# Frequency modulation **Frequency modulation** (FM) is a method of impressing a message on a [[Carrier_wave|carrier wave]] by varying the carrier's instantaneous frequency in proportion to the message, while the carrier's amplitude stays constant throughout. It belongs to the angle-modulation family, alongside phase modulation, and stands apart from [[Amplitude_modulation|amplitude modulation]] precisely because a receiver looking only at the signal's height learns nothing at all; the message lives entirely in how densely the wave's cycles are packed together. The *Frequency modulation* microsim on this page drives one message through a carrier and traces the resulting instantaneous frequency directly, alongside a live readout of the bandwidth the signal actually needs. Frequency modulation is one specific case of the broader idea of [[Signal_modulation|signal modulation]], and it feeds directly into the receiver technology built to track a moving frequency, the [[Phase-locked_loop|phase-locked loop]]. Its central engineering trade is bandwidth for robustness: a frequency-modulated signal typically occupies far more spectrum than the message it carries would need on its own, and in exchange the message becomes far harder for ordinary noise to disturb, a trade the sections below make quantitative. ## FM Signal A general angle-modulated carrier is written `s(t) = A_c·cos(2π f_c t + φ(t))`, where the constant amplitude A_c never changes and the entire message rides in the phase term φ(t). The signal's instantaneous frequency — the quantity an ear or a discriminator actually perceives as "the frequency" from one moment to the next — is the rate of change of the total phase, a formal extension of the same rotating-phasor picture that describes a plain, unmodulated sinusoid.[^john022] Frequency modulation is the special case in which that instantaneous frequency, rather than the phase itself, is set directly proportional to the message; the two are related by an integral, so a phase modulator driven by a message's derivative produces exactly the same waveform as a frequency modulator driven by the message. Formally, the instantaneous frequency is defined from the [[Hilbert_transform|Hilbert transform]] of the real signal, which supplies the analytic signal whose phase derivative is unambiguous even for a message that is not a single tone. ## Sinusoidal baseband signal The clearest case to analyze is a single-tone message, `m(t) = cos(2π f_m t)`. Driving the carrier's frequency with this message, up to a peak deviation Δf away from the center frequency f_c, produces `s(t) = A_c·cos(2π f_c t + β·sin(2π f_m t))`, exactly the equation the sketch computes, with β = Δf/f_m. The instantaneous frequency traces `f_i(t) = f_c + Δf·cos(2π f_m t)`, swinging symmetrically above and below f_c at the message rate f_m — in the sketch, a dedicated panel plots this instantaneous frequency directly, alongside the message and the modulated waveform, so the reader can watch the swing rather than infer it from the waveform's crowding and thinning. ## Modulation index The modulation index β = Δf/f_m compares how far the frequency swings to how fast it swings. In the sketch, β runs from 0 to 10 against a default message frequency of 1 Hz, so the peak deviation Δf shown in the readout tracks β directly. A small index, β ≪ 1, is narrowband frequency modulation: the spectrum looks almost like an amplitude-modulated signal, with one significant sideband pair close to the carrier. A large index, β > 1, is wideband frequency modulation, the regime broadcast FM operates in, where many sideband pairs carry meaningful power and the occupied bandwidth grows well beyond twice the message frequency; the sketch marks β = 1 as the boundary between the two regimes. ## Bessel functions Expanding the wideband FM equation algebraically — the Jacobi–Anger expansion, a piece of [[Fourier_analysis|Fourier analysis]] applied to a cosine of a sinusoid's argument — turns one modulated carrier into an infinite sum of sidebands at f_c + n·f_m for every integer n, each weighted by a Bessel function of the first kind, J_n(β), evaluated at the modulation index. Unlike amplitude modulation's two sidebands, a frequency-modulated tone in principle has infinitely many, though J_n(β) falls off quickly once |n| exceeds roughly β + 1 and the sidebands beyond that carry negligible power. One consequence surprises newcomers: because J_0(β), the carrier's own weight, is an oscillating function rather