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Consider the example *B*(*Z*) = 1 - *Z*/2.
The inverse
| |
(41) |

can be found by a variety of familiar techniques, such as
(1) polynomial division,
(2) Taylor's power-series formula, or
(3) the binomial theorem.
In equation (41) we see that
there are an infinite number of filter coefficients,
but because they drop off rapidly,
approximation in a computer presents no difficulty.
We are not so lucky with the filter *B*(*Z*) = 1 - 2*Z*.
Here we have

| |
(42) |

The coefficients of this series increase without bound.
This is called ``**instability**.''
The outputs of the filter *A*(*Z*) depend infinitely
on inputs of the infinitely distant past.
(Recall that the present output of *A*(*Z*) is *a*_{0} times
the present input *x*_{1}, plus *a*_{1} times the previous input *x*_{t-1},
etc., so *a*_{n} represents memory of *n* time units earlier.)
This example shows that some filters *B*(*Z*)
will not have useful inverses *A*(*Z*)
determined by polynomial division.
Two sample plots of divergence are given in Figure 11.
**diverge
**

Figure 11
Top: the growing time function of a pole
inside the unit circle at zero frequency.
Bottom: at a nonzero frequency.
Where the time axis is truncated,
the signals are growing,
and they will increase indefinitely.

For the filter 1-*Z*/*Z*_{0} with a single zero,
the inverse filter has a single pole at the same location.
We have seen a stable inverse filter
when the pole |*Z*_{p}|>1 exceeds unity and
**instability** when the pole |*Z*_{p}|<1 is less than unity.
Occasionally we see **complex-valued signal**s.

Stability for wavelets with complex coefficients is as follows:
if the solution value *Z*_{0} of *B*(*Z*_{0}) = 0 lies inside the **unit circle**
in the complex plane,
then 1/*B*(*Z*) will have coefficients that blow up;
and if the root lies outside the unit circle,
then the inverse 1/*B*(*Z*) will be bounded.

** Next:** Anticausality
** Up:** Z-plane, causality, and feedback
** Previous:** Rational filters
Stanford Exploration Project

10/21/1998