SAMPLED-DATA SYSTEMS 91 
is generated. The error is in the form of a sequence of error pulses whose 
z transform is H(z), as seen in the figure. There are two continuous ele- 
ments whose transfer functions are G(s) and H(s), representing the feed- 
forward and feedback systems, respectively. In the system illustrated, 
the input and output functions are continuous, and their respective 
Laplace transforms are R(s) and C(s), respectively. The dashed samplers 
shown in Fig. 5.4 are fictitious and represent a process of mathematical 
sampling only. This means that the continuous functions represented 
by R(s), C(s), and B(s) are examined at sampling instants only, and the 
pulse sequences so obtained have z transforms given by R(z), C(z), and 
B(z). The reason for creating these fictitious pulse sequences is that the 
z transformation relates such pulse sequences in terms of a pulse transfer 
function. 
From Fig. 5.4, it is seen that the error pulse sequence F(z) is given by 
E(z) = R(z) — Bz) (5.18) 
Also, the relation between the z transform B(z) and E(z) is given by 
Bz) = GH (z) E(e) (5.19) 
where it is recalled that GH(z) represents the z transform corresponding 
to the Laplace transform G(s)H(s). Substituting (5.19) back into (5.18), 
there results 
E(z) = R) — GH(z)E(z) (5.20) 
Solving (5.20) for H(z), the error-sequence z transform becomes 
ie R(z) 
This error-sequence transform is useful in determining the performance 
of the system; however, the over-all response is of even more interest. 
From the figure it is seen that the output-sequence z transform C(z) is 
related to the error-sequence z transform E(z) by the feedforward pulse 
transfer function G(z), 
Ce) SG@Ee) (5.22) 
Substituting (5.21) in (5.22), there results the expression 
ns) 
This is the relation between the input and output z transforms of the 
pulse sequences. The expression is relatively simple and straightforward 
only because it is restricted to relating the input and output values at 
sampling instants and not continuously. This relation is listed in the 
table*® in Appendix III, along with the resulting relationships for other 
feedback configurations. Note is made that (5.23) contains the z trans- 
form GH (z) corresponding to the loop transfer function G(s) H(s). 
