The *Z*-transform of an arbitrary, time-discretized function *x*_{t} is
defined by

(17) |

Going on to consider numerical values for the delay
operator *Z*, we discover that it is useful to ask whether *X*(*Z*) is
finite or infinite.
Numerical values of *Z* that are of
particular interest are *Z* = +1, *Z* = -1, and
all those complex values of *Z* which are unit magnitude,
say, |*Z*| = 1 or

(18) |

The most straightforward way to say that a filter is causal
is to say that its time domain coefficients vanish
before zero lag, that is *u*_{t} = 0 for *t*<0.
Another way to say it is to say that *U*(*Z*) is finite for *Z*=0.
At *Z*=0 the *Z*-transform would be infinite
if the coefficients etc. were not zero.
For a causal function,
each term in |*U*(*Z*)| will be smaller if *Z* is taken
inside the disk |*Z*|<1 rather than on it.
Thus convergence at *Z*=0 and on the circle |*Z*|=1 implies
convergence everywhere inside the unit disk.
So boundedness combined with causality means convergence in the unit disk.
Convergence at *Z* = 0 but not on the circle |*Z*| = 1 would
refer to a causal function with infinite energy,
a case of no practical interest.
What kind of function converges on the circle, at ,but not at *Z* = 0?
What function converges at all three places,
*Z* = 0, , and |*Z*| = 1?

The filter can be
expanded into powers of *Z* in (at least)
two different ways.
These are

(19) | ||

Let *b*_{t} denote a filter.
Then *a*_{t} is its inverse filter
if the convolution of *a*_{t} with *b*_{t} is a delta function.
In the Fourier domain, we would say that
filters are inverse to one another if their Fourier transforms
are inverse to one another.
*Z*-transforms can be used to define
the inverse filter, say, *A*(*Z*) = 1/*B*(*Z*).
Whether the filter *A*(*Z*) is causal depends on
whether it is finite everywhere inside the unit disk,
or really on whether *B*(*Z*) vanishes
*
anywhere
*
inside the disk.
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 |

10/31/1997