590 BELL SYSTEM TECHNICAL JOURNAL 



on substituting Wc = kPnB, ^ = B - 2Fb, and Fb = 2Nfr. Equation (61) 

 with ^ = 16 was used to obtain the marginal power curve of Fig. 13. The 

 result may be compared with that given by Rauch^^ (See also Landon^) for a 

 rectangular pulse and rectangular gate, which requires a higher value for 

 Ff,. The two systems give the same signal-to-noise ratio when the rectangu- 

 lar pulse and gate of Ranch's system endure for one half of the total channel 

 allotment time. The value of Fb necessary for such a case was estimated by 

 Ranch to be S.SN^fr. A calculation made as above, except that the gate 

 was assumed sinuosoidal instead of rectangular, showed very nearly the same 

 signal-to-noise ratio. 



The case of CW interference with all channels idle is represented by the 

 r.f. wave: 



V{l) = P cos llivfit + J|^ sin Tr/^b/ j + (3 cos lirf-.L 



(62) 



When Q/ P is small, the detected frequency is 



/2)/ + ;^sin7rF5/ 



(63) 



f = /, + _cosxF./-2^-sm 



27r(/i - Z^)/ -f AsiuTrFfe/ 

 Itb 



-/i + ^cosTrFft/ - 5(/).. 

 4 



By Fourier series expansion followed by differentiation, the error b{t) may be 

 written as: 



m = 2 t^ (/, - /, + ^') J,.(x) cos 2, (/, - /, + '4') (, (64) 



where x = ^/2Fb. The baseband filter passes only those components of 5(/) 

 which have frequencies in the range —Fb to Fb- Writing c = (/i — fo) Fb, we 

 find: 



5o(/) = ^ Z (c + l) /,„(.r) cos 2^Fb (c + |) I (65) 



where W2 is the largest integer which does not exceed 2(1 — c) and nii is the 

 smallest integer which is not exceeded by —2(1 + c). The term integer is 

 here understood to include zero and both positive and negative integers. 

 The wave 5o(0 is next multiplied by the gating function G{t) and the com- 

 ponents falling in the audio band —fr/2 to/r/2 selected. We find: 



GiOm =^ f: J: <^n(c + '^) JM) cos 2^[{2c + m)X + u]frt 



^■L m=mi n= — oo \ ^ / 



(66) 

 with a„ = a_„. 



* Loc. cit. 



■** L. L. Rauch, Fluctuation Noise in Pulse-Height Multiplex Radio Links, Proc. I.R.E., 

 Vol. 35, Nov. 1947, pp. 1192-1197. 



