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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 <!-- $Id: ndim_ng_content.html,v 1.6 2011/05/02 19:08:05 luigi Exp $ -->
2
3 <!-- ================================================== -->
4 <!-- BEGIN CONTENT FILE -->
5 <!-- ================================================== -->
6 <!-- ===== link box: Begin ===== -->
7 <p>
8 <table border="1" width="80%">
9 <tr>
10 <td>
11 <table border="0" cellpadding="5" class="categorylist" width="100%">
12 <colgroup>
13 <col width="37%"/>
14 <col width="63%"/>
15 </colgroup>
16 <tbody>
17 <tr valign="top">
18 <td>
19 <a href="#mchspectra">Multichannel Spectra</a>
20 </td>
21 <td>Theoretical background on multichannel spectra.</td>
22 </tr>
23 <tr valign="top">
24 <td>
25 <a href="#NGTheory">Noise generation</a>
26 </td>
27 <td>Theoretical introduction to multichannel noise generation.</td>
28 </tr>
29 <tr valign="top">
30 <td>
31 <a href="#ngMCH">Multichannel Noise Generation</a>
32 </td>
33 <td>Generation of multichannel noise with given cross-spectral density matrix.</td>
34 </tr>
35 <tr valign="top">
36 <td>
37 <a href="#ng1D">Noisegen 1D</a>
38 </td>
39 <td>Generation of one-dimensional noise with given spectral density.</td>
40 </tr>
41 <tr valign="top">
42 <td>
43 <a href="#ng2D">Noisegen 2D</a>
44 </td>
45 <td>Generation of two-dimensional noise with given cross-spectral density.</td>
46 </tr>
47 </tbody>
48 </table>
49 </td>
50 </tr>
51 </table>
52 </p>
53 <!-- ===== link box: End ====== -->
54
55
56
57 <p>
58 </p>
59 <p>
60 The following sections gives an introduction to the generation of model
61 noise with a given cross spectral density. Further details can be found
62 in ref. [1].
63 </p>
64
65 <!-- ===== Multichannel Spectra Theory ====== -->
66 <h2><a name="mchspectra">Theoretical background on multichannel spectra</a></h2>
67 <p>
68 We define the autocorrelation function (ACF) of a stationary multichannel process as:
69 </p>
70 <div>
71 <IMG src="images/ngEqn1.gif" align="center" border="0">
72 </div>
73 <p>
74 </p>
75 <p>
76 If the multichannel process is L dimensional then the kth element of the ACF is a LxL matrix:
77 </p>
78 <div>
79 <IMG src="images/ngEqn2.gif" align="center" border="0">
80 </div>
81 <p>
82 </p>
83 <p>
84 The ACF matrix is not hermitian but have the property that:
85 </p>
86 <div>
87 <IMG src="images/ngEqn3.gif" align="center" border="0">
88 </div>
89 <p>
90 </p>
91 <p>
92 The cross-spectral density matrix (CSD) is defined as the fourier transform of the ACF:
93 </p>
94 <div>
95 <IMG src="images/ngEqn4.gif" align="center" border="0">
96 </div>
97 <p>
98 </p>
99 <p>
100 the CSD matrix is hermitian.
101 </p>
102 <p>
103 A multichannel white noise process is defined as the process whose ACF satisfies:
104 </p>
105 <div>
106 <IMG src="images/ngEqn5.gif" align="center" border="0">
107 </div>
108 <p>
109 </p>
110 <p>
111 therefore the cross-spectral matrix has constant terms as a function of the frequency:
112 </p>
113 <div>
114 <IMG src="images/ngEqn6.gif" align="center" border="0">
115 </div>
116 <p>
117 </p>
118 <p>
119 The individual processes are each white noise processes with power spectral density (PSD) given by
120 <IMG src="images/ngEqn7.gif" align="center" border="0">.
121 The cross-correlation between the processes is zero except at the same time instant
122 where they are correlated with a cross-correlation given by the off-diagonal elements of
123 <IMG src="images/ngEqn8.gif" align="center" border="0">.
124 A common assumption is
125 <IMG src="images/ngEqn9.gif" align="center" border="0">
126 (identity matrix) that is equivalent to assume the white processes having unitary variance
127 and are completely uncorrelated being zero the off diagonal terms of the CSD matrix.
128 Further details can be found in [1 - 3].
129 </p>
130
131 <!-- ===== Multichannel Noise Generation Theory ====== -->
132 <h2><a name="NGTheory">Theoretical introduction to multichannel noise generation</a></h2>
133 <p>
134 The problem of multichannel noise generation with a given cross-spectrum
135 is formulated in frequency domain as follows:
136 </p>
137 <div>
138 <IMG src="images/ngEqn10.gif" align="center" border="0">
139 </div>
140 <p>
141 </p>
142 <p>
143 <IMG src="images/ngEqn11.gif" align="center" border="0"> is a
144 multichannel digital filter that generating colored noise data with given cross-spectrum
145 <IMG src="images/ngEqn12.gif" align="center" border="0">
146 starting from a set of mutually independent unitary variance with noise processes.
