Question

derive an equation for AM wave?​

Answer 1

Answer:

Derivation of Amplitude Modulation

Mathematical representation of Amplitude Modulated waves in time domain

m(t)=Amcos(2πfmt) (modulating signal)

c(t)=Accos(2πfct) (carrier signal)

s(t)=⌊Ac+Amcos(2πfmt)⌋cos(2πfct) (equation of Amplitude Modulated wave)

Where,

Am: amplitude of modulating signal

Ac: amplitude of carrier signal

fm: frequency of modulating signal

fc: frequency of carrier signal

Therefore, above is the derivation of Amplitude Modulation.+

Modulation index derivation

Modulation index is also known as modulation depth is defined for the carrier wave to describe the modulated variable of carrier signal varying with respect to its unmodulated level. It is represented as follows:

μ=AmAc

Consider maximum and minimum amplitudes of the wave as Amax and Amin

Depending upon cos(2fmt) following two equations are derived with maximum and minimum amplitude of the modulated waves.

Amax=Ac+Am

Amin=Ac−Am

Amax+Amin=Ac+Am+Ac−Am=2Ac

⇒Ac=Amax+Amin2

Amax−Amin=Ac+Am−(Ac−Am)=2Am

⇒Am=Amax−Amin2

μ=AmAc

Therefore, this is the derivation of the modulation index.