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Derive the expression for frequency modulated wave.

Subject : Principle of Communication Engineering

Topic : Angle modulation and demodulation

Difficulty : High

pce(44) • 18k  views
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  • If the frequency of carrier signal is varied in accordance with the instantaneous amplitude of the modulating signal, then it is called as frequency modulation.
  • In FM the amplitude of carrier remains constant. The variation in carrier frequency from the umodulated carrier is called frequency deviation (δ).
  • FM is non-linear process & information contain in frequency deviation.

    Mathematical Expression of FM:

  • Instantaneous frequency of frequency modulated wave is;

    Wi=Wc+δ

  • The frequency deviation above or below Wc depends on the instantaneous amplitude of modulating signal i.e

    δ=kVm cos Wmt

    where k= frequency deviation sensitivity of FM

  • Again, instantaneous frequency is:

    Wi=Wc+δ

    Wi=Wc+kVm cos Wmt ---------------------(i)

  • The frequency modulated signal is represented mathematically as

    Vfm=Vc sin θ ------------------------(ii)

    where θ instantaneous phase & it can be determine from instantaneous frequency as

    Wi = dθ /dt

    Wi dt = dθ

    ∫dθ= ∫Wi dt

    Substituting the value of Wi from equation (i) in above equation;

    θ= ∫ (Wc+kVm cos Wmt ) dt

    θ= Wct+ (kVm/Wm) sin Wmt

    θ= Wct+Mf sin Wmt ----------------------------------------(iii)

    where Mf (modulation Index)= (kVm/Wm)

    Substituting the value of θ from equation (iii) in equation (ii);

    Vfm=Vc sin (Wct+Mf sin Wmt)

    Above is the equation of frequency modulation.

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