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