Amplitude Modulation

Let carrier wave be represented by the equation
v~^c=V~^ccos(ω~^ct) ----- (1)
Let Modulating Signal be represented by the equation
v~^m=V~^mcos(ω~^mt) ----- (2)

After Amplitude Modulation Amplitude of the carrier varies according to the instantaneous value of modulating signal. The total Amplitude of the Modulated carrier is given as

V~^A~^M = V~^c+V~^mcos(ω~^mt)

The Expression for AM Wave is

v~^A~^M = V~^A~^Mcos(ω~^ct)
=(V~^c+V~^mcos(ω~^mt))cos(ω~^ct)
=V~^ccos(ω~^ct)+V~^mcos(ω~^mt)cos(ω~^ct)
=V~^c(cos(ω~^ct)+m~^acos(ω~^mt)cos(ω~^ct)) , ( Since V~^m/V~^c = m~^a)



Now use the formula CosA.CosB = $ 1 #_ 2 $ [Cos(A+B)+Cos(A-B)]

v~^A~^M = V~^c(cos(ω~^ct)+ $ m~^a #_ 2 $ [cos(ω~^mt+ω~^ct)+cos(ω~^mt-ω~^ct)]

=V~^ccos(ω~^ct)+ $ m~^aV~^c #_ 2 $ cos2π(f~^m+f~^c)t+ $ m~^aV~^c #_ 2 $ cos2π(f~^m-f~^c)t