322 BELL SYSTEM TECHNICAL JOURNAL 



>\ i-nift' 



= 2v Real Part of \ dt F(Og~'"'^' f dt' F{t')e 



J— 00 J— a: 



■V2W) r '''''' ^' 



J— 00 



= 2p\s{J)\' + 2T{f)h(J) (2.6-4) 



In going from the first equation to the second we have written t' — t -\- t 

 and have considered cos 27r/"r to be the real part of the corresponding ex- 

 ponential. In going from the second equation to the third we have set 



s(f) = [ " F(t)e-'''^' dt (2.6-5) 



J— 00 



and have used 



[ e'"^' dt = d(f) (2.2-9) 



The term in w(f) involving 5(/) represents the average power which would 

 be dissipated by the d.c. component of I(t) in flowing through one ohm. 

 It is in agreement with the concept that the average power in the band 

 0</<€, €>0 but very small, is 



f w(J) dt = 2ntf ( 5(/) df 

 Jo Jo 



_ (2.6-6) 



The expression (2.6-4) for w(f) may also be obtained from the definition 

 (2.5-3) for w(/)plus the additional term due to the d.c. component ob- 

 tained by averaging the expressions (2.2-11). We leave this as an exercise 

 for the reader. He will find it interesting to study the steps in Carson's 

 paper leading up to equation (8). Carson's R{co) is related to our wif) by 



Mf) = 27ri?(co) 



and his f{iu) is equal to our s{f). 



Integrating both sides of (2.6-4) with respect to/ from to oc and using 



72 = f w{f) df 

 Jo 



gives the result 



P-f = 2v( \s{f)fdf (2.6-7) 



Jo 



Jo 



" Loc. cit. 



