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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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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/m-toolbox/html_help/help/ug/franklin_ng.html Wed Nov 23 19:22:13 2011 +0100 @@ -0,0 +1,156 @@ +<!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>Franklin noise-generator (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= + "noisegen.html"><img src="b_prev.gif" border="0" align= + "bottom" alt="Generating model noise"></a> <a href= + "ndim_ng.html"><img src="b_next.gif" border="0" align= + "bottom" alt="Noise generation with given cross-spectral density"></a></td> + </tr> + </table> + + <h1 class="title"><a name="f3-12899" id="f3-12899"></a>Franklin noise-generator</h1> + <hr> + + <p> + The following sections gives an introduction to the <a href="noisegen.html">generation of model noise</a> using the noise generator implemented in LTPDA. +<ul> + <li><a href="#franklin">Franklin's noise generator</a></li> + <li><a href="#description">Description</a></li> + <li><a href="#inputs">Inputs</a></li> + <li><a href="#outputs">Outputs</a></li> + <li><a href="#usage">Usage</a></li> +</ul> +<h2><a name="franklin">Franklin's noise generator</a></h2> +Franklin's noise generator is a method to generate arbitrarily long time series with a prescribed spectral density. +The algorithm is based on the following paper: +</p> +<p>Franklin, Joel N.: + <i> Numerical simulation of stationary and non-stationary gaussian + random processes </i>, SIAM review, Volume {<b> 7</b>}, Issue 1, page 68--80, 1965. +</p> +<p> + The Document <i> Generation of Random time series with prescribed spectra </i> by Gerhard Heinzel (S2-AEI-TN-3034) <br> corrects a mistake in the aforesaid paper and describes the practical implementation. +</p> +<p> + See <a href="noisegen.html">Generating model noise</a> for more general information on this. +</p> +<p> + Franklin's method does not require any 'warm up' period. It starts with a transfer function given as ratio of two polynomials.<br/> + The generator operates on a real state vector y of length n which is + maintained between invocations. It produces samples of the time series in equidistant steps <tt>T = 1/fs</tt>, where <tt>fs</tt> is the sampling frequency. +</p> +<p> + <ul> + <li> y0 = Tinit * r, on initialization + <li> yi = E * yi-1 + Tprop * r, to propagate + <li> xi = a * yi , the sampled time series. + </ul> + r is a vector of independent normal Gaussian random numbers + Tinit, E, Tprop which are real matrices and a which is a real vector are determined once by the algorithm. +</p> + +<h2><a name="description">Description</a></h2> +<p> + When an analysis object is constructed from a pole zero model Franklin's noise generator is called (compare <a href="ao_create.html#pzmodel">Creating AOs from pole zero models</a>). +</p> + + +<h2><a name="inputs">Inputs</a></h2> +for the function call the parameter list has to contain at least: +<ul> + <li> nsecs - number of seconds (length of time series) + <li> fs - sampling frequency + <li> pzmodel with gain +</ul> + +<h2><a name="outputs">Outputs</a></h2> +<ul> + <li> b - analysis object containing the resulting time series +</ul> +</p> +<h2><a name="usage">Usage</a></h2> +The analysis object constructor <a href="ao_create.html">ao</a> calls the following four functions when the input is a pzmodel. +<ul> + <li> ngconv + <li> ngsetup + <li> nginit + <li> ngprop +</ul> +<p> + First a parameter list of the input parameters is to be done. For further information on this look at <a href="plist_create.html#params">Creating parameter lists from parameters</a>.<br/> +</p> +<h2><a name="starting">Starting from a given pole/zero model</a></h2> +<p> + The parameter list should contain the number of seconds the resulting time series should have <tt>nsecs</tt> and the sampling frequency <tt>fs</tt>. <br/> + The constructor call should look like this: +</p> +<div class="fragment"><pre> + f1 = 5; + f2 = 10; + f3 = 1; + gain = 1; + fs = 10; <span class="comment">% sampling frequency</span> + nsecs = 100; <span class="comment">% number of seconds to be generated</span> + p = [pz(f1) pz(f2)]; + z = [pz(f3)]; + pzm = pzmodel(gain, p, z); + a = ao(pzm, plist(<span class="string">'nsecs'</span>, nsecs, <span class="string">'fs'</span>,fs)) + +</pre></div> +The output will be an analysis object <tt>a</tt> containing the time series with the spectrum described by the input pole-zero model. +</p> + + + + + </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="noisegen.html"><img src= + "b_prev.gif" border="0" align="bottom" alt= + "Generating model noise"></a> </td> + + <td align="left">Generating model noise</td> + + <td> </td> + + <td align="right">Noise generation with given cross-spectral density</td> + + <td align="right" width="20"><a href= + "ndim_ng.html"><img src="b_next.gif" border="0" align= + "bottom" alt="Noise generation with given cross-spectral density"></a></td> + </tr> + </table><br> + + <p class="copy">©LTP Team</p> +</body> +</html>