5-5] THE POWER DENSITY SPECTRUM 249 



would not be finite and Fourier transforms could not be defined. Thus we 

 have the spectra Xt(co) : 



/: 



Xt(c^) = / x{i) e-^'^'dt. (5-29) 



Energy spectra will be given by expressions similar to Equation 5-6. If 

 these energy spectra are divided by the observation time 2T, power spectra 

 will be obtained which we denote by A^t(co): 



NtW) = 2^ \Xt{o:)\' = ^ i^e-^'^^drlxit + T)x{t)dt. (5-30) 



The range of the last integral has been denoted by R. Because the elements 

 of the process x{t) are in effect zero for |/| > T, the limits of integration will 

 be from - T + r to T for r > and from - T to T + r for r < 0. In either 

 case, the total range is 2T — |r|. 



We are, of course, primarily interested in the statistical average of 

 the power spectrum since only average values represent meaningful and 

 measurable attributes of the process. To compute the average value of 

 Nrico), we average the product x (t -\- t) x (/) in the expression for 7Vr(aj) 

 given by Equation 5-30. The average of this product is the autocorrelation 

 function of the process which will depend only upon the time difference r 

 if the process is stationary: 



A^r(co) = ;^ e-'^^'dr / <pir)d( 



Letting T— ^ oo, the factor involving Tin the integrand approaches unity, 

 and we obtain the following expression for the average power density 

 spectrum of the process : 



A^ = / ^(r)^-^"Vr. (5-32) 



This expression gives the power density spectrum as the Fourier transform 

 of the autocorrelation function. These two functions form a Fourier 

 transform pair and the knowledge of one is, at least in theory, equivalent 

 to a knowledge of the other. The inverse of the relation in Equation 5-32 

 gives 



^(r) = 2^ / A^^^'^Voj. (5-33) 



When T is set equal to zero in this relation 



^(0) = C72 -f .^2 = i- / W(^) du^. (5-34) 



