Frequency modulation synthesis

**FM synthesis using 2 operators**
A 220 Hz carrier tone f_c modulated by a 440 Hz modulating tone f_m, with various choices of frequency modulation index, β. The time domain signals are illustrated above, and the corresponding spectra are shown below (spectrum amplitudes in dB).
Waveforms for each β
Spectra for each β

In audio and music, frequency modulation synthesis (or FM synthesis) is a form of audio synthesis where the timbre of a simple waveform (such as a square, triangle, or sawtooth) is changed by modulating its frequency with a modulator frequency that is also in the audio range, resulting in a more complex waveform and a different-sounding tone that can also be described as "gritty" if it is a thick and dark timbre. The frequency of an oscillator is altered or distorted, "in accordance with the amplitude of a modulating signal." (Dodge & Jerse 1997, p. 115)

FM synthesis can create both harmonic and inharmonic sounds. For synthesizing harmonic sounds, the modulating signal must have a harmonic relationship to the original carrier signal. As the amount of frequency modulation increases, the sound grows progressively more complex. Through the use of modulators with frequencies that are non-integer multiples of the carrier signal (i.e. non harmonic), atonal and tonal bell-like and percussive sounds can easily be created.

FM synthesis using analog oscillators may result in pitch instability. However, FM synthesis can also be implemented digitally, the latter proving to be more 'reliable' and is currently seen as standard practice. Digital FM synthesis (using the more frequency-stable phase modulation variant) was the basis of several commercial musical instruments beginning as early as 1977. The Synclavier I, manufactured by New England Digital Corporation beginning in 1977, included a 32-voice digital FM synthesizer. Yamaha's groundbreaking DX7 brought FM to the forefront of synthesis in the mid-1980s.

History

The technique of the digital implementation of frequency modulation, which was developed by John Chowning (Chowning 1973, cited in Dodge & Jerse 1997, p. 115) at Stanford University in 1967-68, was patented in 1975 and later licensed to Yamaha.

The implementation commercialized by Yamaha (US Patent 4018121 Apr 1977 or U.S. Patent 4,018,121) is actually based on phase modulation, but the results end up being equivalent mathematically as both are essentially a special case of QAM, with phase modulation simply making the implementation resilient against undesirable drift in frequency of carrier waves due to self-modulation or due to DC bias in the modulating wave.^[1]

As noted earlier, FM synthesis was the basis of some of the early generations of digital synthesizers, most notable being those from New England Digital Corporation and Yamaha. Yamaha's popular DX7 synthesizer was ubiquitous throughout the 1980s. Several other models by Yamaha provided variations and evolutions of FM synthesis during that decade.

Yamaha had patented its hardware implementation of FM in the 1980s, allowing it and New England Digital Corporation (under license from Yamaha) to nearly monopolize the market for that technology until the mid-1990s. Casio developed a related form of synthesis called phase distortion synthesis, used in its CZ range of synthesizers. It had a similar (but slightly differently derived) sound quality to the DX series. Don Buchla implemented FM on his instruments in the mid-1960s, prior to Yamaha's patent. His 158, 258 and 259 dual oscillator modules had a specific FM control voltage input,^[2] and the model 208 (Music Easel) had a modulation oscillator hard-wired to allow FM as well as AM of the primary oscillator.^[3] These early applications used analog oscillators, and this capability was also followed by other modular synthesizers and portable synthesizers including Minimoog and ARP Odyssey.

With the expiration of the Stanford University FM patent in 1995, digital FM synthesis can now be implemented freely by other manufacturers. The FM synthesis patent brought Stanford $20 million before it expired, making it (in 1994) "the second most lucrative licensing agreement in Stanford's history".^[4] FM today is mostly found in software-based synths such as FM8 by Native Instruments or Sytrus by Image-Line, but it has also been incorporated into the synthesis repertoire of some modern digital synthesizers, usually coexisting as an option alongside other methods of synthesis such as subtractive, sample-based synthesis, additive synthesis, and other techniques. The degree of complexity of the FM in such hardware synths may vary from simple 2-operator FM, to the highly flexible 6-operator engines of the Korg Kronos and Alesis Fusion, to creation of FM in extensively modular engines such as those in the latest synthesisers by Kurzweil Music Systems.

New hardware synths specifically marketed for their FM capabilities have not been seen since the Yamaha SY99 and FS1R, and even those marketed their highly powerful FM abilities as counterparts to sample-based synthesis and formant synthesis respectively. However, well-developed FM synthesis options are a feature of Nord Lead synths manufactured by Clavia, the Alesis Fusion range, the Korg Oasys and Kronos and the Modor NF-1. Various other synthesizers offer limited FM abilities to supplement their main engines.

Spectral analysis

The spectrum generated by FM synthesis with one modulator is expressed as follows:^[5]^[6]

For modulation signal $m(t) = B\,\sin(\omega_m t)\,$ , the carrier signal is

\begin{align} FM(t) & \ =\ A\,\sin\left(\,\int_0^t \left(\omega_c + B\,\sin(\omega_m\,\tau)\right)d\tau\right) \\ & \ =\ A\,\sin\left(\omega_c\,t - \frac{B}{\omega_m}\left(\cos(\omega_m\,t) - 1\right)\right) \\ & \ =\ A\,\sin\left(\omega_c\,t + \frac{B}{\omega_m}\left(\sin(\omega_m\,t - \pi/2) + 1\right)\right) \\ \end{align}

