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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_psd.html Wed Nov 23 19:22:13 2011 +0100 @@ -0,0 +1,188 @@ +<!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>Power 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_methods.html"><img src="b_prev.gif" border="0" align= + "bottom" alt="Spectral Estimation Methods"></a> <a href= + "sigproc_cpsd.html"><img src="b_next.gif" border="0" align= + "bottom" alt="Cross-spectral density estimates"></a></td> + </tr> + </table> + + <h1 class="title"><a name="f3-12899" id="f3-12899"></a>Power spectral density estimates</h1> + <hr> + + <p> + <h2>Description</h2> +<p> +The LTPDA method <a href="matlab:doc('ao/psd')">ao/psd</a> estimates the power spectral density of time-series + signals, included in the input <tt>ao</tt>s following the Welch's averaged, modified periodogram method <a href="#references">[1]</a>. + 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 a polinomial of time in order to reduce the impact + of the border discontinuities. The window length is adjustable to shorter lenghts to reduce the spectral + density uncertainties, and the percentage of subsequent window overlap can be adjusted as well. The detrending is + performed on the individual windows. 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). + <br> + <br> + <h2>Syntax</h2> +</p> +<div class="fragment"><pre> + <br> bs = psd(a1, a2, a3, ..., pl) + bs = psd(as, pl) + bs = as.psd(pl) +</pre> </div> +<p> + <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 + the output object(s) and <tt>pl</tt> is an optional parameter list. +</p> +<h2>Parameters</h2> +<p> + The parameter list <tt>pl</tt> includes the following parameters: +</p> +<ul> + <li> <tt>'Nfft'</tt> - number of samples in each fft [default: length of input data] + A string value containing the variable 'fs' can + also be used, e.g., plist('Nfft', '2*fs') </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> + <li> <tt>'Olap'</tt> - segment percent overlap [default: -1, (taken from window function)] </li> + <li> <tt>'Scale'</tt> - scaling of output. Choose from: <ul> + <li> 'ASD' - amplitude spectral density </li> + <li> 'PSD' - power spectral density [default] </li> + <li> 'AS' - amplitude spectrum </li> + <li> 'PS' - power spectrum </li> </ul> </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> + <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> + <li><tt>'Times'</tt> - interval of time to evaluate the calculation on. If empty [default], it will take the whole section.</li> +</ul> +<p> + The length of the window is set by the value of the parameter <tt>'Nfft'</tt>, so that the window + is actually built using only the key features of the window: the name and, for Kaiser windows, the psll. +</p> +<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> +<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> +<h2>Algorithm</h2> +<p> + 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 + 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. +</p> +<h2>Examples</h2> +<p> + 1. Evaluation of the PSD of a time-series represented by a low frequency sinewave signal, superimposed to + white noise. Comparison of the effect of windowing on the estimate of the white noise level and + on resolving the signal. +</p> +<div class="fragment"><pre> + <br> <span class="comment">% create two AOs</span> + 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)); + 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="comment">% add both AOs</span> + x = x1 + x2; + <span class="comment">% compute the psd changing the 'nfft'</span> + y_lf = psd(x); + y_hf = psd(x,plist(<span class="string">'nfft'</span>,1000)); + <span class="comment">% compare </span> + iplot(y_lf, y_hf) +</pre></div> + +<img src="images/psd_1.png" alt="" border="3"> + +<p> + 2. Evaluation of the PSD of a time-series represented by a low frequency sinewave signal, superimposed to + white noise and to a low frequency linear drift. In the example, the same spectrum is computed with different + spectral windows. +</p> +<div class="fragment"><pre> + <br> <span class="comment">% create three AOs</span> + 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>)); + 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>)); + 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>)); + <span class="comment">% add them</span> + x = x1 + x2 + x3; + <span class="comment">% compute psd with different windows</span> + 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>)); + 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>)); + 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)); + <span class="comment">% compare</span> + iplot(y_1, y_2, y_3); +</pre></div> +<p> + <img src="images/psd_2.png" alt="" border="3"> +</p> +<h2><a name="references">References</a></h2> + +<ol> + <li> P.D. Welch, The Use of Fast Fourier Transform for the Estimation of Power Spectra: A Method Based on Time Averaging Over Short, +Modified Periodograms, <i>IEEE Trans. on Audio and Electroacoustics</i>, Vol. 15, No. 2 (1967), pp. 70 - 73.</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_methods.html"><img src= + "b_prev.gif" border="0" align="bottom" alt= + "Spectral Estimation Methods"></a> </td> + + <td align="left">Spectral Estimation Methods</td> + + <td> </td> + + <td align="right">Cross-spectral density estimates</td> + + <td align="right" width="20"><a href= + "sigproc_cpsd.html"><img src="b_next.gif" border="0" align= + "bottom" alt="Cross-spectral density estimates"></a></td> + </tr> + </table><br> + + <p class="copy">©LTP Team</p> +</body> +</html>