BAND WIDTH AND TRANSMISSION PERFORMANCE 



591 



For each value of ;;/, there is only one value of n satisfying the audio filter 

 inequality, which may be written: 



-^ - (2c + nt)X < // < ^ - (2c + m)N (67) 



lIcMue iiiti'rference accepted by the channel filter is: 



/(/) = ^ Z ''" (c + "0 J'nix) COS 27r[(2c + m)X + n\Jrl (68) 



11 m=mi \ " / 



The mean square value of interference is 



(0 = j^ Z^ al [c + I j Jl(x). (69) 



r / = 



The signal-to-interference ratio is 



32 



[/=^(0]"' 



or 



S/I = 



2FtQ 



[t^ a\ [c + fj il(.v)' 



-1/2 



(70) 



(71) 



When a rectangular gate of duration equal to the full channel allotment is 

 used, we substitute fl„ = 2(sin mr/N)/n'K. We then find that the largest 

 values of mean square interference occur when c is an odd multiple of one 

 fourth. If we set 



c= -(2r-M)/4,r=0,±l,±2, 

 it follows that if N is an even integer, 



n = {r -\r h ~ ni)N, 



sm ^ = sm (r + t — ni)Tr, 



sm 



rnr 



= 1. 



Substituting these values in the expression for S/I, we find 



1-1/2 



s/i = 



2Fb Q [_m=T- 



[r+2 -1-1/2 



m=r—l J 



(72) 



(73) 

 (74) 



(75) 

 (76) 



The value of r is to be chosen as the integer which makes S/I a minimum; 

 i.e., we place the CW^ frequency at that part of the band where it does tJie 

 most damage. The curve marked CW(Gate) of Fig. 14 was obtained in 

 this way. 



