SAMPLED DATA SYSTEMS WITH RANDOM INPUTS 255 
result that the spectral density of the output of the system is related to 
the spectral density of the input by the well-known equation 
Syy(s) = Szz(s)H(s)H(—s) (10.19) 
where H(s) is the transfer function of the linear filter or the Laplace trans- 
form of the response of the filter to a unit impulse. 
10.2 Analysis of Sampled Random Signals 
In the study of sampled-data systems one is immediately faced with the 
analysis of the situation shown in Fig. 10.1, where a continuous random 
signal x(t) is modulating the amplitudes of a train of rectangular pulses of 

SSSA SNSSS SNES SSS S SS SSS 
Fic. 10.1. Pulse sequence used for determining statistical parameters of sampled 
signal. 
fixed width y. The modulated pulse train represents the sampled signal 
z*(t), which has a gain factor of 1/y introduced to make the area of the 
pulses equal to the values of the signal at the sampling instants. This 
gain normalization will make the limit process as y tends to zero simpler 
to perform later when impulse modulation is introduced. 
If the origin of time in the modulated pulse train is random, then the 
process x*(t) will be stationary and ergodic if x(t) is stationary and 
ergodic. In this case, the autocorrelation of «*(t) may be found by taking 
a time average of the product x*(t)x*(t + 7) as follows: 
To 
©,+,+(T) = lim a i Babe + T) dt (10.20) 
To 2T 9 —To 
Sketches of a typical x*(¢) and the shifted x*(¢ + 7) are shown in Fig. 10.2. 
The integral of (10.20) may be broken up into a sum of integrals, each 
