The following sections gives an introduction to the generation of model noise with a given cross spectral density.

Multichannel Spectra

We define the autocorrelation function (ACF) of a stationary multichannel process as:

If the multichannel process is L dimensional then the kth element of the ACF is a LxL matrix:

The ACF matrix is not hermitian but have the property that:

The cross-spectral density matrix (CSD) is defined as the fourier transform of the ACF:

the CSD matrix is hermitian.

A multichannel white noise process is defined as the process whose ACF satisfies:

therefore the cross-spectral matrix has constant terms as a function of the frequency:

The individual processes are each white noise processes with power spectral density (PSD) given by . 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 . A common assumption is (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, 2].

Theory

The problem of multichannel noise generation with a given cross-spectrum is formulated in frequency domain as follows:

is a multichannel digital filter that generating colored noise data with given cross-spectrum starting from a set of mutually independent unitary variance with noise processes.

After some mathematics it can be showed that the desired multichannel coloring filter can be written as:

where and are the eigenvectors and eigenvalues matrices of matrix.

References

  1. S. M. Kay, Modern Spectral Estimation, Prentice-Hall, 1999
  2. G. M. Jenkins and D. G. Watts, Spectral Analysis and Its Applications, Holden-Day 1968.