THE Z-TRANSFORM ANALYSIS OF LINEAR SAMPLED-DATA SYSTEMS 75 
applying the second pulse transform in this example, the initial value of 
the output, y(0), can be obtained by application of (4.27) and is found to 
T/2x(0). The value T/2x(0) is to be subtracted from y(nT) in order 
to obtain the correct result. The handling of nonzero initial values is 
treated in Sec. 11.5. As developed here, both 7(0) and y(O) or c(O) and 
r(0) have zero initial values. This is usually the case in the analysis of 
practical control systems containing digital elements. 
Thus, if the output y(n7) of a function which has been operated on by 
D(z) is to be obtained, the correct inversion of Y(z) is, for relaxed 
conditions, 
yur) = = Y (z)z"-1 dz — y(0) 
where y(0) is the initial value of the inversion sequence of Y(z). 
Another point is that in inverting the higher-order integration rules, it 
should be recalled that only values of n compatible with the basic time- 
domain inversion should be used. For instance, in the simplest integra- 
tion formulas, n can take on all integral values. On the other hand, in 
applying Simpson’s 2 rule, only every other integral value of n should be 
used; that is, n can take on only the values of 0, 2, 4, 6, 8, etc., since the 
process of integration progresses two intervals at a time. The values of 
y(nT) at n equal to 3, 5, 7, 9, etc., are not valid, as can be seen by inspec- 
tion of the time-domain recursion formula. Similarly, higher-order 
integration formulas are valid for every mth ordinate, where m is the order 
of the polynomial being fitted to the m most recent values of the time 
function x(t). 
The main reason for emphasizing the similarities between the gener- 
ating function and the z transform and between the digital pulse transfer 
function D(z) and the pulsed-filter pulse transfer function G(z) is that 
practical sampled-data control systems contain both types of element in 
the loop. A digital controller is a small-scale digital computer whose 
output is applied to a pulsed linear plant. It is extremely convenient to 
be able to employ the same operational methods and describe both types 
of element by a unified operational approach, broadly described as the z 
transform. ‘The inversion theorems, the initial- and final-value theorems, 
and other manipulative rules to be developed apply equally to digital and 
pulsed-filter applications. 
4.7 Implementation of Pulse Transfer Functions 
If the pulse transfer function G(z) is one which results from the applica- 
tion of a pulse sequence to a linear filter and the sampling of its output in 
synchronism with the input, there is no problem of implementation. If 
the impulse approximation is acceptable, that is, if the impulsive response 
