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Daniele Nicolodi <nicolodi@science.unitn.it>
date
Wed, 07 Dec 2011 17:24:36 +0100 (2011-12-07)
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+ − 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>
+ −
+ −
+ −