AM vs FM: Sidebands, Bessel Functions and Why FM Sounds Cleaner

Modulation: riding a message on top of a carrier

Radio needs a signal that can actually be transmitted through an antenna and propagate at radio frequencies — an audio waveform at a few hundred hertz to a few kilohertz has a wavelength kilometres long, needing an impractically large antenna. Modulation solves this by varying some property of a high-frequency carrier wave in proportion to a lower-frequency message signal, letting the message ride along at a frequency where practical antennas and spectrum allocation both work.

Amplitude Modulation

AM varies the carrier's amplitude in direct proportion to the message signal m(t), leaving its frequency and phase untouched:

s(t) = [A_c + m(t)] · cos(2π f_c t)
     = A_c·cos(2πf_c t) + m(t)·cos(2πf_c t)

Multiplying a signal by a cosine in the time domain is equivalent, by the modulation theorem, to shifting its entire spectrum up and down by f_c in the frequency domain. If m(t) contains frequencies up to B Hz, the modulated signal occupies a band from f_c−B to f_c+B — the message spectrum appears twice, once mirrored above the carrier (the upper sideband) and once below it (the lower sideband), each an exact (redundant) copy of the message's spectral content. Standard AM broadcasts both sidebands plus the carrier itself, which is simple to demodulate (a diode envelope detector recovers m(t) directly from the envelope of s(t)) but wastes both bandwidth and transmitted power on the redundant sideband and the carrier, which carries no information at all. Single-sideband (SSB) modulation removes one sideband and the carrier to reclaim that bandwidth and power, at the cost of a more complex receiver.

Frequency Modulation

FM instead varies the carrier's instantaneous frequency in proportion to the message, keeping amplitude constant:

s(t) = A_c · cos( 2πf_c t + 2πk_f ∫₀ᵗ m(τ) dτ )

  k_f = frequency sensitivity (Hz per unit of m)
  instantaneous frequency: f(t) = f_c + k_f·m(t)

Because information rides on frequency rather than amplitude, FM is far more resistant to the amplitude noise (static, electrical interference) that plagues AM — the entire reason FM broadcast radio sounds cleaner than AM broadcast radio. The trade-off is bandwidth: unlike AM's fixed 2B, FM's occupied bandwidth depends on both the message bandwidth and how far the frequency is allowed to swing.

Bessel functions and the FM sideband spectrum

For a single-tone message at frequency f_m, FM produces not just two sidebands but an infinite series of them, spaced at every integer multiple of f_m from the carrier, with amplitudes given by Bessel functions of the first kind, J_n(β):

s(t) = A_c · Σ_{n=−∞}^{∞}  J_n(β) · cos(2π(f_c + n·f_m)t)

  β = k_f·A_m / f_m     (modulation index)

The modulation index β is the ratio of peak frequency deviation to the message frequency, and it controls how many sidebands carry meaningful energy: J_n(β) decays rapidly once n exceeds roughly β+1, so higher-order sidebands beyond that point are negligible. Carson’s rule turns this into a practical bandwidth estimate that captures roughly 98% of the signal's power:

BW ≈ 2 (Δf + f_m) = 2 f_m (β + 1)      (Carson's rule)

  Δf = peak frequency deviation = k_f·A_m

FM broadcast radio uses a deviation of ±75 kHz against a maximum audio frequency around 15 kHz, giving β ≈ 5 and a Carson bandwidth around 180 kHz — close to the 200 kHz channel spacing actually allocated, which is why FM stations sound rich and full-bandwidth while sitting comfortably within their assigned slice of spectrum. This is also, incidentally, why FM's noise immunity is not free: spreading the same message over a wider bandwidth is precisely the mechanism (per Shannon's channel capacity formula) that buys the improved robustness to noise.

Frequently asked questions

Try it live

Everything above runs in your browser — open AM/FM Modulation and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.