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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>Noise generation with given cross-spectral density (LTPDA Toolbox)</title>
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21 <p style="font-size:1px;">&nbsp;</p>
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23 <table class="nav" summary="Navigation aid" border="0" width=
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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 "franklin_ng.html"><img src="b_prev.gif" border="0" align=
30 "bottom" alt="Franklin noise-generator"></a>&nbsp;&nbsp;&nbsp;<a href=
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32 "bottom" alt="Parameteric models"></a></td>
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35
36 <h1 class="title"><a name="f3-12899" id="f3-12899"></a>Noise generation with given cross-spectral density</h1>
37 <hr>
38
39 <p>
40
41 <!-- ================================================== -->
42 <!-- BEGIN CONTENT FILE -->
43 <!-- ================================================== -->
44 <!-- ===== link box: Begin ===== -->
45 <p>
46 <table border="1" width="80%">
47 <tr>
48 <td>
49 <table border="0" cellpadding="5" class="categorylist" width="100%">
50 <colgroup>
51 <col width="37%"/>
52 <col width="63%"/>
53 </colgroup>
54 <tbody>
55 <tr valign="top">
56 <td>
57 <a href="#mchspectra">Multichannel Spectra</a>
58 </td>
59 <td>Theoretical background on multichannel spectra.</td>
60 </tr>
61 <tr valign="top">
62 <td>
63 <a href="#NGTheory">Noise generation</a>
64 </td>
65 <td>Theoretical introduction to multichannel noise generation.</td>
66 </tr>
67 <tr valign="top">
68 <td>
69 <a href="#ngMCH">Multichannel Noise Generation</a>
70 </td>
71 <td>Generation of multichannel noise with given cross-spectral density matrix.</td>
72 </tr>
73 <tr valign="top">
74 <td>
75 <a href="#ng1D">Noisegen 1D</a>
76 </td>
77 <td>Generation of one-dimensional noise with given spectral density.</td>
78 </tr>
79 <tr valign="top">
80 <td>
81 <a href="#ng2D">Noisegen 2D</a>
82 </td>
83 <td>Generation of two-dimensional noise with given cross-spectral density.</td>
84 </tr>
85 </tbody>
86 </table>
87 </td>
88 </tr>
89 </table>
90 </p>
91 <!-- ===== link box: End ====== -->
92
93
94
95 <p>
96 </p>
97 <p>
98 The following sections gives an introduction to the generation of model
99 noise with a given cross spectral density. Further details can be found
100 in ref. [1].
101 </p>
102
103 <!-- ===== Multichannel Spectra Theory ====== -->
104 <h2><a name="mchspectra">Theoretical background on multichannel spectra</a></h2>
105 <p>
106 We define the autocorrelation function (ACF) of a stationary multichannel process as:
107 </p>
108 <div>
109 <IMG src="images/ngEqn1.gif" align="center" border="0">
110 </div>
111 <p>
112 </p>
113 <p>
114 If the multichannel process is L dimensional then the kth element of the ACF is a LxL matrix:
115 </p>
116 <div>
117 <IMG src="images/ngEqn2.gif" align="center" border="0">
118 </div>
119 <p>
120 </p>
121 <p>
122 The ACF matrix is not hermitian but have the property that:
123 </p>
124 <div>
125 <IMG src="images/ngEqn3.gif" align="center" border="0">
126 </div>
127 <p>
128 </p>
129 <p>
130 The cross-spectral density matrix (CSD) is defined as the fourier transform of the ACF:
131 </p>
132 <div>
133 <IMG src="images/ngEqn4.gif" align="center" border="0">
134 </div>
135 <p>
136 </p>
137 <p>
138 the CSD matrix is hermitian.
139 </p>
140 <p>
141 A multichannel white noise process is defined as the process whose ACF satisfies:
142 </p>
143 <div>
144 <IMG src="images/ngEqn5.gif" align="center" border="0">
145 </div>
146 <p>
147 </p>
148 <p>
149 therefore the cross-spectral matrix has constant terms as a function of the frequency:
150 </p>
151 <div>
152 <IMG src="images/ngEqn6.gif" align="center" border="0">
153 </div>
154 <p>
155 </p>
156 <p>
157 The individual processes are each white noise processes with power spectral density (PSD) given by
158 <IMG src="images/ngEqn7.gif" align="center" border="0">.
159 The cross-correlation between the processes is zero except at the same time instant
160 where they are correlated with a cross-correlation given by the off-diagonal elements of
161 <IMG src="images/ngEqn8.gif" align="center" border="0">.
162 A common assumption is
163 <IMG src="images/ngEqn9.gif" align="center" border="0">
164 (identity matrix) that is equivalent to assume the white processes having unitary variance
165 and are completely uncorrelated being zero the off diagonal terms of the CSD matrix.
166 Further details can be found in [1 - 3].
167 </p>
168
169 <!-- ===== Multichannel Noise Generation Theory ====== -->
170 <h2><a name="NGTheory">Theoretical introduction to multichannel noise generation</a></h2>
171 <p>
172 The problem of multichannel noise generation with a given cross-spectrum
173 is formulated in frequency domain as follows:
174 </p>
175 <div>
176 <IMG src="images/ngEqn10.gif" align="center" border="0">
177 </div>
178 <p>
179 </p>
180 <p>
181 <IMG src="images/ngEqn11.gif" align="center" border="0"> is a
182 multichannel digital filter that generating colored noise data with given cross-spectrum
183 <IMG src="images/ngEqn12.gif" align="center" border="0">
184 starting from a set of mutually independent unitary variance with noise processes.
