APPLICATION OF CONVENTIONAL TECHNIQUES 14] 
The transfer function for the feedback portion of the pulsed network 
can be shown from (6.33) to satisfy the relation 
z2—1 y Q(s) 1 
: = oe (6.37) 

Since the left-hand side of (6.37) is of exactly the same form as the right 
side of (6.36), it is immediately possible to say: 
1. The zeros of D;(z) are in the range 0 < z; < 1. 
2. The poles of D;(z) are arbitrary. 
From the two sets of conditions on the characteristics of D;(z) and 
D(z) given above it is possible to state rules for the realization of an 
arbitrary physically realizable transfer function in the form of a pulsed 
network as follows: 
Step 1. Factor the desired D(z) into D,(z)D;(z) so that all finite 
zeros outside the range 0 < z; < 1 are contained in D,(z) and all poles 
outside this range are in D,(z). 
Step 2. Assign poles and zeros of D(z) which fall inside the range 
0 < z, < 1 to either D,(z) or D;(z), respectively, so that D,(z) has no 
poles and D,(z) has no zeros for infinite values of z. This is necessary 
in order to prevent the resultant network from being physically 
unrealizable in that it would be required to predict future values of the 
sample sequence. If there are not enough such factors in the desired 
pulse transfer function D(z), add arbitrary poles in the range 0 < a; < 1 
to D,(z) and identical corresponding zeros to D;(z) until D,(«) and 
1/D;(%) are both less than infinity. 
Step 3. Determine P(s) from the inverse relation 
_, 2D,(2) 
z—1 
sZ |. 2 ; : — (6.39) 
(Ss) aes 
and find Q(s) from 

(6.38) 
Q(s) 

where Z~! signifies the Laplace transform corresponding to the z 
transform. 
The simplest way to perform the inversions indicated in (6.38) and (6.39) 
is to expand D,(z)/(z — 1) in partial fractions, multiply by z and look 
up, or write by inspection, the corresponding continuous transform. 
As an example of the synthesis of a pulse transfer function, first con- 
sider D(z) = z-!. This function has a pole at zero which, from step 1, 
must be contained in D;(z). Therefore, after the first step, the two fac- 
