SAMPLED DATA SYSTEMS WITH RANDOM INPUTS Qa 
ters. It is also possible to realize a transfer function of the type required 
here with a tapped delay line, under certain circumstances. 
EXAMPLE 
A block diagram of an illustrative problem is shown in Fig. 10.8. 
The objective of the problem is to design a linear filter which will pre- 
mit) 

Fia. 10.8. Error formation of example problem. 
dict or delay in the absence of noise a message whose spectrum is given 
by 
iG) = a 
w1 
5 (10.86) 
From the results of Sec. 10.4 the sampled power spectrum of this signal is 
1 es e7 2Te: 
(1 — e-T12-1)(1 — e712) 
Srr(z) = (10.87) 
The factors of this sampled power spectrum having zeros and poles 
inside and outside the unit circle respectively, are 
(1 — eaTy1/2 
1 — e-f*iz71 
(1 — e-2aT) V2 
1 — e feizg 
[S,-(z)]* 
I 
(10.88) 
For prediction or delay the ideal transfer function is e%*, which, when 
substituted in (10.83) with (10.86) gives the cross-correlation between 
the input and the desired output as 
20 1e% 
5.6) = : (10.90) 
@1 — § 
where positive values of a correspond to prediction and negative values 
of a correspond to delay. The substitution of (10.90) and (10.89) in 
(10.81) gives 
ae ( — ea?) V2 o 204 an TrXp—Tw ) pdt 
v(t) a Oni De ean? (1 oh Cams C; Je dn (10.91) 
The integral (10.91) can be evaluated over three ranges of time, depend- 
ing on the values of a and T. The evaluation, which is relatively 
