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In amplitude modulation when carrier signal and base band signals are mixed, side band frequencies are generated. Upper side band frequency is $f_c+f_m$ and lower side band frequency is $f_c-f_m$. Wikipedia defines sideband frequencies as follows:

In radio communications, a sideband is a band of frequencies higher than or lower than the carrier frequency containing power as a result of the modulation process.

My doubt is why the definition says band of frequencies when there are only two other specific frequencies (one is upper side band frequency i.e. $f_c+f_m$ and the other one is lower side band frequency $f_c-f_m$).

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In general the modulating signal is a range of frequencies, and indeed that range is a function of time. For example the modulating signal may be music and represented by some function $A(f,t)$, where $A(f,t)$ is the amplitude of the modulating signal at a frequency $f$ and a time $t$.

In that case the extra frequencies will not be two distinct signals at $F_c \pm F_m$, but will be a band of frequencies $F_c \pm A(f,t)$.

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