THE Z-TRANSFORM ANALYSIS OF LINEAR SAMPLED-DATA SYSTEMS 73 
In this context, G(z) is regarded as a weighting sequence rather than the 
z transform of the impulsive response of a linear system. The pulse trans- 
fer function G(z) is referred to as D(z) in this application to emphasize the 
fact that the latter is the result of numerical or arithmetic operations on 
numbers rather than the result of a linear filter. If D(z), as given in 
(4.55), is expanded into a power series in z—!, the summation given above 
will result. 
EXAMPLE 
To demonstrate how a pulse transfer function can be used to describe 
a numerical operation, the example of numerical integration will be 
used. If the integral y(t) of a function x(t) is desired, the following 
integral must be evaluated: 
y(t) = i x(t) dt 
This can be done numerically, using a number of possible numerical 
integration rules, such as Simpson’s 2 rule. Taking the simplest 
possible integration rule first, the following steps are taken. First, 
the integral will be evaluated only at discrete instants of time, a general 
one of whichisn7. Thus, 
y(nT) = ta 2) dt + Hees x(t) dt 
It is recognized that 
Ne = Iie = ‘agp x(t) dt 
so that y(n) = ya — 1)T + [ie oO) dt 
The integral in the above expression is the contribution to the total 
integral of the function x(t) over one quadrature interval. Various 
assumptions of increasing complexity can be made concerning the 
behavior of x(¢) within this interval. The simplest is that x(¢) remains 
constant at the value x(n — 1)T, just at the beginning of the interval 
in question. This is equivalent to the action of a clamp circuit in a 
physical system. With this assumption, the integral has the value 
nT 
[rey 2O H = Tan — YT 
Hence, the relationship between the y’s and x’s becomes 
y(nT) — y(n — 1)T = Tx(n — 1)T 
This is a recursion formula similar to (4.49). Carrying out the pro- 
cedure outline in this section or, more simply, ‘‘taking the z transform”’ 
of both sides, 
VY@Ql = a>) = We Dade 
