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                <td>
                  <a href="#mchspectra">Multichannel Spectra</a>
                </td>
                <td>Theoretical background on multichannel spectra.</td>
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                <td>
                  <a href="#NGTheory">Noise generation</a>
                </td>
                <td>Theoretical introduction to multichannel noise generation.</td>
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		<td>
                  <a href="#ngMCH">Multichannel Noise Generation</a>
                </td>
		 <td>Generation of multichannel noise with given cross-spectral density matrix.</td>
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                  <a href="#ng1D">Noisegen 1D</a>
                </td>
                <td>Generation of one-dimensional noise with given spectral density.</td>
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                <td>
                  <a href="#ng2D">Noisegen 2D</a>
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                <td>Generation of two-dimensional noise with given cross-spectral density.</td>
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  <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">
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  <p>
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  <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">
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  <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">
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  <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">
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  <p>
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  <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">
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  <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>