Editing Amplitude modulation (section)

==Analysis==
[[File:Illustration of Amplitude Modulation.png|thumb|391x391px|Illustration of amplitude modulation]]
The carrier wave ([[sine wave]]) of frequency ''f<sub>c</sub>'' and amplitude ''A'' is expressed by

:<math>c(t) = A \sin(2 \pi f_c t)\,</math>.

The message signal, such as an audio signal that is used for modulating the carrier, is ''m''(''t''), and has a frequency ''f<sub>m</sub>'', much lower than ''f<sub>c</sub>'':

:<math>m(t) = M \cos\left(2\pi f_m t + \phi\right)= Am \cos\left(2\pi f_m t + \phi\right)\,</math>,

where ''m'' is the amplitude sensitivity, ''M'' is the amplitude of modulation. If ''m'' < 1, ''(1 + m(t)/A)'' is always positive for undermodulation. If ''m'' > 1 then overmodulation occurs and reconstruction of message signal from the transmitted signal would lead in loss of original signal. Amplitude modulation results when the carrier ''c(t)'' is multiplied by the positive quantity  ''(1 + m(t)/A)'':

:<math>\begin{align}
  y(t) &= \left[1 + \frac{m(t)}{A}\right] c(t) \\
       &= \left[1 + m \cos\left(2\pi f_m t + \phi\right)\right] A \sin\left(2\pi f_c t\right)
\end{align}</math>

In this simple case ''m'' is identical to the [[#Modulation index|modulation index]], discussed below. With ''m'' = 0.5 the amplitude modulated signal ''y''(''t'') thus corresponds to the top graph (labelled "50% Modulation") in figure 4.

Using [[prosthaphaeresis#The identities|prosthaphaeresis identities]], ''y''(''t'') can be shown to be the sum of three sine waves:

:<math>y(t) = A \sin(2\pi f_c t) + \frac{1}{2}Am\left[\sin\left(2\pi \left[f_c + f_m\right] t + \phi\right) +  \sin\left(2\pi \left[f_c - f_m\right] t - \phi\right)\right].\,</math>

Therefore, the modulated signal has three components: the carrier wave ''c(t)'' which is unchanged in frequency, and two [[sideband]]s with frequencies slightly above and below the carrier frequency ''f<sub>c</sub>''.