SAMPLED DATA SYSTEMS WITH RANDOM INPUTS 261 
The usual substitution 2 = e*? is made in the last step of the reduction 
of (10.38). 
10.4 Random Signals in Sampled Systems 
The principal use of the foregoing analysis is in the study of signals in 
linear sampled-data systems. For this purpose, consider the system 
represented by the block diagram 
of Fig. 10.4 and the problem of caleu-  *!4) 7 x*(t)! His) | yl) y*(t) 
lating the autocorrelation function, - 
cross-correlation functions, and the fFy¢. 10.4. Open-loop sampled-data sys- 
power spectra defined for the various tem with input and output variables 
continuous and sampled signals ap- ‘efned. 
pearing in this elementary system on the assumption that the autocorrela- 
tion function of the input x(¢) and the system function of the network are 
given. It follows immediately from the analysis of Sec. 10.3 and conven- 
tional analysis expressed by (10.19) that the power spectrum of the out- 
put is given by 
Sy,(s) = Sz.(s)H(s)H(—s) (10.39) 
or, in terms of the real frequency, 
Syy(jo) = Sz.(jo)|H (je)? (10.40) 
if one assumes that the sampled signal x*(¢) may be sufficiently weli 
approximated by the periodic impulse modulation of x(t). If pulse 
modulation of finite width y is a better approximation to the sampled 
signal, this effect can be generated by passing the (nonexistent) impulse- 
modulated signal through a filter whose response is a rectangular pulse of 
the proper width. In this case the output of this particular system has 
an autocorrelation function of the type sketched in Fig. 10.3 and a 
spectrum 
Gu | = Gu 
Sa(s) = St) —— (10.41) 
aS: 

A more common situation is the use of a clamp, or zero-order hold, for the 
system or filter, in which case the spectral density of the output is 
iL = ans Lael —— al! 
SO) = S20) === = (10.42) 
Before going further into the analysis of the signals in Fig. 10.4 in 
general terms, it is instructive to observe some of the characteristics of a 
particular random signal and to consider the effects of passing this signal 
through a particular system. For example, consider the signal x(t) with 
