270 SAMPLED-DATA CONTROL SYSTEMS 
spectrum which is rational in the frequency s (or w). If the optimum 
transfer function between r*(t) and c(t) is H(s), then the optimum trans- 
fer function between p*(t) and c(t) must be 
H(s) 
D*(s) 
If (10.68) represents a realizable transfer function, the output, c(é), is 
the same for the two cases. That is, if G(s) in Fig. 10.7 satisfies (10.68), 
then the output c(t) shown in Fig. 10.7 would be the same as the output 
c(t) shown in Fig. 10.6 in the presence of identical inputs. This transfer 
function G(s) is then optimum in the same sense that H(s) is optimum. 
Conversely, if G(s) is realized as the optimum filter with the input 
p*(t), then the optimum filter with input r*(¢) has the transfer function. 
H(s) = D*(s)G(s) (10.69) 
if this is realizable. If both (10.68) and (10.69) are to be the transfer 
functions of realizable filters, then D*(s) must have neither poles nor 
zeros in the right half plane. Equivalently, D(z) must have neither poles 
nor zeros outside the unit circle. The conditions that D(z) have neither 
poles nor zeros outside the unit circle and that the sampled power spec- 
trum of p*(t) be constant are sufficient to determine D(z). 
From (10.53) the sampled power spectrum of p*(é) is 

G(s) = (10.68) 
S222) = DE) DE)s,,©@) (10.70) 
If S,,(z) is to be unity then 
D(2) D(z) = sO (10.71) 
From (10.71) and the conditions that D(z) have neither poles nor zeros 
outside the unit circle, it is clear that 
DE) (10.72) 
whe 
[S--(2)]* 
where the superscript + indicates that factor of S,,(z) whose poles and 
zeros lie inside the unit circle. The optimum form for the transfer func- 
tion between p*(t) and c(t) will, by (10.69), complete the design. 
From (10.66), if the mean-square value of p(t) is unity, the optimum 
impulse response of the filter from p*(t) to c(t) is 
The cross-correlation ®,.,(t) may be determined by the average 
N 
ik 
=) im Gees 10.74 
Spall) = lim pe) PieMe(eP +) (0.78 
n=—N 
