254 SAMPLED-DATA CONTROL SYSTEMS 
formation. In the study of random signals this technique is embodied 
in the concept of the spectral-density function. This function is an 
average similar to the moments defined above but expresses the average 
power contained in a signal in an infinitesimal frequency band. Physical 
interpretations of the spectral density usually speak of averaging the out- 
put of a very narrow band filter, whose input is the signal in question. 
For the purpose of this brief review, the spectral density is defined as the 
transform of the autocorrelation function 
Ss) i ®,,(7)e—*" dr (10.15) 
—7o 
from which the inverse transform gives the result 
re 
on) = =| S.2(s)e" ds (10.16) 
2M i) 70 
The transform used in (10.15) and (10.16) is the bilateral Laplace trans- 
formation, and those familiar with Fourier analysis may replace the 
variable s by jw to obtain possibly more familiar forms for the equations. 
The form given is chosen because it seems pointless to continually rotate 
the same complex plane in order to avoid imaginary limits in certain early 
expressions when one knows they will occur later anyway. By use of the 
bilateral Laplace transform, bounded time functions for positive time are 
represented by singularities of the transform in the left half of the s plane 
and bounded time functions for negative time are represented by singular- 
ities in the right half plane. It is noted that the path of integration for 
the inverse transform (10.16) is the imaginary axis and singularities on 
this axis are not allowed in the present analysis. 
From (10.9) and (10.16) it is seen that the mean-square value of a 
random variable may be obtained by summing (integrating) the spectral 
density over all frequencies. 
je 
®,,(0) = aa / Szz(s) ds (10.17) 
2a page 
The result expressed by (10.17) is the reason for the name “spectral 
density’? because S,,(s) gives the density of mean-square value of the 
variable over the spectrum of real frequencies (imaginary s). The cross- 
spectral density is a function defined for two random variables as the 
transform of the cross-correlation function. 
Soy(s) = [", ®ay(r)e* dr (10.18) 
As mentioned above, the principal use of the spectral density is in the 
analysis of random processes in linear systems, where one can use the 
