SAMPLED DATA SYSTEMS WITH RANDOM INPUTS 253 
reduces to the simple relation that 
$.(0) = (a) = [_ x%f.(z) de (10.9) 
The cross-correlation of two stationary random signals is 
€,,(7) = ie a fiula,y,r)ay dx dy (10.10) 
which has such symmetry that ®,,(7) equals ©,,.(—7). 
A further simplifying assumption which is usually made during the 
analysis of stationary signals is the ergodic property. This hypothesis 
states that under certain conditions present in many physical cases, the 
moments of a stationary process obtained by a process of averaging in 
time on a particular signal are identical to the moments obtained by 
averaging over the probability-density function. For example, where 
the ergodic hypothesis applies, 
EX) =X = <x> 
where, by definition, 
T 
2rs = jim 2 | ota (10.11) 
T- 0 2T —T 
and furthermore, the autocorrelation function is given by 
&,.(r) = ELXOX(t +7) 
= line x(t)a(t + 7) d (10.12) 
and the cross-correlation may be determined from 
Py(7) = E[XQYV( + 7)] 
= en ae a(t)y(t + 7) dt (10.13) 
It will be assumed that all processes considered hereafter are both sta- 
tionary and ergodic where necessary. Another property of random 
signals which is of some interest to the analysis of sampled random signals 
is that the autocorrelation function of a stationary and ergodic process 
may be obtained by an average of samples according to 
+N 
oa iim aN 1 an » a(nT)x(nT + 7) (10.14) 
n=—N 
for positive T. 
An important technique in the analysis of linear systems is the resolu- 
tion of signals into their frequency content by Fourier or Laplace trans- 
