CARRIER AND SIDE-BANDS IN RADIO TRANSMISSION 107 



Phase of the Local Carrier 



Coming now to homodyne reception, the important new factor 

 to be considered is the fact that the carrier component is now per- 

 fectly arbitrary in amplitude and phase. This is true even tho the 

 sending carrier is not suppressed, for, by suitably choosing the local 

 carrier, the resultant of the two may be given any desired value. Since 

 the amplitude of the carrier affects only the magnitude of the re- 

 produced signal as a whole, we need consider here only the effect of 

 arbitrary values of its phase, $. For simplicity we shall assume that 

 the modulated wave reaches the receiver unchanged except for the 

 phase lags involved in undistorted transmission. Let us designate 

 by c/>i the phase lag of the carrier which is received from the trans- 

 mitting station or would be received if it were not suppressed. Then 



= C/>1+??, (25) 



where t/ may be regarded as the phase displacement of the local carrier. 

 Consider first a system in which one side-band is suppressed. From 

 equation (16), the amplitudes of the reproduced signal, componants 

 are independent of the phase of the carrier. From equation (17), 

 the phase, 



^ = + -0 1 -r ? . (26) 



But <f> + —4>\ represents only the phase shifts of undistorted trans- 

 mission ; that is, the delay suffered by the signal as a whole. Hence 

 the net result is that all components have their phases shifted by 

 the same amount; namely, the phase displacement of the carrier, 

 which can never be more than a single cycle. For telephony this is 

 of no practical importance, but it is evident that in a telegraph system 

 using side-band suppression and homodyne reception the phase of 

 the local carrier would have to be very carefully controlled. 



Consider now the case of homodyne reception of both side-bands 

 received without distortion; that is, 



B + =B-=B ± (27) 



and 0+ — 4>\ = 4>\ — 4>- = h q. (28) 



From these relations and equations (13) and (14) we get 



R = 2B B^cos v . (29) 



* = hq. (30) 



This shows that the amplitude of every component varies as cos rj; 

 that is, when the local carrier is in phase or 180° out of phase with the 

 received carrier the reproduced components are maximum; for inter- 



