 |
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.
FM Equation
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.

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.
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(θ−kβ)
]
sin(θ+asinβ)=
J
0
(a)sinθ+
∑
k=1
∞
J
k
(a)[
sin(θ+kβ)+
(−1)
k
sin(θ−kβ)
]
(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.
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.
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.
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
N≥1
N≥1
, 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.

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.
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.
"This online course covers signal processing concepts using music and audio to keep the subject relevant and interesting. Written by Prof. Ed Doering from the Rose-Hulman Institute of Technology, […]"