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Frequency modulation

Short answer

Frequency modulation varies instantaneous carrier frequency in proportion to message amplitude, keeping amplitude constant. Message m(t)=Am cos(2πfmt); carrier c(t)=Ac cos(2πfct). Instantaneous frequency fi=fc+kf m(t), max deviation Δf=kf Am. FM wave: s(t)=Ac cos(2πfct+β sin 2πfmt), β=Δf/fm. Power constant; Carson's rule bandwidth ≈2(Δf+fm).

What is frequency modulation (FM)? Explain with suitable equations and waveforms.

Frequency modulation (FM) is a method in which the instantaneous frequency of the carrier is varied in proportion to the instantaneous amplitude of the message signal, while the amplitude of the carrier stays constant.

Equations

Carrier: c(t) = Ac cos(2πfct). Message: m(t) = Am cos(2πfmt).

Instantaneous frequency: fi(t) = fc + kf·m(t) = fc + kf·Am cos(2πfmt)

where kf is the frequency sensitivity (Hz/V). The maximum frequency deviation is Δf = kf·Am.

FM wave: s(t) = Ac cos(2πfct + β sin(2πfmt))

where the modulation index β = Δf / fm.

  • Narrowband FM: β << 1, bandwidth ≈ 2fm.
  • Wideband FM: bandwidth by Carson's rule, BW = 2(Δf + fm) = 2fm(β + 1).

Power is constant: P = Ac²/2R, independent of the message.

Waveforms

Message m(t):   _.-'''-._       _.-'''-._
              _/        \_____/        \_

FM output:   |||||||||   | | | |   |||||||||
             (dense at positive peaks, sparse at negative
              peaks; amplitude constant)

At the positive peak of the message the carrier frequency is highest (fc + Δf); at the negative peak it is lowest (fc - Δf); at zero message it is fc. Commercial FM broadcasting uses Δf = 75 kHz with a 200 kHz channel spacing.

✓ Answer checked by Claude Sonnet 5.5

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