# 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 xt 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, Z2 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.