> </div> +</p> +<h2>Algorithm</h2> +<p> + The algorithm is implemented according to <a href="#references">[1]</a>. In order to + compute the standard deviation of the mean for each frequency bin, the averaging of the different segments is performed using Welford's + algorithm <a href="#references">[2]</a> which allows to compute mean and variance in one loop. <br> + In the LPSD algorithm, the first frequencies bins are usually computed using a single segment containing all the data. + For these bins, the sample variance is set to <tt>Inf</tt>. +</p> + <b>Example</b> +<p> + Evaluation of the log-scale CPSD of two time-series represented by: a low frequency sinewave signal superimposed to + white noise, and a low frequency sinewave signal at the same frequency, phase shifted and with different + amplitude, superimposed to white noise. +</p> +<div class="fragment"><pre> + <br> <span class="comment">% Parameters</span> + nsecs = 1000; + fs = 10; + + <span class="comment">% Create input AOs</span> + 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)) + ... + 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)); + x.setYunits(<span class="string">'m'</span>); + 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)) + ... + 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)); + y.setYunits(<span class="string">'V'</span>); + + <span class="comment">% Compute log cpsd</span> + z = lcpsd(x,y,plist(<span class="string">'nfft'</span>,1000)); + + <span class="comment">% Plot</span> + iplot(z); + </pre> +</div> + +<img src="images/l_cpsd_1.png" alt="" border="3"> +<br> +<h2><a name="references">References</a></h2> + +<ol> + <li> M. Troebs, G. Heinzel, Improved spectrum estimation from digitized time series +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> + <li> B. P. Weldford, Note on a Method for Calculating Corrected Sums of Squares and Products, + <i>Technometrics<i>, Vol. 4, No. 3 (1962), pp 419 - 420.</li> + </ol> + </p> + + <br> + <br> + <table class="nav" summary="Navigation aid" border="0" width= + "100%" cellpadding="0" cellspacing="0"> + <tr valign="top"> + <td align="left" width="20"><a href="sigproc_lpsd.html"><img src= + "b_prev.gif" border="0" align="bottom" alt= + "Log-scale power spectral density estimates"></a> </td> + + <td align="left">Log-scale power spectral density estimates</td> + + <td> </td> + + <td align="right">Log-scale cross coherence density estimates</td> + + <td align="right" width="20"><a href= + "sigproc_lcohere.html"><img src="b_next.gif" border="0" align= + "bottom" alt="Log-scale cross coherence density estimates"></a></td> + </tr> + </table><br> + + <p class="copy">©LTP Team</p> +</body> +</html>