147 </p>
148 <p>
149 After some mathematics it can be showed that the desired multichannel coloring filter can be written as:
150 </p>
151 <div>
152 <IMG src="images/ngEqn13.gif" align="center" border="0">
153 </div>
154 <p>
155 </p>
156 <p>
157 where <IMG src="images/ngEqn14.gif" align="center" border="0">
158 and <IMG src="images/ngEqn15.gif" align="center" border="0">
159 are the eigenvectors and eigenvalues matrices of
160 <IMG src="images/ngEqn12.gif" align="center" border="0">
161 matrix.
162 </p>
163
164 <!-- ===== Multichannel Noise Generator ====== -->
165 <h2><a name="ngMCH">Generation of multichannel noise with given cross-spectral density matrix</a></h2>
166 <p>
167 <tt>LTPDA Toolbox</tt> provides two methods (<a href="matlab:doc('matrix/mchNoisegenFilter')">mchNoisegenFilter</a> and
168 <a href="matlab:doc('matrix/mchNoisegen')">mchNoisegen</a>) of the class <tt>matrix</tt> for the production
169 of multichannel noise coloring filter and multichannel colored noise data series.
170 Noise data are colored Gaussian distributed time series with given cross-spectral density matrix.
171 Noise generation process is properly initialized in order to avoid starting transients on the data series.
172 Details on frequency domain identification of noisegen filters and on the noise generation process
173 can be found in ref. [1].
174 <a href="matlab:doc('matrix/mchNoisegenFilter')">mchNoisegenFilter</a> needs a model for the one-sided
175 cross-spectral density or power spectral density if we are considering one-dimensional problems.
176 <a href="matlab:doc('matrix/mchNoisegen')">mchNoisegen</a> instead accepts as input the noise generating filter
177 produced by <a href="matlab:doc('matrix/mchNoisegenFilter')">mchNoisegenFilter</a>.
178 Details on accepted parameters can be found on the documentation pages of the two methods:
179 <ul>
180 <li> <a href="matlab:doc('matrix/mchNoisegenFilter')">mchNoisegenFilter</a>
181 <li> <a href="matlab:doc('matrix/mchNoisegen')">mchNoisegen</a>
182 </ul>
183 </p>
184
185
186 <!-- ===== Noisegen 1D ====== -->
187 <h2><a name="ng1D">Generation of one-dimensional noise with given spectral density</a></h2>
188 <p>
189 <tt>noisegen1D</tt> is a coloring tool allowing the generation of colored noise from withe noise with a given spectrum.
190 The function constructs a coloring filter through a fitting procedure to the model provided.
191 If no model is provided an error is prompted. The colored noise provided has one-sided psd
192 corresponding to the input model.
193 The function needs a model for the one-sided power spectral density of
194 the given process. Details on accepted parameters can be found on
195 the <a href="matlab:doc('ao/noisegen1D')">noisegen1D</a> documentation page. <br>
196 <ol>
197 <li> The square root of the model for the power spectral
198 density is fit in z-domain in order to determine a coloring
199 filter.
200 <li> Unstable poles are removed by an all-pass stabilization procedure.
201 <li> White input data are filtered with the identified filter in order to be colored.
202 </ol>
203 </p>
204
205
206 <!-- ===== Noisegen 2D ====== -->
207 <h2><a name="ng2D">Generation of two-dimensional noise with given cross-spectral density</a></h2>
208 <p>
209 <tt>noisegen2D</tt> is a nose coloring tool allowing the generation
210 two data series with the given cross-spectral density from two starting
211 white and mutually uncorrelated data series.
212 Coloring filters are constructed by a fitting procedure to a model
213 for the corss-spectral density matrix provided.
214 In order to work with <tt>noisegen2D</tt> you must provide
215 a model (frequency series analysis objects) for the cross-spectral density
216 matrix of the process.
217 Details on accepted parameters can be found on
218 the <a href="matlab:doc('ao/noisegen2D')">noisegen2D</a> documentation page. <br>
219 <ol>
220 <li> Coloring filters frequency response is calculated by the
221 eigendecomposition of the model cross-spectral matrix.
222 <li> Calculated responses are fit in z-domain in order to identify
223 corresponding autoregressive moving average filters.
224 <li> Input time-series are filtered. The filtering process corresponds to:<br>
225 o(1) = Filt11(a(1)) + Filt12(a(2))<br>
226 o(2) = Filt21(a(1)) + Filt22(a(2))
227 </ol>
228 </p>
229
230
231 <h2>References</h2>
232 <p>
233 <ol>
234 <li> L. Ferraioli et. al., Calibrating spectral estimation for the LISA
235 Technology Package with multichannel synthetic noise generation, Phys. Rev. D 82, 042001 (2010).
236 <li> S. M. Kay, Modern Spectral Estimation, Prentice-Hall, 1999 </li>
237 <li> G. M. Jenkins and D. G. Watts, Spectral Analysis and Its Applications, Holden-Day 1968. </li>
238 </ol>
239 </p>