258 SAMPLED-DATA CONTROL SYSTEMS 
The discussion of sampled signals thus far has been concerned with the 
interpretation of sampling as a process of amplitude modulation of con- 
tinuous signals. For discrete signals which are number sequences only 
the situation is exactly the same as in the case of deterministic signals. 
The autocorrelation of a sequence of numbers r(n7) is defined as 
N 
o* (kT) —— 7 pe oN +1 > r(nT)r(nT = kT) (10.26) 
and the sampled power spectrum of the sequence is defined as the z trans- 
form of the autocorrelation, 
Sle) = ) Oh (eT)e* (10.27) 
k=—@ 
The factor 1/T in (10.26) is arbitrarily introduced to simplify later 
expressions. 
The mean-square value of a sequence of numbers or of a continuous 
signal x(t) may be obtained from the sampled power spectrum using the 
inversion formula derived in Chap. 4 and either (10.27) or (10.25), which 
relate the sampled autocorrelation function to the desired mean-square 
value. By (10.24) and the inversion formula (4.20), 
P22(0) = T7,(0) 
b [ Sz2(z)z dz (10.28) 
2m 
where the path of interration, I’, is the unit circle. The formula (10.28) is 
equivalent to the expression 
» fa/T 
®,.(0) = = ie m Sz.(jw) dw (10.29) 
I 
which is sometimes given for the mean-square value of a discrete variable. 
The equivalence of (10.29) to (10.28) may be shown by a change of vari- 
able 2 = eT in (10.28). 
Although the forms for the sampled power spectrum [given by (10.25) 
and (10.27)] find the greatest use in the analysis of sampled random 
signals it is also possible to view this spectrum as the summation of 
harmonics introduced by the sampling process. If the complex Fourier 
series representation of the modulated impulse train in (10.25) is used, one 
obtains 
oo 
St(3) = fa), Sule + jn) (10.30) 
n=-—2 
where S,,(s) is the spectral density of the continuous signal x(¢). 
