76 SAMPLED-DATA CONTROL SYSTEMS 
of the filter is acceptably similar to the response to the actual pulse, the 
filter which implements the desired continuous transfer function G(s) is 
used. Obviously, if G(s) is physically realizable, then G(z) is too and vice 
versa. Less simple is the implementation of a digital pulse transfer func- 
tion D(z) in which no linear filter is used but rather a numerical operation. 
In the first place, D(z) must be physically realizable; that is, it cannot 
produce an output prior to the application of an input. This condition is 
met by allowing that only those forms of D(z) in which the denominator 
Cy (nT) 

Fia. 4.9. Implementation of digital system to obtain c;(n7’). 
has an equal or higher power in z than the numerator. Thus, upon inver- 
sion, the sequence in z~! would have no positive powers in z. Properly 
interpreted, this means that the output would occur at or after the instant 
of application of the first pulse. Since D(z) is usually expressed in terms 
of the ratio of a power series in z~! rather than z, the requirement for 
physical realizability is met with the following form: 
i) aU SP One ORS Se Uae 
1 + biz! + bee? + - > + + dye * 

D(@)= (4.57) 
where the important feature is that the denominator contain a term in 
z°. If this is the case, expansion of D(z) into a power series in z~! con- 
tains no term with 2! to any power higher than the zeroth. It is noted 
that the numerator of D(z) can contain z~! to any power and that often 
terms like do or a; are zero. 
In setting up a block diagram showing the implementation of D(z), an 
intermediate step is introduced by which D(z) is divided into two factors, 
D,(z) and D.(z), defined as follows: 
if 
Die) = 7 ie = be he sae 
and Do(2) = ao ase! > G22? GS a Ge” (4.59) 
The block diagram for D(z) has been developed by Barker! and is shown 
in Fig. 4.9. The z transform of the output sequence for this element. is 
Cx(z), and the nth pulse in the time domain is c,(nT). Tracing the signal 
