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Import.
| author | Daniele Nicolodi <nicolodi@science.unitn.it> |
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| date | Wed, 23 Nov 2011 19:22:13 +0100 |
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/m-toolbox/html_help/help/ug/ndim_ng.html Wed Nov 23 19:22:13 2011 +0100 @@ -0,0 +1,304 @@ +<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" + "http://www.w3.org/TR/1999/REC-html401-19991224/loose.dtd"> + +<html lang="en"> +<head> + <meta name="generator" content= + "HTML Tidy for Mac OS X (vers 1st December 2004), see www.w3.org"> + <meta http-equiv="Content-Type" content= + "text/html; charset=us-ascii"> + + <title>Noise generation with given cross-spectral density (LTPDA Toolbox)</title> + <link rel="stylesheet" href="docstyle.css" type="text/css"> + <meta name="generator" content="DocBook XSL Stylesheets V1.52.2"> + <meta name="description" content= + "Presents an overview of the features, system requirements, and starting the toolbox."> + </head> + +<body> + <a name="top_of_page" id="top_of_page"></a> + + <p style="font-size:1px;"> </p> + + <table class="nav" summary="Navigation aid" border="0" width= + "100%" cellpadding="0" cellspacing="0"> + <tr> + <td valign="baseline"><b>LTPDA Toolbox</b></td><td><a href="../helptoc.html">contents</a></td> + + <td valign="baseline" align="right"><a href= + "franklin_ng.html"><img src="b_prev.gif" border="0" align= + "bottom" alt="Franklin noise-generator"></a> <a href= + "smodel.html"><img src="b_next.gif" border="0" align= + "bottom" alt="Parameteric models"></a></td> + </tr> + </table> + + <h1 class="title"><a name="f3-12899" id="f3-12899"></a>Noise generation with given cross-spectral density</h1> + <hr> + + <p> + + <!-- ================================================== --> + <!-- BEGIN CONTENT FILE --> + <!-- ================================================== --> + <!-- ===== link box: Begin ===== --> + <p> + <table border="1" width="80%"> + <tr> + <td> + <table border="0" cellpadding="5" class="categorylist" width="100%"> + <colgroup> + <col width="37%"/> + <col width="63%"/> + </colgroup> + <tbody> + <tr valign="top"> + <td> + <a href="#mchspectra">Multichannel Spectra</a> + </td> + <td>Theoretical background on multichannel spectra.</td> + </tr> + <tr valign="top"> + <td> + <a href="#NGTheory">Noise generation</a> + </td> + <td>Theoretical introduction to multichannel noise generation.</td> + </tr> + <tr valign="top"> + <td> + <a href="#ngMCH">Multichannel Noise Generation</a> + </td> + <td>Generation of multichannel noise with given cross-spectral density matrix.</td> + </tr> + <tr valign="top"> + <td> + <a href="#ng1D">Noisegen 1D</a> + </td> + <td>Generation of one-dimensional noise with given spectral density.</td> + </tr> + <tr valign="top"> + <td> + <a href="#ng2D">Noisegen 2D</a> + </td> + <td>Generation of two-dimensional noise with given cross-spectral density.</td> + </tr> + </tbody> + </table> + </td> + </tr> + </table> + </p> + <!-- ===== link box: End ====== --> + + + + <p> + </p> + <p> + The following sections gives an introduction to the generation of model + noise with a given cross spectral density. Further details can be found + in ref. [1]. + </p> + + <!-- ===== Multichannel Spectra Theory ====== --> + <h2><a name="mchspectra">Theoretical background on multichannel spectra</a></h2> + <p> + We define the autocorrelation function (ACF) of a stationary multichannel process as: + </p> + <div> + <IMG src="images/ngEqn1.gif" align="center" border="0"> + </div> + <p> + </p> + <p> + If the multichannel process is L dimensional then the kth element of the ACF is a LxL matrix: + </p> + <div> + <IMG src="images/ngEqn2.gif" align="center" border="0"> + </div> + <p> + </p> + <p> + The ACF matrix is not hermitian but have the property that: + </p> + <div> + <IMG src="images/ngEqn3.gif" align="center" border="0"> + </div> + <p> + </p> + <p> + The cross-spectral density matrix (CSD) is defined as the fourier transform of the ACF: + </p> + <div> + <IMG src="images/ngEqn4.gif" align="center" border="0"> + </div> + <p> + </p> + <p> + the CSD matrix is hermitian. + </p> + <p> + A multichannel white noise process is defined as the process whose ACF satisfies: + </p> + <div> + <IMG src="images/ngEqn5.gif" align="center" border="0"> + </div> + <p> + </p> + <p> + therefore the cross-spectral matrix has constant terms as a function of the frequency: + </p> + <div> + <IMG src="images/ngEqn6.gif" align="center" border="0"> + </div> + <p> + </p> + <p> + The individual processes are each white noise processes with power spectral density (PSD) given by + <IMG src="images/ngEqn7.gif" align="center" border="0">. + The cross-correlation between the processes is zero except at the same time instant + where they are correlated with a cross-correlation given by the off-diagonal elements of + <IMG src="images/ngEqn8.gif" align="center" border="0">. + A common assumption is + <IMG src="images/ngEqn9.gif" align="center" border="0"> + (identity matrix) that is equivalent to assume the white processes having unitary variance + and are completely uncorrelated being zero the off diagonal terms of the CSD matrix. + Further details can be found in [1 - 3]. + </p> + + <!