Frequency Modulation
Spend bandwidth, buy signal-to-noise — at a favourable exchange rate. Then find the threshold below which the whole trade collapses.
Skip to the animationFrequency modulation varies the carrier's instantaneous frequency with the message and leaves its amplitude constant — which lets the receiver discard amplitude noise entirely, buying signal-to-noise ratio at a rate that improves as β² while bandwidth grows only as β.
The constant envelope
Louder means further from f_c, not taller. That constant amplitude is the single property from which all of FM's advantages follow — and it pays twice, once at the receiver and once at the transmitter.
An FM receiver hard-limits the signal, clipping it flat. Amplitude noise is discarded wholesale and the message is untouched, because it was never in the amplitude. AM cannot do this, since throwing away amplitude in AM throws away the message.
Deviation and index
- Frequency deviation Δf
- How far the carrier swings, set by the message amplitude. Broadcast FM uses ±75 kHz.
- Modulation index β
- Δf / f_m. A ratio, not a percentage, and it may freely exceed 1.
- Narrowband FM
- β < 1. Spectrum resembles AM's; used for two-way radio in narrow channels.
- Wideband FM
- β > 1. Broadcast FM runs near 5, which is where the noise advantage lives.
Unlike AM's index, β has no overmodulation cliff — there is no envelope to fold. The limit is the channel allocation, not the physics.
Bandwidth
FM produces an infinite set of sidebands whose amplitudes follow Bessel functions. Most carry negligible power, and Carson's rule B ≈ 2(Δf + f_m) captures about 98% of it.
For broadcast FM: Δf = 75 kHz, f_m = 15 kHz, so B ≈ 180 kHz — which is why channels are spaced at 200 kHz.
The trade, and its floor
Output SNR improves as β² while bandwidth grows only as β. FM uses roughly fifteen times AM's bandwidth and delivers far more than fifteen times the quality — a genuinely favourable exchange rate.
But it has a floor. Below about 10 dB input SNR — the threshold effect — the limiter starts locking onto noise and the output collapses. AM degrades gradually; FM is superb and then suddenly is not, which is why a weak FM station is hiss rather than a noisy signal.
Pre-emphasis
FM's demodulated noise rises with frequency, so treble suffers most. Pre-emphasis boosts the high frequencies before transmission and de-emphasis cuts them afterwards — restoring the audio while attenuating the noise added in between.
Dolby noise reduction on tape and RIAA equalisation on vinyl are the same trick applied to different noise spectra. The time constant is 50 µs in Europe and 75 µs in North America.
Where FM ended up
| Application | Why FM | Or why not |
|---|---|---|
| Broadcast music | Noise immunity, wide bandwidth available | — |
| Two-way radio | Capture effect: the stronger signal wins cleanly | — |
| Satellite links | Power-limited, not bandwidth-limited | — |
| GSM (GMSK), Bluetooth (GFSK) | Constant envelope allows an efficient saturated amplifier | — |
| Long-distance HF | — | Bandwidth too scarce; SSB wins |
| Deep space | — | Threshold effect is fatal at very low SNR |
The constant envelope pays a second time at the transmitter: a saturated class-C amplifier runs at high efficiency and would destroy an AM envelope, while leaving an FM one intact. Two wins from one property.
The numbers you will be asked for
- Instantaneous frequency
f_i = f_c + k_f · m(t)
- Modulation index
β = Δf / f_m
- Carson's rule
B ≈ 2(Δf + f_m)
- Narrowband approximation
B ≈ 2 f_m
when β ≪ 1
- SNR improvement
∝ β²
- Broadcast FM
Δf = 75 kHz · f_m = 15 kHz · B = 180 kHz
Watch it work
Check yourself
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