Amplitude Modulation

Definition. In amplitude modulation (AM), the amplitude of a high-frequency carrier wave is made to vary in direct proportion to the instantaneous value of a lower-frequency message (modulating) signal, while the carrier's frequency and phase are both left unchanged.

The envelope. If the carrier is written as $c(t)=A_c\cos(\omega_c t)$ and the message signal as $m(t)=A_m\cos(\omega_m t)$ (with $\omega_c \gg \omega_m$), the amplitude-modulated wave is $c_m(t) = \left[A_c + A_m\cos(\omega_m t)\right]\cos(\omega_c t)$. The quantity in the square brackets -- the slowly varying amplitude of the fast carrier oscillations -- is called the envelope of the modulated wave, and it is drawn as a smooth curve joining the peaks of the carrier cycles; by construction, this envelope has exactly the same shape as the original message signal $m(t)$, riding on top of a constant level $A_c$. Reading off the envelope of an AM wave therefore reads off the message it carries. …