-- ===== Multichannel Noise Generation Theory ====== --> + <h2><a name="NGTheory">Theoretical introduction to multichannel noise generation</a></h2> + <p> + The problem of multichannel noise generation with a given cross-spectrum + is formulated in frequency domain as follows: + </p> + <div> + <IMG src="images/ngEqn10.gif" align="center" border="0"> + </div> + <p> + </p> + <p> + <IMG src="images/ngEqn11.gif" align="center" border="0"> is a + multichannel digital filter that generating colored noise data with given cross-spectrum + <IMG src="images/ngEqn12.gif" align="center" border="0"> + starting from a set of mutually independent unitary variance with noise processes. + </p> + <p> + After some mathematics it can be showed that the desired multichannel coloring filter can be written as: + </p> + <div> + <IMG src="images/ngEqn13.gif" align="center" border="0"> + </div> + <p> + </p> + <p> + where <IMG src="images/ngEqn14.gif" align="center" border="0"> + and <IMG src="images/ngEqn15.gif" align="center" border="0"> + are the eigenvectors and eigenvalues matrices of + <IMG src="images/ngEqn12.gif" align="center" border="0"> + matrix. + </p> + + <!-- ===== Multichannel Noise Generator ====== --> + <h2><a name="ngMCH">Generation of multichannel noise with given cross-spectral density matrix</a></h2> + <p> + <tt>LTPDA Toolbox</tt> provides two methods (<a href="matlab:doc('matrix/mchNoisegenFilter')">mchNoisegenFilter</a> and + <a href="matlab:doc('matrix/mchNoisegen')">mchNoisegen</a>) of the class <tt>matrix</tt> for the production + of multichannel noise coloring filter and multichannel colored noise data series. + Noise data are colored Gaussian distributed time series with given cross-spectral density matrix. + Noise generation process is properly initialized in order to avoid starting transients on the data series. + Details on frequency domain identification of noisegen filters and on the noise generation process + can be found in ref. [1]. + <a href="matlab:doc('matrix/mchNoisegenFilter')">mchNoisegenFilter</a> needs a model for the one-sided + cross-spectral density or power spectral density if we are considering one-dimensional problems. + <a href="matlab:doc('matrix/mchNoisegen')">mchNoisegen</a> instead accepts as input the noise generating filter + produced by <a href="matlab:doc('matrix/mchNoisegenFilter')">mchNoisegenFilter</a>. + Details on accepted parameters can be found on the documentation pages of the two methods: + <ul> + <li> <a href="matlab:doc('matrix/mchNoisegenFilter')">mchNoisegenFilter</a> + <li> <a href="matlab:doc('matrix/mchNoisegen')">mchNoisegen</a> + </ul> + </p> + + + <!-- ===== Noisegen 1D ====== --> + <h2><a name="ng1D">Generation of one-dimensional noise with given spectral density</a></h2> + <p> + <tt>noisegen1D</tt> is a coloring tool allowing the generation of colored noise from withe noise with a given spectrum. + The function constructs a coloring filter through a fitting procedure to the model provided. + If no model is provided an error is prompted. The colored noise provided has one-sided psd + corresponding to the input model. + The function needs a model for the one-sided power spectral density of + the given process. Details on accepted parameters can be found on + the <a href="matlab:doc('ao/noisegen1D')">noisegen1D</a> documentation page. <br> + <ol> + <li> The square root of the model for the power spectral + density is fit in z-domain in order to determine a coloring + filter. + <li> Unstable poles are removed by an all-pass stabilization procedure. + <li> White input data are filtered with the identified filter in order to be colored. + </ol> + </p> + + + <!-- ===== Noisegen 2D ====== --> + <h2><a name="ng2D">Generation of two-dimensional noise with given cross-spectral density</a></h2> + <p> + <tt>noisegen2D</tt> is a nose coloring tool allowing the generation + two data series with the given cross-spectral density from two starting + white and mutually uncorrelated data series. + Coloring filters are constructed by a fitting procedure to a model + for the corss-spectral density matrix provided. + In order to work with <tt>noisegen2D</tt> you must provide + a model (frequency series analysis objects) for the cross-spectral density + matrix of the process. + Details on accepted parameters can be found on + the <a href="matlab:doc('ao/noisegen2D')">noisegen2D</a> documentation page. <br> + <ol> + <li> Coloring filters frequency response is calculated by the + eigendecomposition of the model cross-spectral matrix. + <li> Calculated responses are fit in z-domain in order to identify + corresponding autoregressive moving average filters. + <li> Input time-series are filtered. The filtering process corresponds to:<br> + o(1) = Filt11(a(1)) + Filt12(a(2))<br> + o(2) = Filt21(a(1)) + Filt22(a(2)) + </ol> + </p> + + + <h2>References</h2> + <p> + <ol> + <li> L. Ferraioli et. al., Calibrating spectral estimation for the LISA + Technology Package with multichannel synthetic noise generation, Phys. Rev. D 82, 042001 (2010). + <li> S. M. Kay, Modern Spectral Estimation, Prentice-Hall, 1999 </li> + <li> G. M. Jenkins and D. G. Watts, Spectral Analysis and Its Applications, Holden-Day 1968. </li> + </ol> + </p> + + </p> + + <br> + <br> + <table class="nav" summary="Navigation aid" border="0" width= + "100%" cellpadding="0" cellspacing="0"> + <tr valign="top"> + <td align="left" width="20"><a href="franklin_ng.html"><img src= + "b_prev.gif" border="0" align="bottom" alt= + "Franklin noise-generator"></a> </td> + + <td align="left">Franklin noise-generator</td> + + <td> </td> + + <td align="right">Parameteric models</td> + + <td align="right" width="20"><a href= + "smodel.html"><img src="b_next.gif" border="0" align= + "bottom" alt="Parameteric models"></a></td> + </tr> + </table><br> + + <p class="copy">©LTP Team</p> +</body> +</html>