Luckily, unstable filters can be made stable as follows:

(43) |

Examining equations
(42) and
(43),
we see that the filter 1/(1-2*Z*) can be
expanded into powers of *Z* in (at least) two different ways.
Which one is correct?
The theory of complex variables shows that,
given a particular numerical value of *Z*,
only one of the sums
(42) or
(43)
will be finite.
We must use the finite one, and since
we are interested in Fourier series,
we want the numerical value |*Z*|=1
for which the first series diverges and the second converges.
Thus the only acceptable filter is *anticausal.*

The spectra plotted in Figure 11 apply to the anticausal expansion. Obviously the causal expansion, which is unbounded, has an infinite spectrum.

We saw
that a polynomial *B*(*Z*) of degree *N* may be factored
into *N* subsystems, and that the ordering of subsystems is unimportant.
Suppose we have factored *B*(*Z*) and found that some of its roots lie outside
the unit circle and some lie inside.
We first invert the outside roots with equation (41)
and then invert the inside roots with equation (43).
If there are any roots exactly on the unit circle, then we have
a special case in which we can try either inverse,
but neither may give a satisfactory result in practice.
Implied zero division is nature's way of telling us that
what we are trying to do cannot be done that way (if at all).

10/21/1998