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author Daniele Nicolodi <nicolodi@science.unitn.it>
date Wed, 23 Nov 2011 19:22:13 +0100
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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>
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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