SAMPLED DATA SYSTEMS WITH RANDOM INPUTS 265 
If (10.50) is substituted in (10.51), 
Diy) = e Y keatyxr = nt) Y AGT)xOr + eT — iT)» 
n=0 j=0 
» h(nT) ) AGT) (aT — nT)x (IT + kT —jT)) (10.52) 
j=0 
I 
n=0 
oo 
I 
» h(nT) ) h(jT)®.AnT + kT — jT) 
n=0 j=0 
where, in the last step, the ergodic nature of the processes is used along 
with (10.14). 
By the definition (10.27), the sampled power spectrum of y*() is the 
2 transform of &*,(kT) = Fb (kT). ane rerare® 
S,,(2) = LS y h(nT?) E h(jT)®2(nT + kT — jT) 
— (10.53) 
= y h(nT) y h(jT) y OF (nT + kT — jT)z* 
n=0 k=—o 
= J(@\VEle 1)8..(2) 
which is the desired result. As before, the sampled power spectrum, 
S,,(z), is the power spectrum of the impulse-modulated y(t). The mean- 
square value of y(t) may be obtained from S,,(z) by the use of either 
(10.28) or (10.29). 
The cross-correlation between a sampled and a continuous signal may 
also be computed directly. Let x*(t) = > x(t)d(t — kT) represent a 
k=—o0 
sampled signal and y(t) represent the nonsampled signal. Then 
To = 
@,+,(7) = Ha all a(t)y(t + 7) » 5(t — kT) di (10.54) 
To 
k=—-—0 
The right side of (10.54) contains an integral over the limits from — 7") to 
T of a train of uniformly spaced impulses extending over all time. Those 
impulses falling outside the limits of integration cannot contribute to the 
value of the integral, and one can terminate the summations on k at 
N = +(T /T), the largest integer contained in 7/7. If the summation 
is terminated at these limits, then the integration limits may be extended 
