96 BELL SYSTEM TECHNICAL JOURNAL 



heterodyne beat notes resulting from all the pairs of component 

 sinusoids of the modulated wave. 



The carrier component, C cos pt, beating with a component of 

 the upper side-band, \ k C S cos [{p + q) t + 0], equation (4), gives 

 the beat note or reproduced component, 



r+ =\k C 2 Scos (qt + e), (7) 



which is identical in frequency and phase with the corresponding 

 component of the signal, and has an amplitude proportional to that 

 of the signal component. Exactly the same expression results from 

 beating the carrier and the corresponding component of the lower 

 side-band. These two low frequency components, being in phase, 

 add directly to give 



r = k C 2 S cos (qt+e) (8) 



as the reproduced component. As the factor k C 2 is independent 

 of q, all of the signal components are reproduced with the same rela- 

 tive amplitudes and phases, as in the original signal. Their sum is 

 therefore k C 2 F(t), and the signal is accurately reproduced. 



However, there are still other components of the modulated wave 

 to be considered. Every pair of components in one side-band beat 

 to give the difference of their frequencies, which is also the difference 

 of the corresponding signal components. The corresponding pair of 

 components of the other side-band yield an identical component and 

 the two add in phase. Similarly every component of one side-band 

 beats with every component of the other, giving in each case the 

 sum of two component frequencies of the signal wave. Like the 

 difference frequencies, each of these sum frequencies is produced 

 twice. The combination of the components of the two side-bands 

 which were derived from the same signal component yields a com- 

 ponent of twice the frequency of the signal component. The addi- 

 tion of these extraneous components serves to distort the reproduced 

 wave in a manner quite similar to that of external interference. It 

 is of interest therefore to consider the magnitude of these distorting 

 components relative to the reproduced signal. The product of two 

 side-band components of amplitudes \ k C S and \ k C S', equation 

 (4), gives as the amplitude of one of the two components of the differ- 

 ence frequency, ——-, \ k 2 C 2 S S'. Comparing this with the ampli- 

 tude, \ k C 2 S, equation (7), of one of the two reproduced signal com- 

 ponents of frequency^-, the ratio of the undesired to desired compo- 

 nent is found to be \ k S'. It is evident that this type of distortion 



