252 SAMPLED-DATA CONTROL SYSTEMS 
Where the analysis of two related random processes is involved, as, for 
example, in the consideration of the input to a filter, x(¢), and the output 
of the filter, y(¢), one must define joint probability-density functions to 
describe the simultaneous characteristics of the two variables. The first 
joint density function is defined so that the product fii(2,t1;y,te) dx dy 
gives the probability that X has a value in the range x < X < 2+ dzat 
time t; and Y has a value in the range y < Y < y + dy at timete. The 
most elementary moment of the first-order joint density function is the 
cross-correlation function, defined as 
®,.,,(t1,t2) = EX (1) Y (t2)] 
= sad he fir(w,trjy,te)cy da dy (10.5) 
If the variables X and Y are independent, then the joint density function 
is the product of the individual first-order density functions for X and Y, 
and the cross-correlation reduces to 
Pry (tite) = E[X(t1)] BLY (t2)] 
= X(h) Y(t) (10.6) 
Therefore, if the mean value of either of two independent random vari- 
ables is zero, their cross-correlation is zero. It is important to note that 
the converse of this statement is false. That is, a zero cross-correlation 
between two variables, one or both of which have zero mean value, does 
not imply the independence of the variables. The cross-correlation is 
the average of the product of values of X at ¢; times the values of Y at te. 
Since this average is unchanged if X is replaced by Y and é1 by és, it 
follows that ®,,,(t1,t2) equals ®,,(t2,t1). 
In many cases of practical interest it is possible to make the simplifying 
assumption that the random process is stationary. That is, the prob- 
ability-density functions do not depend on the origin of time, they are 
stationary in time. If this assumption can be made, then the expected 
value given by (10.2) becomes independent of ¢; and is written simply as 
E(X) =X (10.7) 
The second-order density function of a stationary process is written 
fe(x1,22,7) and expresses the probability that the variable X has a value 21 
at some time and a value x27 sec later. The autocorrelation function of a 
stationary process becomes 
®,,(7) = ines ee X1Lof o(21,22,7) Axi dre (10.8) 
and has even symmetry since ©®,,(r) equals ®,,(—7). The mean-square 
value of the stationary variable is independent of the time, and (10.4) 
