Frequency and Phase Modulation

Angle modulation varies the carrier's phase or frequency. It uses more bandwidth than AM but gives much better noise immunity.

FM and PM

• PM: phase ∝ message: s(t) = A cos(ω_c t + k_p m(t)).
• FM: instantaneous frequency f_i = f_c + k_f m(t).
• For a tone m(t) = A_m cos ω_m t in FM: frequency deviation Δf = k_f A_m and modulation index β = Δf/f_m.
• In FM, Δf depends only on the message amplitude; in PM, the deviation also grows with message frequency.
• The amplitude is constant, so the power is A²/2 regardless of β.

Spectrum

s(t) = A Σ J_n(β) cos((ω_c + nω_m)t): sidebands at every multiple of f_m, with Bessel function amplitudes.

• Narrowband FM (β ≪ 1): like AM, bandwidth ≈ 2f_m.
• Carson's rule: B ≈ 2(Δf + f_m) = 2f_m(β + 1).
• At certain β (2.405, 5.52, …) J₀(β) = 0: the carrier vanishes.
• Commercial FM broadcast: Δf = 75 kHz, f_m = 15 kHz → B ≈ 180 kHz.

Generation and detection

• Direct (VCO, varactor) and indirect (Armstrong) methods: NBFM followed by frequency multiplication. Frequency multiplication by n multiplies both f_c and Δf by n; mixing shifts f_c without changing Δf.
• Detection: slope detector, Foster–Seeley discriminator, ratio detector, PLL.

Noise

• For a tone, wideband FM gives an output SNR of about (3/2)β² times the baseband SNR, as long as the input stays above the threshold. Below threshold, performance collapses.
• Noise at the FM detector output rises with frequency (parabolic PSD), so FM uses pre-emphasis (boost high frequencies before transmission) and de-emphasis at the receiver.

Worked example

An FM signal has Δf = 50 kHz and f_m = 10 kHz.

• β = 5; Carson bandwidth = 2(50 + 10) = 120 kHz.

Practice