MATHEMATICAL ANALYSIS OF RANDOM NOISE 317 



where we have assumed T so large and w{f) of such a nature that the summa- 

 tion may be replaced by integraiion. 



If / remains finite, then as T -^ oo with t held fixed, the correction term 

 on the left becomes negligibly small and we have, upon using the definitions 

 (2.1-4) for the correlation function \J/{t), the second of the fundamental 

 inversion formulas (2.1-6). The first inversion formula may be obtained 

 from this at once by using Fourier's double integral for w{f). 



Incidentally, the relation (2.3-6) between w{f) and the coefficients a„ and 

 bn is in agreement with the definition (2.1-3) for w{f) as a limit involving 

 I S(f) I \ From the expressions (2.3-2) for a„ and 6„, the spectrum 6'(/„) 

 given by (2.1-2) is 



T 

 S(Ju) = -^{an - ibn) 



Then, from (2.1-3) w(/„) is given by the limit, as T— > <», of 



= l^idn + On) 



(?) 



and this is the expression for «/ ( - 1 given by (2.3-6). 



2.4 Discussion of Results of Section One — Parseval's Theorem 



The use of Parseval's theorem enables us to derive the results of section 

 2.1 more directly than the method of the preceding section. This theorem 

 states that 



\ FiiJ)F2if) df = f Gr{t)G2{-t) dt (2.4-1) 



J— oo J— 00 



where Fi , d and F2 , G2 are Fourier mates related by 



Fif) = [ C(Oe"*''^' di 

 J— 00 



GH) = r'^ F(f)e''"'' df 



J— CO 



(2.4-2) 



It may be proved in a formal manner by replacing the F\ on the left of 

 (2.4-1) by its expression as an integral involving G\(t). Interchanging the 



" E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Oxford (1937). 



