** Next:** LEAKY INTEGRATION
** Up:** Table of Contents

# Z-plane, causality, and feedback

All physical systems share the property that they do not respond
before they are excited.
Thus the impulse response of any physical system
is a one-sided time function (it vanishes before *t* = 0).
In system theory such a filter function is called ``**realizable**"
or ``**causal**.''
In wave propagation this property is associated with *causality*
in that no wave may begin to arrive before it is transmitted.
The lag-time point *t* = 0 plays a peculiar and an important role.
Thus many subtle matters can be more clearly understood
with sampled time than with continuous time.
When a filter responds at and after lag time *t* = 0,
we say the filter is realizable or causal.
The word
``causal" is appropriate in physics, where stress causes
instantaneous strain and vice versa,
but one should return to the less pretentious
words ``realizable" or ``one-sided"
when using filter theory to describe economic or social systems
where simultaneity is different from cause and effect.
The other new concept in this chapter is ``**feedback**."
Ordinarily a filter produces its output using only past *inputs*.
A filter using feedback uses also its past *outputs*.
After digesting the feedback concept, we will look at a wide variety
of filter types, at what they are used for, and at how to implement them.

First a short review:
the *Z*-transform of an arbitrary,
time-discretized signal *x*_{t} is defined by

| |
(1) |

In chapter
we saw that (1)
can be understood as a Fourier sum (where ).
It is not necessary for *Z* to take on numerical values,
however, in order
for the ideas of convolution and correlation to be useful.
In chapter
we defined *Z* to be the unit delay operator.
Defined thus, *Z*^{2} delays two time units.
Expressions like and are
useful because they imply convolution and crosscorrelation
of the time-domain coefficients.
Here we will be learning how to interpret 1/*A*(*Z*) as a feedback filter,
i.e., as a filter that processes not only past inputs, but past outputs.
We will see that this approach brings with it
interesting opportunities
as well as subtle pitfalls.

** Next:** LEAKY INTEGRATION
** Up:** Table of Contents
Stanford Exploration Project

10/21/1998