SAMPLED DATA SYSTEMS WITH RANDOM INPUTS 277 
given by 
(e2(t))min © ke oT K1 (10.100) 
(e7(t) min © asia! GALS 1 (10.101) 
wT" 
1 
It is interesting to work out a numerical example using (10.100). Sup- 
pose one is required to reproduce from samples a signal whose spectrum 
is given by (10.86) with a bandwidth w;/2m of 1 eps to an accuracy of 
1 per cent, that is, it is necessary to maintain the rms error of the repro- 
duction less than or equal to 10-?.. The mean-square error must then 
be equal to or less than 10-4. Substitution of these values in (10.100) 
gives 
2rT 
ay pee ae 
10 ; 
— 4 
iP = ee = DOG S< 10 gee 
1/T = f. ~ 20,000 cps 
In words, to be able to reproduce a signal described by the spectrum of 
(10.86) with an rms error of 1 per zent, it is necessary to sample approxi- 
mately 20,000 times the bandwidth frequency of the signal! Later 
calculation will show how a simple clamp compares with the optimum 
filter on a mean-square-error basis. 
The simplest filter which will realize the absolute minimum error 
given by (10.99) is that designed for a delay of one sampling period. 
The optimum transfer function could be obtained for arbitrary delays, 
of course, but the case of an ideal delay of exactly one sampling period 
illustrates clearly all the important details with a minimum of unneces- 
sary mathematics. Fora = —T, (10.92) reduces to 
y(t) = 0 = 28 SiS 
= (a — oa (eet — ee) Davern 
= (Ul = eae) ug om ri acon en (iO 102) 
substitution of (10.102) and (10.88) in (10.81) leads to the optimum 
transfer function 
2w eit Gl poe Cuomo) alt eis Gmc) 
Ele ®) = ae raya (10.103) 
The z transform of (10.103) reveals 
Te) em ; (10.104) 
which indicates that this filter delays each sample by one sampling 
period, as indicated by the design requirement that the output equal 
