328 BELL SYSTEM TECHNICAL JOURNAL 



The signal input current is 



is = —2^ ^"~'^ cos St, ' (4) 



while the second order output signal sideband is 



ic±s = ^„_t' C"-'S cos {c ± 5)/. (5) 



Similarly, if the output signal voltage at second order sideband 

 frequency (c ± s) had been applied along with the carrier in place of 

 the input signal, and of an equal amplitude, then the following currents 

 of the output and input signal frequencies would result: 



ic±s = 2„_/ C"-'S cos {c ± s)t, (6) 



n-l 



GnflK 



n—2 

 2 



i. = ^^^C--'S COS St. (7) 



If both signal input and output frequency voltages are applied simul- 

 taneously, equation (3) then becomes 



i « an[(C cos c/)" + n{C cos ct)''~^-S cos st 



+ n(C cos c/)"-i-5 cos (c ± 5)/]. (8) 



The current responses obviously are the sum of the separate responses 

 from independent application of the two frequencies. Even if a 

 complex array of terminating impedances are supplied so that voltages 

 appear across the non-linear element at all the modulation product 

 frequencies, each new voltage will individually produce its own current 

 response, quite independently of the responses that are being produced 

 by the other voltages. It can readily be seen then that superposition 

 does not depend on any assumptions about what the terminating 

 impedances may be. 



Reciprocal Theorem 



Equations (5) and (7) show that the sideband response to an input 

 signal voltage is exactly equal in magnitude to the input signal re- 

 sponse to the same amplitude sideband voltage. It can readily be seen 

 that any two modulation products also bear such a reciprocal relation- 



