316 BELL SYSTEM TECHNICAL JOURNAL 



where the last term represents correction terms which must be added be- 

 cause the series (2.3-3) does not represent /(/ + r) in the interval {T — t, T) 

 when r > 0, or in the interval (0, — r) if t < 0. 



If l{t) were a current and if it were to flow through one ohm for the in- 

 terval (0, T), each component would dissipate a certain average amount 

 of power. The average power dissipated by the component of frequency 

 /„ = w/r cycles per second would be, from the Fourier series and elementary 

 principles, 



i (al + h\) watts, n 9^ Q 



(2.3-5) 



— watts, « = 



4 



The band width associated with the wth component is the difference in 

 frequency between the « -|- 1 th and wth components : 



/ f - ^ + ^ ^ _ 1 >, 



Jn-^\ ~~ Jn — J, -=, — — CpS 



Hence if the average power in the band f,f-\- df is defined as w(J)df, the 

 average power in the band /„+i — /„ is 



w(/J (Jn+i - /n) = IV l^j - 



and, from (2.3-5), this is given by 



z.(0)l = ^, ^^=0 



(2.3-6) 



"W'Tien the coefficients in (2.3^) are replaced by their expressions in terms 

 of 7i'(/) we get 





lirnr 

 ^ , cos -7=- 

 TJ T 



2iriiT dn 

 ^ I cos -^ ^ 



= / w(J) cos IttJt df 

 Jo 



(2.3-7) 



