Let c(t) = A C sin ω c t represent carrier wave and m(t) = A m sin ω m t represent the message or the modulating signal where ω m = 2 π ƒ m is the angular frequency of the message signal. The modulated signal c m (t) can be written as

c m (t) = (A C + A m sin ω m t) sin ω c t

= A C sin ω c t ..........(i)

Note that the modulated signal now contains the message signal. From Eq. (i), we can write,

c m (t) = A c sin ω c t + μ A c sin ω m t sin ω c t ..........(ii)

Here μ = A m /A c is the modulation index; in practice, μ is kept ≤ 1 to avoid distortion.

Using the trignomatric relation sin A sin B = 1/2 (cos (A – B) – cos (A + B),

we can write c m (t) of Eq. (ii) as

c m (t) = A c sin ω c t + cos ( ω C – ω m ) t – cos ( ω C + ω m ) t ..........(iii)

In amplitude modulated wave, the frequencies contained are ω c – ω m , ω c , ω c + ω m .

The frequency of ω m is not contained in A.M. wave