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  

$$X(Z) \eq \cdots\ + x_{-2} \,Z^{-2} \ +\ x_{-1} \,Z^{-1} \ +\ x_0 \ +\ x_1 \,Z \ +\ x_2 \,Z^2 \ +\ \cdots$$ (1)

In chapter we saw that (1) can be understood as a Fourier sum (where $Z=e^{i\omega}$). 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 $X(Z)\,B(Z)$ and $X(Z)\,\bar B(1/Z)$ 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.