SAMPLED DATA SYSTEMS WITH RANDOM INPUTS 271 
and, by convolution, the output of the shaping filter at sampling instants 
is given by 
i-} 
p(nT) = » d(kT)r(nT — kT) (10.75) 
k=0 
Substitution of (10.75) in (10.74) yields, after some rearrangement, 
N e) 
®,,,() = lim al » » d(kT)r(nT — kT)c,(nT + 1) (10.76) 
1 
N- «© 
n=—N k=0 
The system function of the optimum filter is implicit in (10.69), (10.72), 
(10.73), and (10.76). This transfer function H(s) may be made explicit 
by expressing all these equations in their equivalent frequency-domain 
forms. 
The cross power spectrum of the signal p(t) and the desired output 
ca(t) is the transform of (10.76): 
Spea(s) = [", Breelterme at 
= [7 Y ahyg,.(kT 44+ at (10.77) 
k=0 
Integrating term by term and substituting x = t + kT, 
Syea(8) = » d(kT)ek?s sla &,.,(x)e-* dx 
k=0 
= DGS) Src (S) 
ea Srea(S) 
USF (8) J- 
where, in the last step, use has been made of (10.71) and (10.72) and the 
symmetry of the sampled power spectrum about the imaginary axis of 
the s plane. 
From (10.78) and (10.73), the optimum transfer function between 
»*(t) and c(¢) in Fig. 10.7 is 
(10.78) 
) oi) 
G(s) = i dt os | _ D*(=2)Srea(d)O™ OD (10.79) 
a 
* i >? SL Oee 
= —st 
| dt € Sn} ie S=A)E dy (10.80) 
The superscript — in (10.78) and (10.80) indicates the factor of the input 
sampled power spectrum whose poles and zeros lie in the right half plane. 

