584 BELL SYSTEM TECHNICAL JOURNAL 



Then 



(,/;V)' = f (0(l-2 0(^-|.^-lJ. (21) 



The signal-to-noise ratio is found to be maximum when 



To calculate the CW interference with an idle channel we assume that the 

 carrier wave of the system is represented by 



Fi(0 = P cos [iTTJit + ^ (01 (23) 



where 



7r,3 ( / + -4- sin TrFb A , < / < ^ 

 4>i.t) = \ (24) 



^(_/) = -0(0, (^^ ± ^) = 0(0, m = 1, 2, . . . . (25) 



We have chosen </>(/) so that the phase is a continuous function of time 

 with a derivative representing the correct frequency. This gives a sinusoi- 

 dal change in the instantaneous frequency 4>'{t)/2-K starting with the value 



/^ at / = — =r , reaching the peak/i 4- /3 at / = 0, and subsiding to/i at / = 

 Fh 



— . By making the wave repeat with frequency Nfr, we assume all channels 



Fh 



of the system are idle since all pulses are at their central positions. It seems 



reasonable to neglect the effect of variations in adjacent channel loading on 



CW interference in one channel. The interfering CW wave is represented 



by 



V^{t) = Q cos (27r/2/ -I- 6) (26) 



To a first approxunation the resulting error in frequency at the output of 

 the frequency detector is : 



^W = ^ J, ^i^ t27r(/i - /,)/ + 0(0 - e] (27) 



lirr at 



By straghtforward Fourier series expansion and differentiation: 



5(0 = p i^6 Z (c + "^)^« cos {lirFic -I- n\)t - 6] (28) 



Jr n=— 00 



