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+ − 10
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+ − 11 <title>Franklin noise-generator (LTPDA Toolbox)</title>
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+ − 14 <meta name="description" content=
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+ − 15 "Presents an overview of the features, system requirements, and starting the toolbox.">
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+ − 16 </head>
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+ − 19 <a name="top_of_page" id="top_of_page"></a>
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+ − 20
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+ − 21 <p style="font-size:1px;"> </p>
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+ − 22
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+ − 23 <table class="nav" summary="Navigation aid" border="0" width=
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+ − 24 "100%" cellpadding="0" cellspacing="0">
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+ − 25 <tr>
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+ − 26 <td valign="baseline"><b>LTPDA Toolbox</b></td><td><a href="../helptoc.html">contents</a></td>
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+ − 27
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+ − 28 <td valign="baseline" align="right"><a href=
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+ − 29 "noisegen.html"><img src="b_prev.gif" border="0" align=
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+ − 30 "bottom" alt="Generating model noise"></a> <a href=
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+ − 31 "ndim_ng.html"><img src="b_next.gif" border="0" align=
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+ − 32 "bottom" alt="Noise generation with given cross-spectral density"></a></td>
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+ − 33 </tr>
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+ − 34 </table>
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+ − 35
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+ − 36 <h1 class="title"><a name="f3-12899" id="f3-12899"></a>Franklin noise-generator</h1>
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+ − 37 <hr>
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+ − 38
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+ − 39 <p>
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+ − 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.
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+ − 41 <ul>
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+ − 42 <li><a href="#franklin">Franklin's noise generator</a></li>
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+ − 43 <li><a href="#description">Description</a></li>
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+ − 44 <li><a href="#inputs">Inputs</a></li>
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+ − 45 <li><a href="#outputs">Outputs</a></li>
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+ − 46 <li><a href="#usage">Usage</a></li>
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+ − 47 </ul>
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+ − 48 <h2><a name="franklin">Franklin's noise generator</a></h2>
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+ − 49 Franklin's noise generator is a method to generate arbitrarily long time series with a prescribed spectral density.
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+ − 50 The algorithm is based on the following paper:
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+ − 51 </p>
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+ − 52 <p>Franklin, Joel N.:
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+ − 53 <i> Numerical simulation of stationary and non-stationary gaussian
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+ − 54 random processes </i>, SIAM review, Volume {<b> 7</b>}, Issue 1, page 68--80, 1965.
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+ − 55 </p>
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+ − 56 <p>
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+ − 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.
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+ − 58 </p>
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+ − 59 <p>
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+ − 60 See <a href="noisegen.html">Generating model noise</a> for more general information on this.
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+ − 61 </p>
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+ − 62 <p>
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+ − 63 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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+ − 64 The generator operates on a real state vector y of length n which is
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+ − 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.
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+ − 66 </p>
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+ − 67 <p>
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+ − 68 <ul>
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+ − 69 <li> y0 = Tinit * r, on initialization
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+ − 70 <li> yi = E * yi-1 + Tprop * r, to propagate
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+ − 71 <li> xi = a * yi , the sampled time series.
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+ − 72 </ul>
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+ − 73 r is a vector of independent normal Gaussian random numbers
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+ − 74 Tinit, E, Tprop which are real matrices and a which is a real vector are determined once by the algorithm.
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+ − 75 </p>
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+ − 76
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+ − 77 <h2><a name="description">Description</a></h2>
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+ − 78 <p>
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+ − 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>).
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+ − 80 </p>
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+ − 81
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+ − 82
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+ − 83 <h2><a name="inputs">Inputs</a></h2>
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+ − 84 for the function call the parameter list has to contain at least:
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+ − 85 <ul>
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+ − 86 <li> nsecs - number of seconds (length of time series)
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+ − 87 <li> fs - sampling frequency
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+ − 88 <li> pzmodel with gain
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+ − 89 </ul>
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+ − 90
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+ − 91 <h2><a name="outputs">Outputs</a></h2>
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+ − 92 <ul>
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+ − 93 <li> b - analysis object containing the resulting time series
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+ − 94 </ul>
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+ − 95 </p>
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+ − 96 <h2><a name="usage">Usage</a></h2>
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+ − 97 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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+ − 98 <ul>
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+ − 99 <li> ngconv
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+ − 100 <li> ngsetup
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+ − 101 <li> nginit
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+ − 102 <li> ngprop
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+ − 103 </ul>
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+ − 104 <p>
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+ − 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/>
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+ − 106 </p>
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+ − 107 <h2><a name="starting">Starting from a given pole/zero model</a></h2>
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+ − 108 <p>
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+ − 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/>
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+ − 110 The constructor call should look like this:
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+ − 111 </p>
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+ − 112 <div class="fragment"><pre>
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+ − 113 f1 = 5;
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+ − 114 f2 = 10;
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+ − 115 f3 = 1;
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+ − 116 gain = 1;
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+ − 117 fs = 10; <span class="comment">% sampling frequency</span>
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+ − 118 nsecs = 100; <span class="comment">% number of seconds to be generated</span>
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+ − 119 p = [pz(f1) pz(f2)];
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+ − 120 z = [pz(f3)];
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+ − 121 pzm = pzmodel(gain, p, z);
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+ − 122 a = ao(pzm, plist(<span class="string">'nsecs'</span>, nsecs, <span class="string">'fs'</span>,fs))
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+ − 123
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+ − 124 </pre></div>
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+ − 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.
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+ − 126 </p>
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+ − 127
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+ − 130
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+ − 131 </p>
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+ − 132
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+ − 133 <br>
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+ − 134 <br>
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+ − 135 <table class="nav" summary="Navigation aid" border="0" width=
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+ − 136 "100%" cellpadding="0" cellspacing="0">
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+ − 137 <tr valign="top">
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+ − 138 <td align="left" width="20"><a href="noisegen.html"><img src=
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+ − 139 "b_prev.gif" border="0" align="bottom" alt=
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+ − 140 "Generating model noise"></a> </td>
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+ − 141
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+ − 142 <td align="left">Generating model noise</td>
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+ − 143
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+ − 144 <td> </td>
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+ − 145
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+ − 146 <td align="right">Noise generation with given cross-spectral density</td>
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+ − 147
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+ − 148 <td align="right" width="20"><a href=
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+ − 149 "ndim_ng.html"><img src="b_next.gif" border="0" align=
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+ − 150 "bottom" alt="Noise generation with given cross-spectral density"></a></td>
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+ − 151 </tr>
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+ − 152 </table><br>
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+ − 153
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+ − 154 <p class="copy">©LTP Team</p>
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+ − 155 </body>
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+ − 156 </html>