274 SAMPLED-DATA CONTROL SYSTEMS 
simple, leads to the result that 

Teale Mabe hese ecu. (10.92a) 
= ele — mtn] G+ 1) StS a 
(10.920) 
Se ee i on —ax<t<o (10.92c) 
A sketch of y(t) is shown in Fig. 10.9 with no designation of the time 
origin since this depends on the particular value of a. From (10.81) it 
is noted that the optimum transfer function contains the unilateral 
Laplace transform of y¥/(¢) so that only the portion of the function over 
positive time is of interest. 
For positive a, the ideal operation is one of prediction and, in this 
case, the entire positive time axis is contained in (10.92c). Substitu- 
tion of this expression in (10.81) gives the optimum transfer function 
1 — eof e—sT 
HENS) = = enti e7eie(1 im e— 2:7) vf evite—st dt 
1s — eel e—sT 
It is interesting to compare the optimum filter whose transfer function 
is given by (10.93) with the transfer function of the optimum filter for 
the prediction (without sampling) of the signal whose spectrum is given 
by (10.86). In the absence of sampling, the Wiener theory gives 
H,,.(s) = ea (10.94) 
At sampling instants, the operation of the filter whose transfer function 
is given by (10.93) is expressed by the z transform of this transfer func- 
tion; that is, by 
ACG) enc (10.95) 
Comparison of (10.95) with (10.94) indicates at once that the outputs 
of the two filters at sampling instants are identical. Between sampling 
instants, the output of the filter whose input is sampled is an exponen- 
tial decay of time constant 1/w1. 
The mean-square error of prediction of the sampled-data filter 
calculated from (10.92c) and (10.85) is 
(e?(t)) je al e214 (1 — e217) e—2e1t dt 
0 
= e7 wT 
201T 
it! nae e7 2414 
Darrel (2w1T)? 
Dine wtot 


1 — e212 + cme spo | (10.96) 
