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Let *b*_{t} denote a filter.
Then *a*_{t} is its ``**inverse filter**''
if the convolution of *a*_{t} with *b*_{t} is an impulse function.
Filters are said to be inverse to one another if their Fourier transforms
are inverse to one another.
So in terms of *Z*-transforms,
the filter *A*(*Z*) is said to be inverse to the signal of *B*(*Z*)
if *A*(*Z*)*B*(*Z*)=1.
What we have seen so far is that the inverse filter can be stable
or unstable depending on the location of its poles.
Likewise, if *B*(*Z*) is a filter, then *A*(*Z*) is a usable filter
inverse to *B*(*Z*), if *A*(*Z*)*B*(*Z*)=1 and if *A*(*Z*) does not have coefficients
that tend to infinity.
Another approach to inverse filters lies in the Fourier domain.
There a filter inverse to *b*_{t} is the *a*_{t} made by taking the
inverse Fourier
transform of .If *B*(*Z*) has its zeros outside the unit circle,
then *a*_{t} will be causal; otherwise not.
In the Fourier domain the only danger is dividing by a zero,
which would be a pole on the unit circle.
In the case of *Z*-transforms,
zeros should not only be off the circle
but also outside it.
So the -domain seems safer than the *Z*-domain.
Why not always use the Fourier domain?
The reasons we do not always inverse filter in the
-domain, along with many illustrations,
are given in chapter .

** Next:** The unit circle
** Up:** INSTABILITY
** Previous:** Anticausality
Stanford Exploration Project

10/21/1998