5-2
Two time series Figure 1 x and _{1}x input to a
matrix of four filters illustrates the general linear model of
multichannel filtering.
_{2} |

The filtering operation in the figure can be expressed
as a matrix times vector operation,
where the elements of the matrix and vectors are *Z* transform polynomials.
That is,

(39) |

Now we can address ourselves to the inverse problem;
given a filter **B** and the outputs **Y** how
can we find the inputs **X**?
The solution is analogous to that of single time series.
Let us regard as a matrix of polynomials.
One knowns, for example, that the inverse of
any matrix

When one generalizes to many time series,
the numerator matrix is the so-called adjoint matrix
and the denominator is the determinant.
The adjoint matrix can be formed without the use of any division operations.
In other words,
elements in the adjoint matrix are in the form of sums of products.
For this reason,
we may say that the criterion for a minimum-phase matrix wavelet
is that the determinant of its *Z* transform has no zeros
inside the unit circle.

Equation (39) is a useful description of Figure 1 in most
applications.
However in some applications
(where the filter is an unknown to be determined),
a transposed form of (39) is more useful.
If *b _{12}* was interchanged with

(40) |

Now that we have generalized the concept of filtering
from scalar-valued time series to vector-valued series,
it is natural to generalize the idea of spectrum.
For vector-valued time functions,
the spectrum is a matrix called the *spectral matrix*
and it is given by

(41) | ||

(42) |

(43) |

Single-channel spectral factorization gives insight into numerous important problems in mathematical physics. We have seen that the concepts of filter and spectrum extend in quite a useful fashion to multichannel data. It was only natural that a great deal of effort should have gone to spectral factorization of multichannel data. This effort has been successful. However, in retrospect, from the point of view of computer modeling and interpretation of observed waves, it must be admitted that multichannel spectral factorization has not been especially useful. Nevertheless a brief summary of results will be given.

**The root method.** I extended the single-channel root
method to the multichannel case.^{}
The method is even more cumbersome in the multichannel case.
A most surprising thing about the
solution is that it includes a much broader result: that a polynomial with
matrix coefficients may be factored.
For example,

**The Toeplitz method.** The only really practical method for
finding an invertible matrix wavelet with a given spectrum is the multichannel
Toeplitz method.
The necessary algebra is developed in a later section on
multichannel time series prediction.

**The exp-log and Hilbert transform methods.** A number of famous
mathematicians including Norbert Wiener have worked on the problem from the
point of view of extending the exp-log or the Hilbert transform method.
The principal stumbling block is that exp(*A* + *B*) does not equal exp(*A*)
exp(*B*) unless *A* and *B* happen to commute,
that is, *AB* = *BA*.
This is usually not the case.
Although many difficult papers have appeared on the subject
(some stating that they solved the problem),
the author is unaware
of anyone who ever wrote a computer program which works at fast Fourier
transform speeds as does the single-channel Hilbert transform method.

- Think up a matrix filter where the two outputs
*y*(_{1}*t*) and*y*(_{2}*t*) are the same but for a scale factor. Clearly**X**cannot be recovered from**Y**. Show that the determinant of the filter vanishes. Find another example in which the determinant is zero at one frequency but nonzero elsewhere. Explain in the time domain in what sense the input cannot be recovered from the output. - Given a thermometer which measures temperature plus times
pressure and a pressure gage which measures pressure plus times the
time rate of change of the temperature,
find the matrix filter which converts
the observed series to temperature and pressure.
[HINT: Use either
the time derivative approximation 1 -
*Z*or 2 (1 -*Z*)/(1 +*Z*).] - Let
Identify coefficients of powers of
*Z*in ,to develop recursively the coefficients of . - Express the inverse of in a Taylor or Laurent series as is necessary.
- The determinant of a polynomial with matrix coefficients may be
independent of
*Z*. Applied to matrix filters, this may mean that an inverse filter may have only a finite number of powers in*Z*instead of the infinite series one always has with scalar filters. What is the most nontrivial example you can find?

10/30/1997