MULTIRATE SAMPLED SYSTEMS 231 
14 transforms which are included in (9.24) are simple examples of the 
““modified”’ transform discussed in the last chapter in connection with 
the intersample ripple problem. The solution of (9.24) for one of the 
unknowns—L,(z), for example—is largely a matter of algebra. For 

Fig. 9.8. Block diagram for example of closed-loop multirate system analysis. 
example, substitution of the known functions for 7’ = 1 sec into (9.24) 
and elimination of Cy, Co, and C; leads to the two equations 
1 -- e—l/3,-1 + e—2/8g-1 



F(z) a R(z) a Ey(z) ( am zy(1 ar e-lz-1) 
eo l/2g—1 a e7Vby—-1 + e—9/6y—2 
= Ble) hs BNC = ate) 
1 —1/3 —2/384—1 
E.(z) = Z[R(s)es?/2] — E,(z) a a ay To =e 
—T/2y—-1 + e— /8g—-1 “+ e—/6g—1 
(Gl = ey = E-s) 
Finally, if R(s) = 1/s, the response of the error is given by the solution 
of (9.25) to be 
E\(z2) = 
= JI) = (9.25) 

i dt (G48 = gy 
2 (FT =e eV 26 Fe VO TL G88) 2-1 4 e-iz-2 
(9.26) 

JEG) = Ra (=) = I SAO HB ele oye Le 

