248 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 



^2 = ^(0) _ ^(<x>). (5_26) 



The normalized autocorrelation function can be defined in terms of the 

 function ^(t) by subtracting the d-c term and dividing by the variance: 



<p{t) - v?(°o) 



P{t) = —77^ 7— T- (5-27) 



(p(0) - <p(oo) 



For a Gaussian process with the joint probability density given in 

 Equation 5-28, the autocorrelation would be computed in the following 

 manner: 



^W = o__2./l ~^j_^j_^ '''''' 



exp I 2a'^(l - 2) J^.vi^.V2 (5-28) 



[ 



(t^p{t). 



This integral can be evaluated by completing the square in the exponent for 

 one of the variables and transforming to a standard form. In the next 

 paragraph, it will be shown that the autocorrelation function is very closely 

 related to the power spectrum of the process. 



5-5 THE POWER DENSITY SPECTRUM 



It is possible to decompose random processes into frequency components 

 in a certain sense, and this will provide a powerful analytic technique. For 

 instance, it was previously mentioned that a Gaussian random process could 

 be constructed by the superposition of a large number of sinusoids of 

 varying frequency and random phase. This sort of a process can certainly 

 be decomposed into frequency components. Of course, the average values 

 of the in-phase and quadrature components at a given frequency will be 

 zero because of the introduction of a random phase angle. The power at a 

 given frequency, though, will be independent of phase and in general have 

 a non-zero value. Thus a frequency decomposition could be carried out on 

 a power basis. This possibility turns out to be valid for more general 

 random processes and leads to the useful concept of the power density 

 s-pectrum. Physically, the power density spectrum of a noise process corre- 

 sponds to the average power outputs of a bank of narrow filters covering 

 the frequency range of the process. 



To develop this idea, consider a stationary random process x{t). Subject 

 to the restrictions noted in Paragraph 5-2, the portions of the elements of 

 the process between — T and T possess Fourier transform spectra. 



By limiting the range we can ensure that the integrals of the squares ofthe 

 elements of the process are finite. Over an infinite range these integrals 



