SAMPLED DATA SYSTEMS WITH RANDOM INPUTS 279 
(r2{t — (T'/2)]), is the same as the mean-square value of r(¢). Also, from 
the results of Sec. 10.4, it is clear that the mean-square value of the out- 

Fig. 10.12. Error formulation for evaluation of clamp as a filter. 
put of a zero-order hold equals the mean-square value of the unsampled 
input as shown in (10.47). Therefore, 
pea) oO) aes (0) (10.106) 
If the input is assumed to have the spectrum of (10.86) then ©,,(0) equals 
unity. Also 
®..,(0) = (e(t)ca(t)) 
= (c()r[é — (1'/2)]) 


= ©,,(T/2) (10.107) 
From (10.59) 
ON, 1 ae 1 2041 1 — es 
Se) a= 7 S,(s)H(s) = pn Bea ge FT gr (10.108) 
and, by definition, 
1 je F 
— 8T/2 
(YD) om (ee S,-(s)e*?/? ds 
atl SO | Bey sT/2 
nj jae RE Gye = SF Cae 
1 gee (8/1) (@1T/2) _ g—(8/;) (w,T/2) 2 ds 
= = ss ———., — (10.109 
21) —jeo w1T (s/w1) 1 — (s/w)? W1 ( ) 
With the substitutions s/w; = \ and w,7' = a, (10.109) becomes 
Ik J° gad/2 _ p-ad/2 9 
EAE) = 5 ie Tae ea 
1 — ew? 
= 2 z (10.110) 
The mean-square value of the error is found by the substitution of 
(10.106) and (10.110) in (10.105): 
1 me e7eir/2 
(2(t)) = 2-4 Sar Nas (10.111) 
The expression (10.111) is plotted in Fig. 10.10 for comparison with 
(10.99). For small values of w:7, (10.111) is approximately 
(e(t)) = Be (10.112) 
