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** Up:** Waveform applications of least
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Multichannel filters are frequently useful.
For example,
with a vector-prediction filter
one might wish to predict a time series,
using its past and the past of a group of other series.
With a matrix-prediction filter
one could predict a group of series,
using the past of the whole group.
If the series are related,
the group prediction should be better than
self-prediction of individual channels.
For definiteness,
let us take two time series
*x*_{t} and *y*_{t} and suppose we are to find a vector filter
which converts them into a third series
*d*_{t}.
If *d*_{t} is *x*_{t+1},
this is a unit time-span prediction for filter for *x*_{t}.
If *d*_{t} is a vertical seismogram and
*x*_{t} and *y*_{t} are horizontals,
then the two-channel filter might be called an
extrapolation filter.
The set of equations which we wish to solve
by least squares takes the form

| |
(29) |

If this set of equations is abbreviated

| |
(30) |

then,
as we have seen in an earlier chapter,
the solution is of the form

| |
(31) |

We wish to inspect the matrix being inverted,
call it **R**. For a filter with three time lags we get

| |
(32) |

If we define

and likewise for *r*_{yx}(*i*) and *r*_{yy}(*i*)
the matrix (32) becomes

| |
(33) |

We may take the 6 x 6 matrix of (33)
and partition it into a 3 x 3 matrix of
2 x 2 submatrices. If we define the submatrix
blocks as

| |
(34) |

then (33) in terms of the blocks
defined in (34) is

| |
(35) |

The matrix in (35) is called
*block Toeplitz* or *multichannel Toeplitz*.
As with the ordinary Toeplitz matrix
there is a trick method of solution.
It will be taken up in the next section.

The reader should note that the matrix **R**
does not depend on the desired output **d**.
This results in a computational saving when there is more
than one possible output.
An example would be when it is desired to predict several
different series or distances into the future on a given series.

## EXERCISES:

- In the exercises of Chapter 2,
we determined
*B*(*Z*) and *A*(*Z*)
such that some given power series *C*(*Z*) was expressed as
*C*(*Z*) = *B*(*Z*)/*A*(*Z*).
Write normal equations (do not solve them)
for doing this in an approximate way by minimizing
where
subject to the constraint
*A*_{0} - 1.
(It can be proved that *A*(*Z*) comes out minimum-phase
by examining the Levinson recursion.)

** Next:** LEVINSON RECURSION
** Up:** Waveform applications of least
** Previous:** ADAPTIVE FILTERS
Stanford Exploration Project

10/30/1997