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+ − 1 The following sections gives an introduction to the <a href="noisegen.html">generation of model noise</a> using the noise generator implemented in LTPDA.
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+ − 2 <ul>
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+ − 3 <li><a href="#franklin">Franklin's noise generator</a></li>
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+ − 4 <li><a href="#description">Description</a></li>
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+ − 5 <li><a href="#inputs">Inputs</a></li>
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+ − 6 <li><a href="#outputs">Outputs</a></li>
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+ − 7 <li><a href="#usage">Usage</a></li>
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+ − 8 </ul>
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+ − 9 <h2><a name="franklin">Franklin's noise generator</a></h2>
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+ − 10 Franklin's noise generator is a method to generate arbitrarily long time series with a prescribed spectral density.
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+ − 11 The algorithm is based on the following paper:
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+ − 12 </p>
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+ − 13 <p>Franklin, Joel N.:
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+ − 14 <i> Numerical simulation of stationary and non-stationary gaussian
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+ − 15 random processes </i>, SIAM review, Volume {<b> 7</b>}, Issue 1, page 68--80, 1965.
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+ − 16 </p>
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+ − 17 <p>
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+ − 18 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.
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+ − 19 </p>
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+ − 20 <p>
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+ − 21 See <a href="noisegen.html">Generating model noise</a> for more general information on this.
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+ − 22 </p>
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+ − 23 <p>
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+ − 24 Franklin's method does not require any 'warm up' period. It starts with a transfer function given as ratio of two polynomials.<br/>
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+ − 25 The generator operates on a real state vector y of length n which is
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+ − 26 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.
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+ − 27 </p>
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+ − 28 <p>
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+ − 29 <ul>
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+ − 30 <li> y0 = Tinit * r, on initialization
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+ − 31 <li> yi = E * yi-1 + Tprop * r, to propagate
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+ − 32 <li> xi = a * yi , the sampled time series.
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+ − 33 </ul>
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+ − 34 r is a vector of independent normal Gaussian random numbers
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+ − 35 Tinit, E, Tprop which are real matrices and a which is a real vector are determined once by the algorithm.
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+ − 36 </p>
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+ − 37
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+ − 38 <h2><a name="description">Description</a></h2>
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+ − 39 <p>
Daniele Nicolodi <nicolodi@science.unitn.it>
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+ − 40 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>).
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+ − 41 </p>
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+ − 42
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+ − 43
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+ − 44 <h2><a name="inputs">Inputs</a></h2>
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+ − 45 for the function call the parameter list has to contain at least:
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+ − 46 <ul>
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+ − 47 <li> nsecs - number of seconds (length of time series)
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+ − 48 <li> fs - sampling frequency
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+ − 49 <li> pzmodel with gain
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+ − 50 </ul>
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+ − 51
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+ − 52 <h2><a name="outputs">Outputs</a></h2>
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+ − 53 <ul>
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+ − 54 <li> b - analysis object containing the resulting time series
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+ − 55 </ul>
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+ − 56 </p>
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+ − 57 <h2><a name="usage">Usage</a></h2>
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+ − 58 The analysis object constructor <a href="ao_create.html">ao</a> calls the following four functions when the input is a pzmodel.
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+ − 59 <ul>
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+ − 60 <li> ngconv
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+ − 61 <li> ngsetup
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+ − 62 <li> nginit
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+ − 63 <li> ngprop
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+ − 64 </ul>
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+ − 65 <p>
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+ − 66 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/>
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+ − 67 </p>
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+ − 68 <h2><a name="starting">Starting from a given pole/zero model</a></h2>
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+ − 69 <p>
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+ − 70 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/>
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+ − 71 The constructor call should look like this:
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+ − 72 </p>
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+ − 73 <div class="fragment"><pre>
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+ − 74 f1 = 5;
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+ − 75 f2 = 10;
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+ − 76 f3 = 1;
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+ − 77 gain = 1;
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+ − 78 fs = 10; <span class="comment">% sampling frequency</span>
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+ − 79 nsecs = 100; <span class="comment">% number of seconds to be generated</span>
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+ − 80 p = [pz(f1) pz(f2)];
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+ − 81 z = [pz(f3)];
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+ − 82 pzm = pzmodel(gain, p, z);
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+ − 83 a = ao(pzm, plist(<span class="string">'nsecs'</span>, nsecs, <span class="string">'fs'</span>,fs))
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+ − 84
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+ − 85 </pre></div>
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+ − 86 The output will be an analysis object <tt>a</tt> containing the time series with the spectrum described by the input pole-zero model.
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+ − 87 </p>
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+ − 88
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+ − 89
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+ − 90