102 BELL SYSTEM TECHNICAL JOURNAL 



that no new frequencies are introduced. Then any possible distor- 

 tion in the modulated wave may be represented by assigning the 

 proper amplitudes and phases to all of the components. Correspond- 

 ing to a single component of the signal we may write for the received 

 wave 



m = B cos (pt-<t>)+B + cos [(p+q) Z4-e-<£+]4- 



B-.cos[(p-q)t-e-4>-] (10) 



where the amplitude, B } and phase lag, </>, may vary in any arbitrary 

 manner for the different components of the modulated wave. We 

 shall assume that B is always large enough compared with B + and 

 -B_ that the interaction between the side-band components may be 

 neglected. It will be seen that the single frequency components 

 reproduced from the two side-bands are not in general equal nor in 

 phase and may either aid or tend to neutralize each other. They 

 will be of the form, 



r = B\B+cos [qt + Q-(<j> + -<f>)]+B_cos [q t + e- (</>-</>-)] | • (11) 



Taking the resultant of these two gives as the component of the 

 reproduced wave, 



r = Rcos (qt + e-V) (12) 



where 



R = BVB> + +^_ + 2B + B_cos[(ct> + -<'>)- (</>-</,_)] (13) 



tan y= B + sin &+ -4)+B-sin (<fr-*_) (14) 



B + cos ((/>+ -4>)+B_cos (0-0_)' 



It is evident that both the amplitude, R, and the phase shift, ty, 

 of the reproduced component depend upon both the amplitudes and 

 phases of the corresponding components of both side-bands and on 

 the phase of the carrier. The amplitude depends also on the ampli- 

 tude B of the carrier, but as variations in this affect all components 

 alike, they do not alter the wave form of the reproduced signal, but 

 only its magnitude. 



The expressions for the reproduced wave become much simpler 

 for a system in which one side-band, say the lower, is suppressed. 

 Then 



5_ = (15) 



and equations (13) and (14) reduce to 



R = BB+ (16) 



^ = + -0. (17) 



The amplitude of the reproduced component is independent of the 



