MATHEMATICAL ANALYSIS OF RANDOM NOISE 315 



The definition (2.1-3) for w(/) gives the continuous part of the power 

 spectrum. In order to get the part due to the d.c, and periodic com- 

 ponents, which is exemplified by the expression (2.2-6) for w(f) involving 

 the 8 functions, we must supplement (2.1-3) by adding terms of the type 



2A'8{f) + I' 8{f - /o) = [Limit ^-1^'] 8{f) 



(2.2-11) 



2^2 



+ [urnit^-^^^\8{f-fo) 



The correctness of this expression may be verified by calculating S{f) for 

 the /(/) of this section given by (2.2-2), and actually carrying out the limiting 

 process. 



2.3 Discussion of Results of Section One — Fourier Series 



The fact that the spectrum of power w(/) and the correlation function 

 i/'(t) are related by Fourier inversion formulas is closely connected with 

 Parseval's theorems for Fourier series and integrals. In this section we shall 

 not use Parseval's theorems explicitly. We start with Fourier's series and 

 use the concept of each component dissipating its share of energy inde- 

 pendently of the behavior of the other components. 



Let that portion of /(/) which lies in the interval < t < The expanded 

 in the Fourier series 





<^o , v^ / 2irnt , . . 27r«A 



2 +Sr'°'-7-+^"'"-r-j 



Hi) = o + 2^ I <^« cos -— -j- bn sin -— - (2.3-1) 



where 



2irnt ,, 

 cos -—- at 



JQ T 



(2.3-2) 



. 2 [^ ^,^. . 2-wnl -^ 



f^n = f J ^(0 sm ^Y- ^i 



Then for the interval —T<t<T—T, 



Hi + r) = - + 2^^ U„ cos + bn sm ^ 'j (2.3-3) 



Multiplying the series for /(/) and /(/ -f t) together and integrating with 

 respect to / gives, after some reduction, 



i fj I(t)I{l + r) dt 



flO , V 1 / 2 , i2 X „ „ 27r« ^ / ^-^ \ 



