CHAPTER 10 
SAMPLED DATA SYSTEMS WITH RANDOM INPUTS 
In the previous chapters all the input signals have been considered as 
deterministic, or known, functions of the time. In many cases of prac- 
tical interest, however, this is not the case, and some or all of the signals 
must be considered to be random functions of time. That is, one is not 
able to make a statement such as ‘‘r(¢) is unity for all positive time and 
zero for all negative time,” but instead must be content with a descrip- 
tion in probability such as ‘‘r(¢) is between 0.9 and 1.1 in 80 per cent of 
the signals in this class at this time.” 
Several rather broad treatments of random signals in control systems 
are readily available (see, for example, ref. 34), and the present chapter 
contains only a brief review of certain aspects of the theory and an appli- 
cation of these techniques to sampled-data systems. As in the previous 
chapters, the point of view here is influenced greatly by the simultaneous 
presence of discrete and continuous signals which characterizes sampled- 
data systems. The attention of this chapter is accordingly focused on a 
combination of the analysis of strictly discrete systems and the analysis 
of strictly continuous systems and signals. 
10.1 Review of the Analysis of Random Signals 
One can imagine many possible sources or generators of random signals, 
as, for example, a vacuum tube whose plate current is subject to fluctua- 
tions because of the random emission of electrons from the cathode (a 
phenomenon known as shot noise) or a conductor which contains ther- 
mally excited electrons which give rise to thermal noise. A random input 
to a control system might arise as the command signal from a gyro 
mounted in an aircraft or ship which is being rocked by wind or waves. 
All of these signals are examples of random variables which must be 
described in probabilistic terms. 
The mathematical description of a random process depends upon the 
concepts of a probability-distribution function and probability-density 
functions of various orders. The first-order probability-density function 
associated with a random signal X(t) is designated fi(z,t). This function 
is defined so that the product f1(z,t) dx is the probability that the variable 
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