CARRIER AND SIDE-BANDS IN RADIO TRANSMISSION 109 



If the static be assumed to approximate an impulse, the ampli- 

 tudes of all its components will be sensibly the same. If we call this 

 amplitude S, then, in the case where the receiving circuit admits 

 only one side-band, the amplitude of the reproduced interfering cur- 

 rent of the frequency of the signal component is 



Ii^gS, (36) 



and its power, 



W 1 = ^g i S i . (37) 



With both side-bands this interfering current is made up of two equal 

 components derived from the static components of frequencies — - 



2,TT 



P — Q 

 and — — respectively. The phase difference e between these two will 



27T 



be accidental, so for any one case the resultant amplitude, 



h = 2gScost, (38) 



and the power, 



W 2 = 2 g 2 S 2 cos 2 e. (39) 



As all values of e are equally probable, we may average W 2 with 

 respect to e, whence 



W 2 = g 2 S i = 2W 1 . (40) 



There is then no choice between one and two side-bands on the 

 basis of the ratio of signal to interference. With a single side-band, 

 the major advantage of economy in frequency range is secured at the 

 expense of the minor disadvantage that to give the same response 

 the amplification of the receiving set must be greater by a factor of 

 two in power, or about three miles of standard cable. 



Use of Non-synchronous Local Carrier 



In practice, however, unless the receiving carrier frequency is con- 

 trolled by the same source as the sending carrier, it is rather difficult 

 to maintain even the frequencies alike, to say nothing of the phases. 

 Let us suppose that the local carrier is out of synchronism by a small 

 amount, n. Consider first the simplest case where the carrier is 

 suppressed and one side-band only is transmitted. The local carrier 

 beating with each component of this side-band gives a component 

 of normal amplitude, but of a frequency differing from that of the 

 original signal component by n. That is, all the frequencies of the 

 speech are raised or lowered by the same amount, n. This must alter 



