# Frequency modulation synthesis

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 A 220 Hz carrier tone fc modulated by a 440 Hz modulating tone fm, with various choices of 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. As a result, digital FM synthesis (using the more frequency-stable phase modulation variant) was the basis of Yamaha's groundbreaking DX7, which 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, 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 from Yamaha, with Yamaha's flagship DX7 synthesizer being ubiquitous throughout the 1980s and several other models by Yamaha providing variations of FM synthesis. The most advanced FM synths produced by Yamaha were the 6-operator keyboard SY99 and the 8-operator module FS1R: each features Yamaha's Advanced FM (AFM) alongside and able to be layered or interfaced with other synthesising technologies, respectively AWM2 (Advanced Wave Memory 2) sample-based synthesis in the SY99 and formant synthesis in the FS1R, neither of which combinations have ever been duplicated, as neither have some of the other advanced FM features of these Yamaha devices.

Yamaha had patented its hardware implementation of FM in the 1980s, allowing it 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, and the model 208 (Music Easel) had a modulation oscillator hard-wired to allow FM as well as AM of the primary oscillator. These early applications used analog oscillators.

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".[2] 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.

However, new hardware synths that were specifically marketed for their FM capabilities have not been seen since the Yamaha SY99, which itself was split evenly between this and sample-based synthesis in terms of marketing. Yamaha's later FS1R was marketed chiefly as a device for formant synthesis, despite the fact that it was the only standalone 8-operator FM synth that they, or any other hardware manufacturer, had produced, and had other unique features in its FM engine. Yamaha also produced the FVX1, an 8-op FM expander for certain of their organs, but these are rare, and information on how to actually program them has never been released or derived. Yamaha re-released the engine of the DX7 II in the form of the PLG150DX and PLG100DX expansion cards for certain of their synths, including a standalone model enclosed in the table-top unit DX200, but this was simply a re-release of an existing platform. Thus, since the SY99 and FS1R by Yamaha, no manufacturer has released any new hardware synth dedicated solely/mainly to FM synthesis, though it is a main feature of Nord Lead synths manufactured by Clavia.

## Spectral analysis

The spectrum generated by FM synthesis with one modulator is expressed as following:[3][4]

\begin{align} &\sin(f_c\cdot t+I\cdot\sin(f_m\cdot t)) \\ &\quad = J_0(I)\sin(f_c\cdot t) + \sum_{k=1}^{\infty} J_k(I)\left[\sin(f_c+k f_m)t+(-1)^{k}\sin(f_c-k f_m)t\right] \end{align}
where $f_c\,,\,f_m\,$ are frequencies of carrier and modulator, $I\,$ is modulation index, and $J_k(I)\,$  is  $k\,$-th Bessel function of first kind, respectively.

## References

1. ^ Rob Hordijk. "FM synthesis on Modular". Nord Modular & Micro Modular V3.03 tips & tricks. Clavia DMI AB.
2. ^ Stanford University News Service (06/07/94), Music synthesis approaches sound quality of real instruments
3. ^ Chowning 1973, pp. 1–2
4. ^ Doering, Ed. "Frequency Modulation Mathematics". Retrieved 2013-04-11.