185 </p>
186 <p>
187 After some mathematics it can be showed that the desired multichannel coloring filter can be written as:
188 </p>
189 <div>
190 <IMG src="images/ngEqn13.gif" align="center" border="0">
191 </div>
192 <p>
193 </p>
194 <p>
195 where <IMG src="images/ngEqn14.gif" align="center" border="0">
196 and <IMG src="images/ngEqn15.gif" align="center" border="0">
197 are the eigenvectors and eigenvalues matrices of
198 <IMG src="images/ngEqn12.gif" align="center" border="0">
199 matrix.
200 </p>
201
202 <!-- ===== Multichannel Noise Generator ====== -->
203 <h2><a name="ngMCH">Generation of multichannel noise with given cross-spectral density matrix</a></h2>
204 <p>
205 <tt>LTPDA Toolbox</tt> provides two methods (<a href="matlab:doc('matrix/mchNoisegenFilter')">mchNoisegenFilter</a> and
206 <a href="matlab:doc('matrix/mchNoisegen')">mchNoisegen</a>) of the class <tt>matrix</tt> for the production
207 of multichannel noise coloring filter and multichannel colored noise data series.
208 Noise data are colored Gaussian distributed time series with given cross-spectral density matrix.
209 Noise generation process is properly initialized in order to avoid starting transients on the data series.
210 Details on frequency domain identification of noisegen filters and on the noise generation process
211 can be found in ref. [1].
212 <a href="matlab:doc('matrix/mchNoisegenFilter')">mchNoisegenFilter</a> needs a model for the one-sided
213 cross-spectral density or power spectral density if we are considering one-dimensional problems.
214 <a href="matlab:doc('matrix/mchNoisegen')">mchNoisegen</a> instead accepts as input the noise generating filter
215 produced by <a href="matlab:doc('matrix/mchNoisegenFilter')">mchNoisegenFilter</a>.
216 Details on accepted parameters can be found on the documentation pages of the two methods:
217 <ul>
218 <li> <a href="matlab:doc('matrix/mchNoisegenFilter')">mchNoisegenFilter</a>
219 <li> <a href="matlab:doc('matrix/mchNoisegen')">mchNoisegen</a>
220 </ul>
221 </p>
222
223
224 <!-- ===== Noisegen 1D ====== -->
225 <h2><a name="ng1D">Generation of one-dimensional noise with given spectral density</a></h2>
226 <p>
227 <tt>noisegen1D</tt> is a coloring tool allowing the generation of colored noise from withe noise with a given spectrum.
228 The function constructs a coloring filter through a fitting procedure to the model provided.
229 If no model is provided an error is prompted. The colored noise provided has one-sided psd
230 corresponding to the input model.
231 The function needs a model for the one-sided power spectral density of
232 the given process. Details on accepted parameters can be found on
233 the <a href="matlab:doc('ao/noisegen1D')">noisegen1D</a> documentation page. <br>
234 <ol>
235 <li> The square root of the model for the power spectral
236 density is fit in z-domain in order to determine a coloring
237 filter.
238 <li> Unstable poles are removed by an all-pass stabilization procedure.
239 <li> White input data are filtered with the identified filter in order to be colored.
240 </ol>
241 </p>
242
243
244 <!-- ===== Noisegen 2D ====== -->
245 <h2><a name="ng2D">Generation of two-dimensional noise with given cross-spectral density</a></h2>
246 <p>
247 <tt>noisegen2D</tt> is a nose coloring tool allowing the generation
248 two data series with the given cross-spectral density from two starting
249 white and mutually uncorrelated data series.
250 Coloring filters are constructed by a fitting procedure to a model
251 for the corss-spectral density matrix provided.
252 In order to work with <tt>noisegen2D</tt> you must provide
253 a model (frequency series analysis objects) for the cross-spectral density
254 matrix of the process.
255 Details on accepted parameters can be found on
256 the <a href="matlab:doc('ao/noisegen2D')">noisegen2D</a> documentation page. <br>
257 <ol>
258 <li> Coloring filters frequency response is calculated by the
259 eigendecomposition of the model cross-spectral matrix.
260 <li> Calculated responses are fit in z-domain in order to identify
261 corresponding autoregressive moving average filters.
262 <li> Input time-series are filtered. The filtering process corresponds to:<br>
263 o(1) = Filt11(a(1)) + Filt12(a(2))<br>
264 o(2) = Filt21(a(1)) + Filt22(a(2))
265 </ol>
266 </p>
267
268
269 <h2>References</h2>
270 <p>
271 <ol>
272 <li> L. Ferraioli et. al., Calibrating spectral estimation for the LISA
273 Technology Package with multichannel synthetic noise generation, Phys. Rev. D 82, 042001 (2010).
274 <li> S. M. Kay, Modern Spectral Estimation, Prentice-Hall, 1999 </li>
275 <li> G. M. Jenkins and D. G. Watts, Spectral Analysis and Its Applications, Holden-Day 1968. </li>
276 </ol>
277 </p>
278
279 </p>
280
281 <br>
282 <br>
283 <table class="nav" summary="Navigation aid" border="0" width=
284 "100%" cellpadding="0" cellspacing="0">
285 <tr valign="top">
286 <td align="left" width="20"><a href="franklin_ng.html"><img src=
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288 "Franklin noise-generator"></a>&nbsp;</td>
289
290 <td align="left">Franklin noise-generator</td>
291
292 <td>&nbsp;</td>
293
294 <td align="right">Parameteric models</td>
295
296 <td align="right" width="20"><a href=
297 "smodel.html"><img src="b_next.gif" border="0" align=
298 "bottom" alt="Parameteric models"></a></td>
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301
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