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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.
In terms of *Z*-transforms,
an inverse is simply defined by *A*(*Z*) = 1/*B*(*Z*).
Whether the filter *A*(*Z*) is causal
depends on whether it is finite everywhere inside the unit circle,
or really on whether *B*(*Z*) vanishes
*
anywhere
*
inside the circle.
For example, *B*(*Z*)=1-2*Z* vanishes at *Z* = 1/2.
There *A*(*Z*)=1/*B*(*Z*) must be infinite,
that is to say,
the series *A*(*Z*) must be nonconvergent at *Z* = 1/2.
Thus, as we have just seen, *a*_{t} is noncausal.
A most interesting case, called
``**minimum phase**,"
occurs when both a filter *B*(*Z*) and its inverse are causal.
In summary,
`
causal: for causal inverse: for
minimum phase: both above conditions
`

The reason the interesting words ``minimum phase'' are used
is given in chapter .

** Next:** Mechanical interpretation
** Up:** Z-plane, causality, and feedback
** Previous:** Causality and the unit
Stanford Exploration Project

10/21/1998