FM signals theoretically have infinite bandwidth — the Bessel function series never fully reaches zero. In practice, engineers need a practical rule for how much spectrum an FM signal actually occupies. Carson's rule provides exactly that: a simple formula that captures 98% of the signal's power in a single expression.
Because FM generates an infinite series of sideband pairs, a strict definition of bandwidth (the range of frequencies with nonzero power) is technically infinite. For real engineering, we need a practical bandwidth — the frequency range that contains essentially all the signal's power.
The standard convention is to include all sideband pairs whose amplitude (Bessel coefficient) exceeds 1% of the unmodulated carrier. This rule was codified by John Carson at Bell Labs in 1922, making it one of the oldest practical guidelines in radio engineering.
This can also be written using the modulation index β = Δf/f_m:
When β is very small, (β + 1) ≈ 1, so BW ≈ 2f_m. The bandwidth is approximately the same as double-sideband AM. The frequency deviation is much smaller than the message bandwidth — only the carrier and first sideband pair carry significant power. NBFM is used in voice communications where spectrum efficiency matters more than audio quality.
When β is very large, (β + 1) ≈ β, so BW ≈ 2β·f_m = 2Δf. The bandwidth is approximately twice the frequency deviation, independent of the message frequency. The signal's bandwidth is dominated by the deviation, and improving SNR (by increasing Δf) costs proportional bandwidth. This is the fundamental bandwidth-vs-noise tradeoff of wideband FM.
| Application | Δf | f_m | β | BW (Carson) |
|---|---|---|---|---|
| Broadcast FM radio | 75 kHz | 15 kHz | 5 | 180 kHz |
| Two-way voice radio | 5 kHz | 3 kHz | 1.67 | 16 kHz |
| Police/fire land-mobile (see note) | 2.5 kHz | 3 kHz | 0.83 | 11 kHz |
| Satellite telemetry | 200 kHz | 50 kHz | 4 | 500 kHz |
A note on “narrowband”. M4-L1 defines narrowband FM strictly, as β ≪ 1 (typically β < 0.3), where the bandwidth collapses to about 2fm just like AM. The land-mobile row above has β = 0.83, so it does not meet that test: Carson’s rule gives 11 kHz against 2fm = 6 kHz. The radio industry calls those channels “narrowband” because of their channel spacing (12.5 kHz instead of 25 kHz), not because β is small. Both usages are standard; they are simply measuring different things, so check which one a datasheet means.
Commercial FM check: Broadcast FM uses Δf = 75 kHz and f_m = 15 kHz (full audio bandwidth). Carson's rule gives BW = 2(75 + 15) = 180 kHz. The FCC allocates 200 kHz channels, leaving 10 kHz guard band on each side. The math works out perfectly.
The key insight of wideband FM is that you can trade bandwidth for noise performance. For a single-tone message, the standard result is that a wideband FM receiver's output SNR beats AM's by a factor of 3β²:
State what you held constant, or the formula is meaningless. A noise ratio is only defined once you say what it is measured against, and this one has three parts. (1) Both systems receive the same power — FM's win is bought with bandwidth, not watts. (2) Both carry the same message bandwidth W = fm, and the noise is referred to that message band, not to the RF channel. (3) AM is idealised as neither gaining nor losing on detection, so its output SNR is just the carrier-to-noise ratio in its own 2fm channel, SR/(2N0W).
On that basis the textbook figure of merit for tone-modulated FM — output SNR divided by the baseband reference SNR γ = SR/(N0W) — is (3/2)β², while AM's reference above is γ/2. Dividing one by the other gives (3/2)β² ÷ (1/2) = 3β². See Haykin, Communication Systems, 4th ed., ch. 2 ("Noise in CW modulation systems"), or Proakis & Salehi, Communication Systems Engineering, 2nd ed., ch. 6, for the derivation of the figure of merit.
Why you will also see 3β²(1 + β). That is the same result with the noise referred to FM's own Carson bandwidth 2fm(1 + β) instead of to the message band — which is wider by exactly the factor (1 + β), the bandwidth expansion. Neither form is wrong; quoting either without its basis is. This course always refers noise to the message band, so 3β² is the figure used everywhere in M4. And note 3β² is the conservative statement: measured against real 100%-modulated envelope AM, whose own figure of merit is m²/(2 + m²) = 1/3 (M4-L4, Axis 3), the ratio would be 4.5β², or 20.5 dB at β = 5.
This 3β² improvement factor is why FM radio sounds so much cleaner than AM radio, despite using more bandwidth. For broadcast FM with β = 5: SNR improvement = 3 × 25 = 75 (≈ 18.75 dB better than AM).
In practice, FM noise is not flat — it increases with frequency (FM noise has a parabolic spectrum shape). High audio frequencies suffer more noise than low ones. To compensate: