BAND WIDTH AND TRANSMISSION PERFORMANCE 589 



When the interference is fluctuation noise of mean power Pndf in 

 bandwidth df, the mean square frequency error in the output of the fre- 

 quency detector is Wn(f)df at frequency/, where 



Wn(/) = PnP/Wc (54) 



The baseband filter accepts the portion of this spectrum between/ = and 

 / = F,. 



The action of the rectangular gate on this spectrum may be calculated by 

 multiplication of the typical spectral component by the gating function: 



00 



Gil) = ^^ + T. am COS lirmfrl (55) 



2 m=l 



where for G{t) = 1 throughout the channel allotment time, 



.ll2Nf 



2 sm 



^iliiyjr AT 



= 2fr \ COS Innrfrt dl = (56) 



J~l/2fff. WIT 



a 



' -V2Nf 



Each harmonic of G{l) beats with the noise spectrum on either side to produce 

 audio components which sum up to total mean square audio noise: 



1 r"^"'^ r •> -^ 2 



4 Jo L ' '"=1 



2A' -| 



+ E alwnimfr - f)\df (57) 



m=l J 



The summations stop at 2N because the baseband filter cuts off at 

 f = Fb ^ 2Nfr. The contribution of the Oq term is negligible. Then 



. 2 WZTT 



2N sm — - /,/2 



1F„ = -^ Z —f- [« + f? + imfr - ff] df . 



tP- Wc m=i m^ Jo P^j 



T^Wcm=i \ 12m-/ N 



When N is large, the sum approaches 



Wn ^ ^' (59) 



and 



