CARRIER AND SIDE-BANDS IN RADIO TRANSMISSION 95 



The actual magnitude of the side-band currents relative to the carrier 

 depends on the degree of modulation, k, of equation (2). For com- 

 mercial telephony the limits of the essential band may be taken rough- 

 ly as 200 and 2,000 cycles. If high quality speech or music is to be 

 transmitted, a wider band is required. For telegraphy the band 

 width required varies widely with the speed of sending and the type 

 of apparatus used. In general, it is desirable to preserve very low 

 frequencies, which means that the two side-bands practically meet 

 at the carrier frequency. 



Reproduction of the Signal Wave 



Having arrived at a picture of the modulated wave as given by 

 equation (4), we shall first discuss the reproduction of the signal from 

 this as it stands, and then consider the effect on this reproduction 

 of various modifications to which the modulated wave may be sub- 

 jected before or during the process of detection. While any device 

 in which the current-voltage characteristic is non-linear may be used 

 as a detector, the operation of the vacuum tube lends itself to analysis 

 because of its approximation to a parabolic current-voltage relation. 

 That is, we may write, 



i = a -\-(iiV -\-a-iV 2 , (5) 



where v is the voltage impressed on the grid, in this case the modu- 

 lated wave, and i is the resulting current. As the first term is in- 

 dependent of v and the second represents simple amplification, detec- 

 tion 3 can result only from the third term, a 2 v 2 . Since a 2 multiplies 

 all components of v 2 alike, we may neglect it and simply consider the 

 square of the expression for the modulated wave. This results in a 

 series of terms which are the squares of the individual components 

 and another which are their products taken in pairs. Since 



cos 2 x = \ (1+cos 2x), (6) 



the square terms will yield only direct current, and currents of ap- 

 proximately twice the carrier frequency. The product terms, each 

 of which contains the product of two cosines, may, as in the case of 

 the modulated wave above, be transformed into the sum of two 

 cosine terms the frequencies of which are respectively the sum and 

 difference of the component frequencies. Of these only the difference 

 frequencies can lie in the range of the original signal. In other words, 

 we may think of the reproduced wave as made up of the sum of all the 



3 In practice this parabolic law seldom holds strictly, and secondary contributions 

 are made to the detected wave by terms of higher power. 