than one that simply shrinks, the carrier line can vanish completely at particular values of β — the first such null falls at β ≈ 2.405 — even though the carrier itself is very much still being transmitted at full power. ## Carson's rule Carson's rule collapses that infinite sideband cascade into a single practical number: essentially all of a frequency-modulated signal's power fits within a bandwidth `BW ≈ 2(Δf + f_m) = 2·f_m·(β + 1)`, the same quantity the sketch reads out live as the two sliders move. The rule is named for the Bell System engineer John R. Carson, whose early analysis of frequency modulation is remembered in most textbook accounts as having concluded that the technique offered no fundamental bandwidth advantage over amplitude modulation — a conclusion the field only overturned once wideband FM's noise performance, not its bandwidth economy, was demonstrated to be the real advantage.[^cncarson] At narrowband indices Carson's rule collapses toward 2 f_m, the same bandwidth an amplitude-modulated tone would need; at large β it grows toward 2Δf, dominated by the deviation rather than the message rate. ## Noise reduction Because the message lives in frequency rather than amplitude, a frequency-modulated receiver can run its incoming signal through a hard limiter that clips away any amplitude variation — including most of the amplitude noise added by the channel — before a discriminator ever measures the frequency, a step with no equivalent in amplitude-modulated reception. The result is that wideband frequency modulation can trade the extra bandwidth Carson's rule demands for a substantial improvement in output [[Signal-to-noise_ratio|signal-to-noise ratio]], improving roughly with the modulation index, provided the carrier-to-noise ratio at the receiver stays above a threshold typically stated in [[Decibel|decibels]]. Edwin Armstrong's original paper on wideband frequency modulation presented exactly this noise-reducing property as the technique's central practical justification, against the prevailing assumption that it had none.[^arm1936] Below that threshold the advantage collapses abruptly rather than gracefully: instead of noise rising smoothly as in amplitude modulation, a frequency demodulator starts producing sharp clicks as the [[Noise_(electronics)|noise]] occasionally overwhelms the wanted carrier entirely. ## Implementation ### Modulation A frequency modulator can generate deviation directly, by driving an oscillator whose frequency depends on an applied voltage through a voltage-sensitive [[Semiconductor_device|semiconductor device]], or indirectly, by phase-modulating a highly stable reference oscillator and then multiplying the result up in frequency, which multiplies the achieved deviation right along with it — the historical route to a frequency-stable wideband signal before direct oscillators were stable enough on their own. Digital data can frequency-modulate a carrier the same way, switching between a small set of discrete frequencies once per symbol instead of sweeping continuously; the closest two such tones can sit while staying mathematically distinguishable is a spacing of one over twice the symbol period, exactly the spacing minimum-shift keying uses.[^ell157] ### Demodulation The oldest demodulator is a slope detector: a tuned circuit converts frequency variation into amplitude variation by riding the flank of its own resonance curve, after which an ordinary envelope detector recovers the message, at the cost of needing careful tuning and giving mediocre linearity, which shows up as harmonic [[Distortion|distortion]] on the recovered message. Modern receivers instead demodulate with a [[Phase-locked_loop|phase-locked loop]], whose internal oscillator is continuously steered to track the incoming instantaneous frequency; the steering voltage that keeps it locked is, by construction, the recovered message, and the technique tracks frequency far more linearly than a slope detector can. ## Applications Beyond broadcasting, [[Telecommunications|telecommunications]] and measurement systems put the same instantaneous-frequency idea to several other uses. The Doppler effect, in which a source's motion relative to a receiver shifts the received frequency, is itself a naturally occurring instance of frequency modulation: a [[Radar|radar]] or sonar system reads a target's velocity directly from that shift, essentially running the frequency-modulation relationship in reverse to measure motion rather than