Frequency & Phase Modulation (FM/PM)

Angle modulation varies the carrier's frequency or phase. Learn FM and PM equations, frequency deviation and modulation index β, narrowband vs wideband FM, Carson's rule bandwidth, Bessel sidebands, noise advantage and pre-emphasis, FM generation (VCO) and demodulation (PLL, discriminator).

FM (1933, Armstrong) gave high-fidelity, noise-resistant radio. Its constant envelope also makes it ideal for efficient non-linear power amplifiers — which is why FSK, GMSK and PM variants are used in digital radio too.

An analogy

FM is like a singer who keeps the same loudness but changes pitch to encode the message — static noise (which affects loudness) is largely ignored.

FM vs PM

FM: instantaneous frequency proportional to the message. PM: phase proportional to the message. They are related — FM is PM of the integrated message. Both have constant amplitude.

Bandwidth

Narrowband FM (β < 0.3) needs ≈ 2f_m like AM. Wideband FM spreads energy into many Bessel sidebands; Carson's rule captures ≈ 98 % of power. Broadcast FM: Δf = 75 kHz, f_m = 15 kHz → ≈ 180 kHz (200 kHz channels).

Noise advantage

Above a threshold SNR, wideband FM trades bandwidth for SNR improvement (≈ 3β² (β + 1)). Pre-emphasis boosts high audio frequencies before transmission and de-emphasis restores them, reducing high-frequency noise.

Implementation

Generation: VCOs (direct) or Armstrong indirect method; PLL synthesisers. Demodulation: frequency discriminators, quadrature detectors, PLL demodulators, and digital I/Q demodulation (arctangent of I/Q phase difference) in SDRs.

Formulas

• FM signal: s(t) = A_c cos(2πf_c t + β sin 2πf_m t)
• Modulation index: β = Δf / f_m
• Carson's rule: B_T ≈ 2(Δf + f_m) = 2f_m(β + 1)
• PM signal: s(t) = A_c cos(2πf_c t + k_p m(t))

Worked example: Broadcast FM

Given: Δf = 75 kHz, f_m = 15 kHz

Find: β and bandwidth.

• β = 75/15 = 5
• B_T = 2(75 + 15) = 180 kHz

Answer: β = 5, ≈ 180 kHz

Worked example: Narrowband FM radio

Given: Δf = 5 kHz, f_m = 3 kHz

Find: Bandwidth.

• B_T = 2(5 + 3) = 16 kHz

Answer: 16 kHz (fits 25 kHz or 12.5 kHz channels with care)

Common mistakes

• Wrong: Using AM bandwidth for wideband FM. — Right: Apply Carson's rule.
• Wrong: Ignoring the FM threshold. — Right: Below threshold SNR, FM quality collapses.

Applications

• FM broadcast: 88–108 MHz stereo radio.
• Two-way radio: Narrowband FM for voice.
• Digital variants: FSK, GFSK (Bluetooth), GMSK (GSM).

Key takeaways

• β = Δf/f_m.
• B ≈ 2(Δf + f_m).
• Constant envelope, noise advantage.
• PM and FM are closely related.

Going further

• Bessel spectra: J_n(β) sideband amplitudes.
• Capture effect: Stronger FM signal suppresses weaker.
• I/Q demodulation: Digital FM detection in SDR.