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Frequency Modulation (FM) Mathematics

Module by: Ed Doering. E-mail the authorEdited By: Erik Luther, Sam Shearman

Summary: Frequency modulation (FM) in the audio frequency range can create very rich spectra from only two sinusoidal oscillators, and the spectra can easily be made to evolve with time. The mathematics of FM synthesis is developed, and the spectral characteristics of the FM equation are discussed. Audio demonstrations as implemented by LabVIEW VIs illustrate the relationships between the three fundamental FM synthesis parameters (carrier frequency, modulation frequency, modulation index) and the synthesized spectra.

Table 1
LabVIEWq.png This module refers to LabVIEW, a software development environment that features a graphical programming language. Please see the LabVIEW QuickStart Guide module for tutorials and documentation that will help you:
• Apply LabVIEW to Audio Signal Processing
• Get started with LabVIEW
• Obtain a fully-functional evaluation edition of LabVIEW

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 communications systems 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 radio band 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, and the 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 (Figure 1), the first commercially successful instrument based on FM synthesis. The DX7 was also a milestone by introducing two other new technologies: digital synthesis and MIDI (Musical Instrument Digital Interface). The "FM sound" defines much of the pop music styles of the 1980s.

Figure 1: Yamaha DX7 synthesizer, the first commercially successful instrument to offer FM synthesis, digital synthesis, and MIDI compatibility. The instrument pictured here is packaged in a road case. Photographer: schoschie (http://www.flickr.com/photos/schoschie/51653026/). Copyright holder has granted permission to display this image under the Creative Commons Attribution-ShareAlike license.
Figure 1 (51653026_5d7e9ba315_o.jpg)

FM Equation

The basic FM equation is presented in Equation 1:

y(t)=Asin(2π f c t+Isin(2π f m t)), y(t)=Asin(2π f c t+Isin(2π f m t)),
(1)

where the parameters are defined as follows:

  • f c = f c = carrier frequency (Hz)
  • f m = f m = modulation frequency (Hz)
  • I= I= modulation index

The Figure 2 screencast video continues the discussion by explaining the significance of each part of Equation 1, and demonstrates in a qualitative fashion how the different parameters of the equation influence the spectrum of the audio signal.

LabVIEW.png Download the LabVIEW VI demonstrated in the video: fm_demo1.vi. Refer to TripleDisplay to install the front-panel indicator used to view the signal spectrum.

Figure 2: [video] Significance of each part of the basic FM equation, and audio demonstration
Figure 2 (mod_fm-math-eqn.html)

FM Spectrum

The following trigonometric identity facilitates quantitative understanding of the spectrum produced by the basic FM equation of Equation 1:

sin(θ+asinβ)= J 0 (a)sinθ+ k=1 J k (a)[ sin(θ+kβ)+ (1) k sin(θkβ) ] sin(θ+asinβ)= J 0 (a)sinθ+ k=1 J k (a)[ sin(θ+kβ)+ (1) k sin(θkβ) ]
(2)

The term J k (a) J k (a) defines a Bessel function of the first kind of order k k evaluated at the value a a .

Note how the left-hand side of the identity possesses exactly the same form as the basic FM equation of Equation 1. Therefore, the right-hand side of the identity explains where the spectral components called sidebands are located, and indicates the amplitude of each spectral component. The Figure 3 screencast video continues the discussion by explaining the significance of each part of Equation 2, especially the location of the sideband spectral components.

Figure 3: [video] Trig identity of Equation 2 and location of sideband spectral components
Figure 3 (mod_fm-math-sidebands.html)

As discussed Figure 3 video, the basic FM equation produces an infinite number of sideband components; this is also evident by noting that the summation of Equation 2 runs from k=1 to infinity. However, the amplitude of each sideband is controlled by the Bessel function, and non-zero amplitudes tend to cluster around the central carrier frequency. The Figure 4 screencast video continues the discussion by examining the behavior of the Bessel function J k (a) J k (a) when its two parameters are varied, and shows how these parameters link to the modulation index and sideband number.

Figure 4: [video] Discussion of the Bessel function J k (a) J k (a) and its relationship to modulation index and sideband number
Figure 4 (mod_fm-math-bessel.html)

Now that you have developed a better quantitative understanding of the spectrum produced by the basic FM equation, the Figure 5 screencast video revisits the earlier audio demonstration of the FM equation to relate the spectrum to its quantitative explanation.

Figure 5: [video] FM audio demonstration revisited
Figure 5 (mod_fm-math-eqn-revisit.html)

Harmonicity Ratio

The basic FM equation generates a cluster of spectral components centered about the carrier frequency

f c f c with cluster density controlled by the modulation frequency

f m f m . Recall that we perceive multiple spectral components to be a single tone when the components are located at integer multiples of a fundamental frequency, otherwise we perceive multiple tones with different pitches. The harmonicity ratio H H provides a convenient way to choose the modulation frequency to produce either harmonic or inharmonic tones. Harmonicity ratio is defined as:

H= f m f c H= f m f c
(3)

Harmonicity ratios that involve an integer, i.e., H=N H=N or H=1/N H=1/N for N1 N1 , result in sideband spacing that follows a harmonic relationship. On the other hand, non-integer-based harmonicity ratios, especially using irrational numbers such as π π and 2 2 , produce interesting inharmonic sounds.

LabVIEW.png Try experimenting with the basic FM equation yourself. The LabVIEW VI fm_demo2.vi provides front-panel controls for carrier frequency, modulation index, and harmonicity ratio. You can create an amazingly wide variety of sound effects by strategically choosing specific values for these three parameters. The Figure 6 screencast video illustrates how to use the VI and provides some ideas about how to choose the parameters. Refer to TripleDisplay to install the front-panel indicator used to view the signal spectrum.

Figure 6: [video] Demonstration of fm_demo2.vi
Figure 6 (mod_fm-math-demo2.html)

References

  • Moore, F.R., "Elements of Computer Music," Prentice-Hall, 1990, ISBN 0-13-252552-6.
  • Dodge, C., and T.A. Jerse, "Computer Music: Synthesis, Composition, and Performance," 2nd ed., Schirmer Books, 1997, ISBN 0-02-864682-7.

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