MATHEMATICAL ANALYSIS OF RANDOM NOISE 31"3" 



It is seen that ^(t) is an even function of r and that 



lA(0) = p (2.1-7) 



When either ^(r) or w(/) is known the other may be obtained provided the 

 corresponding integral converges. 



2.2 Power Spectrum for D.C. and Periodic Components 



As mentioned in section 2.1, when /(/) has a d.c. or a periodic com- 

 ponent the limit in the definition (2.1-3) for w(/) does not exist for /equal 

 to zero or to the frequency of the periodic component. Perhaps the most 

 satisfactory method of overcoming this difficulty, from the mathematical 

 point of view, is to deal with the integral of the power spectrum.^^ 



/ 

 wC?) dg (2.2-1) 



/ 



Jo 



Jo 



instead of with iv(f) itself. 



The definition (2.1-4) for \P(t) still holds. If, for example, 



/(/) = yl + C cos (Iw/ol - if) (2.2-2) 



\J/(t) as given by (2.1-4) is 



^(r) = A' + '^cos27r/or (2.2-3) 



The inversion formulas (2.1-5) and (2.1-6) give 



Jo TT Jo T 



^P{r) = J cos iTTfrd j w{g) dg 



(2.2-4) 



1* This is done by Wiener,*^ loc. cit., and by G. \V. Kenrick, "The Analysis of Irregular 

 Motions with Applications to the Energy Frequency Spectrum of Static and of Telegraph 

 Signals," Phil. Mag., Ser. 7, Vol. 7, pp. 176-196 (Jan. 1929). Kenrick appears to be one 

 of the first to apply, to noise problems, the correlation functian method of computing the 

 power spectrum (one of his problems is discussed in Sec. 2.7). He bases his work on re- 

 sults due to Wiener. Khintchine, in "Korrelationstheorie der station iren stochastischen 

 Prozesse," Malh. Annalen, 109 (1934), pp. 604-615, proves the following theorem: A 

 necessary and sutlicient condition that a function R{1) may be the correlation function of 

 a continuous, stationary, stochastic process is that R{1) may be expressed as 



= / 



K{1) = I cos Ix dF{x) 



J— 00 



where F{x) is a certain distribution function. This expression for R{t) is essentially the 

 second of equations (2.2-4). Khintchine's work has been etteniei by H. Cra n -r, "On 

 the theory of stationary random processes," Ann. of Mat't., Ser. 2, Vol. 41 (IJtJ), pp. 

 215-230. However, Khintchine and Cramir appear to be interested pri nxrily in ques- 

 tions of existence, representation, etc., and do not stress the concept of the povver spectrum. 



