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A useful model of *single*-channel time-series analysis is that random
numbers *x*_{t} enter a filter *f*_{t} and come out as a signal *y*_{t}.
A useful model of *multiple*-channel
**time-series analysis**--with two channels, for example--is to
start with independent random
numbers in both the
*x*_{1}(*t*) channel and the
*x*_{2}(*t*) channel.
Then we need *four* filters,
*f*_{11}(*t*),
*f*_{12}(*t*),
*f*_{21}(*t*), and
*f*_{22}(*t*),
which produce two output signals defined by the *Z*-transforms

| |
(53) |

| (54) |

These signals have realistic characteristics.
Each has its own spectral color.
Each has a partial relationship to the other
which is characterized by a spectral amplitude and phase.
Typically we begin by examining the **covariance matrix**.
For example, consider two time series, *y*_{1}(*t*) and *y*_{2}(*t*).
Their *Z*-transforms are *Y*_{1}(*Z*) and *Y*_{2}(*Z*).
Their covariance matrix is
| |
(55) |

Here *Z*-transforms represent the components of the matrix
in the frequency domain.
In the time domain, each of the four elements in the matrix
of (55) becomes a **Toeplitz** matrix,
a matrix of correlation functions
(see page ).
The expectations in equation (55)
are specified by theoretical assertions
or estimated by sample averages
or some combination of the two.
Analogously to spectral **factorization**,
the covariance matrix can be factored into two parts,
,where is an upper triangular matrix.
The factorization might be done by the well known **Cholesky** method.
The factorization is a multichannel generalization
of spectral factorization
and raises interesting questions about minimum-phase
that are partly addressed in FGDP.

** Next:** Bispectrum
** Up:** CROSSCORRELATION AND COHERENCY
** Previous:** Coherency
Stanford Exploration Project

10/21/1998