CARRIER AND SIDE-BANDS IN RADIO TRANSMISSION 93 



It is obvious that if either the amplitudes or the phases of the 

 components be distorted, their instantaneous sum will be changed; 

 that is, the wave resulting from their re-combination will be a dis- 

 torted reproduction of the original wave. Also, those parts of the 

 frequency range in which the amplitude is negligibly small can con- 

 tribute little to the reproduced wave, and the elimination of all com- 

 ponents in those ranges will have little effect on the quality of repro- 

 duction. Just what ranges it is essential to retain depends upon the 

 nature of the signal and the standard of reproduction that is set up. 

 What is important for present purposes is the fact that the faithfulness 

 with which a system will reproduce any arbitrary signal disturbance 

 is deducible, in theory at least, from a knowledge of its transmission 

 of sustained single frequencies. By this is meant a knowledge of how 

 the relation, both in amplitude and phase, between the inpfut and 

 output sinusoidal wave varies as the frequency of the wave is pro- 

 gressively varied thruout the frequency range. 



Analysis of a Modulated Wave 



Let us assume now that a radio system is called on to transmit 

 such a signal wave, F(t), which may be either a telephone or a tele- 

 graph signal. If, as is commonly assumed, the modulator causes 

 the amplitude of the carrier wave, C cos p t, to be varied in accordance 

 with the signal, the resulting modulated wave may be expressed as 



m=C[l+kF(t)]cos'pt, (2) 



where k is a factor which measures the so-called degree of modulation. 

 If the largest negative value of k F(t) is just equal to unity, so that 

 the instantaneous amplitude of the carrier wave just falls to zero, 

 the modulation is said to be complete. The significance of complete 

 modulation will be discussed later. 



Now let us resolve the signal wave into its infinite series of com- 

 ponents, each of the form 5 cos (qt-\-Q), where S and 9 vary with the 



frequency^-. Neglecting non-essential frequencies, q may be con- 



sidered to cover a range from qi to q 2 . If this value of F{t) be sub- 

 stituted in (2) we get 



m = C cos pt-\-k C cos pi ' I S cos (qt-\-e) dq. (3) 



J q\ 



The first term, which is independent of the signal, represents a com- 

 ponent having the carrier frequency, ¥-. The second term represents 



2w 



an infinite series of terms each derived from only one component of 



