MULTIRATE SAMPLED SYSTEMS 233 
of Fig. 9.9 consists of the specification of a desired over-all transfer func- 
tion K(zn) such that the response will have suitable transient and steady- 
state behavior and also such that the necessary controller pulse transfer 
function will be physically realizable. The first step in the design 

Fic. 9.9. Block diagram of system with multirate controller. 
requires the relation between the controller pulse transfer function, 
D(z,), and the system transfer function, K(z,). For the variables as 
defined in Fig. 9.9 one can readily write 
Ex) = Re) — Ce) (9.27) 
and, using the results expressed in (9.7), 
Ge) = Pie ADE NEG) (9.28) 
In order to eliminate H, from (9.27) and (9.28), it is necessary to take the 
z transform of (9.28), as defined in (9.19). 
C(z) 
I|> 
ZC (2n)] 
= Z[Ey(2n") D(en)G@(2n)] (9.29) 
Ey(z)Z[D (2n)G@(Zn)] 
I 
where C(z) is the single-rate z transform corresponding to the multirate 
transform C(z,). 
The last step leading to (9.29) makes use of the fact that the z transform 
of a product which includes a factor depending on z alone is the product 
of z transforms. In the case of (9.29), E(z2n”) is the factor which depends 
on 2 alone and is therefore factored out. If (9.29) is substituted in (9.27), 
one can solve for £,(z) as 
R(z) 
DO) eEriD eee) 
the expression (9.30) may now be substituted into (9.28) to relate the out- 
put to the input 
(9.30) 
R(2n”) D(2n)G (en) 
Cen) = 1 + Z[D@n)GEn)] 
(9.31) 
