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author Daniele Nicolodi <nicolodi@science.unitn.it>
date Wed, 23 Nov 2011 19:22:13 +0100
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11 <title>Franklin noise-generator (LTPDA Toolbox)</title>
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15 "Presents an overview of the features, system requirements, and starting the toolbox.">
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19 <a name="top_of_page" id="top_of_page"></a>
20
21 <p style="font-size:1px;">&nbsp;</p>
22
23 <table class="nav" summary="Navigation aid" border="0" width=
24 "100%" cellpadding="0" cellspacing="0">
25 <tr>
26 <td valign="baseline"><b>LTPDA Toolbox</b></td><td><a href="../helptoc.html">contents</a></td>
27
28 <td valign="baseline" align="right"><a href=
29 "noisegen.html"><img src="b_prev.gif" border="0" align=
30 "bottom" alt="Generating model noise"></a>&nbsp;&nbsp;&nbsp;<a href=
31 "ndim_ng.html"><img src="b_next.gif" border="0" align=
32 "bottom" alt="Noise generation with given cross-spectral density"></a></td>
33 </tr>
34 </table>
35
36 <h1 class="title"><a name="f3-12899" id="f3-12899"></a>Franklin noise-generator</h1>
37 <hr>
38
39 <p>
40 The following sections gives an introduction to the <a href="noisegen.html">generation of model noise</a> using the noise generator implemented in LTPDA.
41 <ul>
42 <li><a href="#franklin">Franklin's noise generator</a></li>
43 <li><a href="#description">Description</a></li>
44 <li><a href="#inputs">Inputs</a></li>
45 <li><a href="#outputs">Outputs</a></li>
46 <li><a href="#usage">Usage</a></li>
47 </ul>
48 <h2><a name="franklin">Franklin's noise generator</a></h2>
49 Franklin's noise generator is a method to generate arbitrarily long time series with a prescribed spectral density.
50 The algorithm is based on the following paper:
51 </p>
52 <p>Franklin, Joel N.:
53 <i> Numerical simulation of stationary and non-stationary gaussian
54 random processes </i>, SIAM review, Volume {<b> 7</b>}, Issue 1, page 68--80, 1965.
55 </p>
56 <p>
57 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.
58 </p>
59 <p>
60 See <a href="noisegen.html">Generating model noise</a> for more general information on this.
61 </p>
62 <p>
63 Franklin's method does not require any 'warm up' period. It starts with a transfer function given as ratio of two polynomials.<br/>
64 The generator operates on a real state vector y of length n which is
65 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.
66 </p>
67 <p>
68 <ul>
69 <li> y0 = Tinit * r, on initialization
70 <li> yi = E * yi-1 + Tprop * r, to propagate
71 <li> xi = a * yi , the sampled time series.
72 </ul>
73 r is a vector of independent normal Gaussian random numbers
74 Tinit, E, Tprop which are real matrices and a which is a real vector are determined once by the algorithm.
75 </p>
76
77 <h2><a name="description">Description</a></h2>
78 <p>
79 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>).
80 </p>
81
82
83 <h2><a name="inputs">Inputs</a></h2>
84 for the function call the parameter list has to contain at least:
85 <ul>
86 <li> nsecs - number of seconds (length of time series)
87 <li> fs - sampling frequency
88 <li> pzmodel with gain
89 </ul>
90
91 <h2><a name="outputs">Outputs</a></h2>
92 <ul>
93 <li> b - analysis object containing the resulting time series
94 </ul>
95 </p>
96 <h2><a name="usage">Usage</a></h2>
97 The analysis object constructor <a href="ao_create.html">ao</a> calls the following four functions when the input is a pzmodel.
98 <ul>
99 <li> ngconv
100 <li> ngsetup
101 <li> nginit
102 <li> ngprop
103 </ul>
104 <p>
105 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/>
106 </p>
107 <h2><a name="starting">Starting from a given pole/zero model</a></h2>
108 <p>
109 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/>
110 The constructor call should look like this:
111 </p>
112 <div class="fragment"><pre>
113 f1 = 5;
114 f2 = 10;
115 f3 = 1;
116 gain = 1;
117 fs = 10; <span class="comment">% sampling frequency</span>
118 nsecs = 100; <span class="comment">% number of seconds to be generated</span>
119 p = [pz(f1) pz(f2)];
120 z = [pz(f3)];
121 pzm = pzmodel(gain, p, z);
122 a = ao(pzm, plist(<span class="string">'nsecs'</span>, nsecs, <span class="string">'fs'</span>,fs))
123
124 </pre></div>
125 The output will be an analysis object <tt>a</tt> containing the time series with the spectrum described by the input pole-zero model.
126 </p>
127
128
129
130
131 </p>
132
133 <br>
134 <br>
135 <table class="nav" summary="Navigation aid" border="0" width=
136 "100%" cellpadding="0" cellspacing="0">
137 <tr valign="top">
138 <td align="left" width="20"><a href="noisegen.html"><img src=
139 "b_prev.gif" border="0" align="bottom" alt=
140 "Generating model noise"></a>&nbsp;</td>
141
142 <td align="left">Generating model noise</td>
143
144 <td>&nbsp;</td>
145
146 <td align="right">Noise generation with given cross-spectral density</td>
147
148 <td align="right" width="20"><a href=
149 "ndim_ng.html"><img src="b_next.gif" border="0" align=
150 "bottom" alt="Noise generation with given cross-spectral density"></a></td>
151 </tr>
152 </table><br>
153
154 <p class="copy">&copy;LTP Team</p>
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156 </html>