If we were to ignore the constant phase terms on the carrier $\phi_c = B/\omega_m\,$ and the modulator $\phi_m = - \pi/2\,$ , finally we would get the following expression, as seen on Chowning 1973 and Roads 1996, p. 232:

\begin{align} FM(t) & \ \approx\ A\,\sin\left(\omega_c\,t + \beta\,\sin(\omega_m\,t)\right) \\ & \ =\ A\left( J_0(\beta) \sin(\omega_c\,t) + \sum_{n=1}^{\infty} J_n(\beta)\left[\,\sin((\omega_c+n\,\omega_m)\,t)\ +\ (-1)^{n}\sin((\omega_c-n\,\omega_m)\,t)\,\right] \right) \\ & \ =\ A\sum_{n=-\infty}^{\infty} J_n(\beta)\,\sin((\omega_c+n\,\omega_m)\,t) \end{align}

where $\omega_c\,,\,\omega_m\,$ are angular frequencies ( $\,\omega = 2\pi f\,$ ) of carrier and modulator, $\beta = B / \omega_m\,$ is frequency modulation index, and amplitudes $J_n(\beta)\,$ is $n\,$ -th Bessel function of first kind, respectively.^{[note 1]}

Footnote

↑ above expression is transformed using trigonometric addition formulas
$\begin{align} \sin(x \pm y) &= \sin x \cos y \pm \cos x \sin y \end{align}$
and a lemma of Bessel function,
$\begin{align} \cos(\beta\sin \theta) & = J_0(\beta) + 2\sum_{n=1}^{\infty}J_{2n}(\beta)\cos(2n\theta) \\ \sin(\beta\sin \theta) & = 2\sum_{n=0}^{\infty}J_{2n+1}(\beta)\sin((2n+1)\theta) \end{align}$

(Source: Kreh, Martin (2012), "Bessel Function" (PDF), Gottingen Summer School on Number Theory, Gottingen, Germany, July 29, 2012–August 18, 2012, pp. 5–6 )
as following:
$\begin{align} & \sin\left(\theta_c + \beta\,\sin(\theta_m)\right) \\ & \ =\ \sin(\theta_c)\cos(\beta\sin(\theta_m)) + \cos(\theta_c)\sin(\beta\sin(\theta_m)) \\ & \ =\ \sin(\theta_c)\left[J_0(\beta) + 2\sum_{n=1}^{\infty}J_{2n}(\beta)\cos(2n \theta_m)\right] + \cos(\theta_c)\left[2\sum_{n=0}^{\infty}J_{2n+1}(\beta)\sin((2n+1)\theta_m)\right] \\ & \ =\ J_0(\beta) \sin(\theta_c) + J_1(\beta) 2\cos(\theta_c)\sin(\theta_m) + J_2(\beta) 2\sin(\theta_c)\cos(2\theta_m) + J_3(\beta) 2\cos(\theta_c)\sin(3\theta_m) + ... \\ & \ =\ J_0(\beta) \sin(\theta_c) + \sum_{n=1}^{\infty} J_n(\beta)\left[\,\sin(\theta_c + n\theta_m)\ +\ (-1)^{n}\sin(\theta_c - n\theta_m)\,\right] \\ & \ =\ \sum_{n=-\infty}^{\infty} J_n(\beta)\,\sin(\theta_c + n\theta_m)\qquad(\because\ J_{-n}(x) = (-1)^n J_{n}(x)) \end{align}$

References

Chowning, J. (1973). "The Synthesis of Complex Audio Spectra by Means of Frequency Modulation" (PDF). Journal of the Audio Engineering Society 21 (7). (also available in PDF as digital version 2/13/2007)
Chowning, John; Bristow, David (1986). FM Theory & Applications - By Musicians For Musicians. Tokyo: Yamaha. ISBN 4-636-17482-8.
Roads, Curtis (1996). The Computer Music Tutorial. MIT Press. ISBN 978-0-262-68082-0.
Dodge, Charles; Jerse, Thomas A. (1997). Computer Music: Synthesis, Composition and Performance. New York: Schirmer Books. ISBN 0-02-864682-7.

↑ Rob Hordijk. "FM synthesis on Modular". Nord Modular & Micro Modular V3.03 tips & tricks. Clavia DMI AB.
↑ Dr. Hubert Howe (1960s). Buchla Electronic Music System: Users Manual written for CBS Musical Instruments (Buchla 100 Owner's Manual). Educational Research Department, CBS Musical Instruments, Columbia Broadcasting System. p. 7. At this point we may consider various additional signal modifications that we may wish to make to the series of tones produced by the above example. For instance, if we would like to add frequency modulation to the tones, it is necessary to patch another audio signal into the jack connected by a line to the middle dial on the Model 158 Dual Sine-Sawtooth Oscillator. ...
↑ Atten Strange (1974). Programming and Metaprogramming in the Electro-Organism - An Operating Directive for the Music Easel. Buchla and Associates.
↑ Stanford University News Service (06/07/94), Music synthesis approaches sound quality of real instruments
↑ Chowning 1973, pp. 1–2
↑ Doering, Ed. "Frequency Modulation Mathematics". Retrieved 2013-04-11.

External links

An Introduction To FM, by Bill Schottstaedt
FM tutorial
Synth Secrets, Part 12: An Introduction To Frequency Modulation, by Gordon Reid
Synth Secrets, Part 13: More On Frequency Modulation, by Gordon Reid
Paul Wiffens Synth School: Part 3
F.M. Synthesis including complex operator analysis
Part 1 of a 2-part YouTube tutorial on FM synthesis with numerous audio examples

Sound synthesis types

Frequency modulation synthesis Phase distortion synthesis Scanned synthesis Subtractive synthesis Additive synthesis Distortion synthesis Linear Arithmetic synthesis

Sample-based synthesis	Table-lookup synthesis Wavetable synthesis Granular synthesis Vector synthesis

Physical modelling synthesis	Digital waveguide synthesis Banded waveguide synthesis Karplus–Strong string synthesis Formant synthesis

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