derive an equation for AM wave?
Answers
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.