6.3 Frequency modulation (fm) mathematics

Overview

Frequency modulation ( FM ) is most often associated with communications systems; for example, you can find all sorts of music stations on the FM band of your radio. In communicationssystems the baseband signal has a bandwidth similar to that of speech or music (anywhere from 8 kHz to 20 kHz), and the modulating frequency is several orders of magnitude higher; the FM radioband is 88 MHz to 108 MHz.

When applied to audio signals for music synthesis purposes, the modulating frequency is of the same order as the audio signals to be modulated. FM can create very rich spectra, andthe spectra can easily be made to evolve with time. The ability of FM to produce a wide variety of interesting spectra from only two sinusoidal oscillators makes FM a fascinating synthesis technique.

Brief history of fm synthesis

John Chowning was the first to systematically evaluate FM in the audio spectrum, and along with Stanford University, filed for a patent on the technique in 1975 (see U.S. Patent 4,018,121 at U.S. Patent and Trademark Office or at Google Patent Search ). The patent was issued in 1977, and Stanford University licensed the technology to Yamaha Corporation. Six years later in 1983, Yamaha introduced the revolutionary DX7 synthesizer ( ), the first commercially successful instrument based on FM synthesis. The DX7 was also a milestone by introducing two other new technologies: digitalsynthesis and MIDI (Musical Instrument Digital Interface). The "FM sound" defines much of the pop music styles of the 1980s.

Fm equation

The basic FM equation is presented in :

where the parameters are defined as follows:

• $f_c=$ carrier frequency (Hz)
• $f_m=$ modulation frequency (Hz)
• $I=$ modulation index

The screencast video continues the discussion by explaining the significance of each part of , and demonstrates in a qualitative fashion how the different parameters of the equation influence the spectrum of the audio signal.