to transmit a message. Analog magnetic tape reproduces a signal's frequency far more faithfully than its amplitude at the extremes of its range, particularly near direct current, so instrumentation tape recorders and early analog videotape formats both modulated the frequency of a recording carrier instead of recording the source signal's level directly, then demodulated it back on playback. In sound synthesis, modulating one audio-frequency oscillator's frequency with a second audio-rate oscillator generates a rich cluster of sidebands that the ear fuses into a single complex tone, a direct musical use of the same Bessel-weighted sideband structure derived above. Broadcast radio remains frequency modulation's largest application, using a wide deviation, for example an 88 to 108 megahertz band in the United States, chosen specifically to buy the noise immunity that made high-fidelity, static-free music broadcasting practical. Personal FM systems built for hearing assistive technology apply the same noise immunity at a small scale: a talker's microphone frequency-modulates a short-range transmitter, and a listener's receiver recovers clear speech over background noise and distance that would otherwise defeat a hearing aid's own microphone. ## Microsims The *Frequency modulation* sketch stacks four panels: the message, the unmodulated carrier, the frequency-modulated output, and the instantaneous frequency f_i(t) traced on its own axis. Carrier frequency f_c (2–20 Hz) and message frequency f_m (0.2–4 Hz) set the two tones, and the modulation index β (0–10) sets how far the frequency swings; a live readout reports f_c, f_m, β, the resulting peak deviation Δf = β·f_m, and the Carson bandwidth 2(Δf + f_m). The space bar pauses the animation and "r" resets it. *Try:* Hold the two frequencies fixed and raise β from 0 toward 10, watching the instantaneous-frequency panel swing further from f_c and the Carson-bandwidth readout grow in step with it, well past the bandwidth a same-frequency amplitude-modulated signal would need. A three.js companion renders the same carrier as a helix whose winding tightens as the modulation index rises, tying that winding rate directly to the Carson bandwidth computed here. ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Frequency_modulation) : [Wikitube](https://en.wikitube.io/wiki/Frequency_modulation) Skeleton mirrored at revision 1371123279. Prose, emphasis and the microsims are Wikitube's own. ## See also - [[Amplitude_modulation]] - [[Signal_modulation]] - [[Phase-locked_loop]] - [[Carrier_wave]] - [[Frequency-shift_keying]] - [[Doppler_effect]] ## References [^john022]: Johnson, D. *Fundamentals of Electrical Engineering I*. 2014, pp. 21–22 (PDF pages), Eqs. 2.19 and 2.22: a sinusoid as the real part of a rotating phasor, and the phasor's complex frequency. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/fundamentals-of-electrical-engineering-1 . CC BY. [^ell157]: Ellingson, S. *Radio Systems Engineering*, Revised 1st ed. 2023, p. 157 (PDF page): minimum-shift keying's tone spacing Δf = 1/(2T), the minimum separation at which two FSK tones stay mathematically orthogonal. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/radio-systems-engineering . CC BY-NC. [^arm1936]: Armstrong, E. H. "A Method of Reducing Disturbances in Radio Signaling by a System of Frequency Modulation." *Proceedings of the Institute of Radio Engineers*, vol. 24, no. 5, May 1936, pp. 689–740. [^cncarson]: Citation needed: a primary-source citation (paper, year, and page) for John R. Carson's original bandwidth analysis of frequency modulation and the specific conclusion attributed to it here. ## Further reading - Steven Ellingson. *Radio Systems Engineering*, Revised 1st Edition (2023). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/radio-systems-engineering - Don Johnson. *Fundamentals of Electrical Engineering I* (2014). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/fundamentals-of-electrical-engineering-1 - Christian Tiberius; Max Mulder. *Engineering Signal Analysis: From Fourier to filtering: Theory* (2026). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/engineering-signal-analysis-from-fourier-to-filtering-theory ## External links - Live sketch: https://editor.p5js.org/sciencenibber/full/CK7IqVmBT - Editor source: https://editor.p5js.org/sciencenibber/sketches/CK7IqVmBT <!-- Hubs: Signal_processing. Portals: PORTAL_Signal_Processing. Signal Processing portal wave 1 · 2026-09-17 · drafted. -->