534 



BELL SYSTEM TECHNICAL JOURNAL 



or taking 5- = 1/2, 



1 



1 + 3cOiVaJa' + 3XV2cOa'- 



(80) 



Since in practice coi ^ Wa and X ^ Wa, amplitude limitation results in a 

 very substantial reduction in low frequency noise power in the receiver 

 proper. Reference to formula (70) shows that, as compared with 

 parallel rectification and recombination, amplitude limitation reduces 

 the noise power by the factor 



1 + 3XV2co„2 ■ ^^^^ 



It should be observed that without amplitude limitation little reduc- 

 tion in the noise-to-signal power ratio results from increasing the 

 modulation index X (and consequently the high frequency transmission 

 band width). On the other hand, with amplitude limitation, the ratio 

 p of noise-to-signal power is 



p = P^/Ps ^l""" 



^^aN\ 



(82) 



The ratio p is then (within limits) inversely proportional to the 

 square of the modulation index X, so that a large value of X is indicated. 

 It should be noted that, within limits (X « coc), the power transmitted 

 from the sending station is independent of the modulation index X. 



It might be inferred from formula (82) that the noise power ratio p 

 can be reduced indefinitely by indefinitely increasing the modulation 

 index X. Actually there are practical limits to the size of X. First, if X 

 is made sufficiently large, the variable frequency oscillator generating 

 the frequency-modulated wave may become unstable or function 

 imperfectly. Secondly, the frequency spread of the frequency modu- 

 lated wave is 2X (from coc — X to co^ + X) and, if this is made too large, 

 interference with other stations will result. Finally, the stationary 

 distortion of the recovered low frequency signal s{t) increases rapidly 

 with the size of X. 



To summarize the results of the foregoing analysis the potential 

 advantages of frequency-modulation depend on two facts. (1) By 

 increasing the modulation index X it is possible to increase the recovered 

 low frequency signal power at the receiving station without increasing 

 the high frequency power transmitted from the sending station. 

 (2) It is possible to employ amplitude limitation (inherently impossible 

 with amplitude-modulation) whereby the effect of interference or noise 

 is reduced to a phase or "instantaneous frequency" variation of the 

 high frequency wave. 



