266 SAMPLED-DATA CONTROL SYSTEMS 
to infinity in both directions without affecting the results. It is also 
noticed that the interval length 7p is within one period of (2N + 1)T. 
In view of the above remarks, 
N 
®,+,(7) = ae ON EDF De i x(t)y(t ala T)d(t +7 kT) dt 
N 
=f tim > a(kT)y(kT +1) (10.55) 
For the special case where y(¢) is the output of a linear filter whose input is 
x*(t) and whose impulse response is A(t), the relation 
i} 
y) = ) kG = nTe(nT) (10.56) 
applies, and therefore : 
N © 
itt 
®,+,(7T) = 7 lim aN 1 > x(kT) », h(r + kT — nT)x(nT) 
k=-—N n=—o 
If a change of index is made on the second summation from n tol + k 
then 
a N 
6) =7 > he = 1) lim se i : y (kT) x(kT + IT) 
[=o ASN 
Wwe 
s 7d h(t — IT)@,,(1T) 
z h(t — IT) @*,(IT) (10.57) 
l=—oa 
the transform of (10.57) yields 
S.+,(s) = Sz,(s)H(s) (10.58) 
which states simply that the cross power spectrum between the sampled 
input to a linear filter and the output of the filter depends only on the 
sampled power spectrum of the input and the continuous transfer func- 
tion of the filter. 
In a manner similar to that used to find (10.58) one can obtain the 
other spectra relating the variables which are defined in Fig. 10.4. A 
tabulation of these spectra is given in (10.59). The variable z is used in 
the second and the last of the expressions in (10.59) because all of the 
functions involved in these particular expressions are sampled functions. 
