Figure 1 A continuous time function sampled at uniform time intervals.
Consider the time function in Figure 1. To analyze such an observed time function in a computer it is necessary to approximate it in some way by a list of numbers. The usual way to do this is to evaluate or observe b(t) at a uniform spacing of points in time. For this example, such a digital approximation to the continuous function could be denoted by the vector
Figure 2 Coefficients of ZB(Z) are shifted version of the coefficients of B(Z)
We see that the time function bt is delayed n time units when B(Z) is multiplied by Zn. The delay operator Z is very important in analyzing waves simply because waves take a certain amount of time to get from place to place.
Another value of the delay operator is that it may be used to build up more complicated time functions from simpler ones. Suppose b(t) represents the acoustic pressure function or the seismogram observed after a distant explosion. Then b(t) is called the impulse response. If another explosion occurs at t = 10 time units after the first, we expect the pressure function y(t) depicted in Figure 3.
Figure 3 Response to two explosions.
In terms of Z transforms this would be expressed as Y(Z) = B(Z) + Z10 B(Z). If the first explosion were followed by an implosion of half strength, we would have .If pulses overlap one another in time [as would be the case if B(Z) was of degree greater than 10], the waveforms would just add together in the region of overlap. The supposition that they just add together without any interaction is called the linearity assumption. This linearity assumption is very often true in practical cases. In seismology we find that--although the earth is a very heterogeneous conglomerations of rocks of different shapes and types--when seismic waves (of usual amplitude) travel through the earth, they do not interfere with one another. They satisfy linear superposition. The plague of nonlinearity arises from large amplitude disturbances. Nonlinearity does not arise from geometrical complications.
Now suppose there was an explosion at t = 0, a half-strength implosion at t = 1, and another, quarter-strength explosion at t = 3. This sequence of events determines a ``source'' time series, .The Z transform of the source is .The observed yt for this sequence of explosions and implosions through the seismometer has a Z transform Y(Z) given by
There are many examples of linear systems. A wide class of electronic circuits is comprised of linear systems. Complicated linear systems are formed by taking the output of one system and plugging it into the input of another. Suppose we have two linear systems characterized by B(Z) and C(Z), respectively. Then the question arises whether the two combined systems of Figure 4 are equivalent. The use of Z transforms makes it obvious that these two systems are equivalent since products of polynomials commute, i.e.,
Consider a system with an impulse response B(Z) = 2 - Z - Z2. This polynomial can be factored into ,and so we have the three equivalent systems in Figure 5. Since any polynomial can be factored, any impulse response can be simulated by a cascade of two-term filters (impulse responses whose Z transforms are linear in Z).
What do we actually do in a computer when we multiply two Z transforms together? The filter 2 + Z would be represented in a computer by the storage in memory of the coefficients (2, 1). Likewise, for 1 - Z the numbers (1, -1) are stored. The polynomial multiplication program should take these inputs and produce the sequence (2, -1, -1). Let us see how the computation proceeds in a general case, say
Identifying coefficients of successive powers of Z, we get
DIMENSION X(LX), B(LB), Y(LY) DO 10 IY=1,LY 10 Y(IY) = 0. DO 20 IX=1,NX DO 20 IB=1,NB IY = IX+IB-1 20 Y(IY) = Y(IY) + X(IX)*B(IB)
The reader should notice that X(Z) and Y(Z) need not strictly be polynomials; they may contain both positive and negative powers of Z; that is