276 SAMPLED-DATA CONTROL SYSTEMS 
this example of course, and for another problem the error will normally 
decrease as the delay is increased. 
The minimum mean-square error for all delay less than —T is given 
by the substitution of (10.92) in (10.85) 
—a 
et1 
(e?(~) min = 1 mom ‘ G — ery etal) V7? [ew? oe FCB sd tent) dt 
—a—T rs 
aes ih [1 a er) W/2g— sine eine dt (10.98) 
After some simplification, (10.98) reduces to 
; seo 1 
2 =S—- —_-—« 
(e?(t))min ncaa (10.99) 
The mean-square error given by (10.99) is independent of the actual 
value of delay a, as was expected, and represents the absolute minimum 
error for this particular example. That is, no value of a outside the 


Mean square error <«* > 
oO 
[o>) 
4 — Optimum filter 
--- Clamp 
0 ana | 
0.1 1 10 100 
Normalized sampling period, wT 

Fic. 10.10. Plot of normalized mean-square error for optimum filter and for clamp for 
illustrative example. 
range a < —T will result in an error as small as that given by (10.99). 
A plot of this error as a function of w;7' is given in Fig. 10.10, which 
could be used to select the sampling period T for the reproduction after 
sampling, with specified error, and at least 7’ sec delay of some process 
whose spectrum is given by (10.86). For values of w:7' outside the 
range of the plot in Fig. 10.10, the asymptotic behavior of the error is 
