Sinusoidal Frequency Modulation (FM)
Frequency Modulation (FM) is well known as
the broadcast
signal format for FM radio. It is also the basis of the
first commercially successful method for
digital sound synthesis.
Invented by John Chowning [
14], it was the method used in
the the highly successful Yamaha
DX-7 synthesizer, and later the
Yamaha OPL chip series, which was used in all ``SoundBlaster
compatible'' multimedia sound cards for many years. At the time of
this writing, descendants of the OPL chips remain the dominant
synthesis technology for ``ring tones'' in cellular telephones.

A general formula for frequency modulation of one
sinusoid by another
can be written as
![$\displaystyle x(t) = A_c\cos[\omega_c t + \phi_c + A_m\sin(\omega_m t + \phi_m)], \protect$](http://www.dsprelated.com/josimages_new/mdft/img516.png) |
(4.5) |
where the parameters

describe the
carrier sinusoid, while the parameters

specify the
modulator sinusoid. Note that, strictly speaking,
it is not the frequency of the carrier that is modulated sinusoidally,
but rather the
instantaneous phase of the carrier. Therefore,
phase modulation would be a better term (which is in fact used).
Potential confusion aside, any modulation of phase implies a
modulation of frequency, and vice versa, since the instantaneous
frequency is always defined as the time-derivative of the
instantaneous phase. In this book, only phase modulation will be
considered, and we will call it FM, following common
practice.
4.8
Figure
4.14 shows a unit generator patch diagram [
42]
for brass-like FM synthesis. For brass-like sounds, the modulation
amount increases with the amplitude of the signal. In the patch, note
that the amplitude
envelope for the carrier
oscillator is scaled and
also used to control amplitude of the modulating oscillator.
figure[htbp]
It is well known that sinusoidal frequency-modulation of a sinusoid
creates sinusoidal components that are uniformly spaced in frequency
by multiples of the modulation frequency, with amplitudes given by the
Bessel functions of the first kind [
14].
As a special case, frequency-modulation of a sinusoid by itself
generates a
harmonic spectrum in which the

th
harmonic amplitude is
proportional to

, where

is the
order of the
Bessel function and

is the
FM index. We will derive
this in the next section.
4.9
Bessel Functions
The
Bessel functions of the first kind may be defined as the
coefficients

in the two-sided
Laurent expansion
of the so-called
generating function
[
84, p. 14],
4.10
 |
(4.6) |
where

is the integer
order
of the Bessel function, and

is its argument (which
can be complex, but we will only consider real

).
Setting

, where

will interpreted as the
FM modulation frequency and

as time in seconds, we obtain
 |
(4.7) |
The last expression can be interpreted as the Fourier superposition of the
sinusoidal harmonics of
![$ \exp[j\beta\sin(\omega_m t)]$](http://www.dsprelated.com/josimages_new/mdft/img528.png)
,
i.e., an
inverse Fourier series sum. In other words,

is
the amplitude of the

th
harmonic in the
Fourier-series expansion of
the
periodic signal 
.
Note that

is real when

is real. This can be seen
by viewing Eq.

(
4.6) as the product of the
series expansion for
![$ \exp[(\beta/2) z]$](http://www.dsprelated.com/josimages_new/mdft/img529.png)
times that for
![$ \exp[-(\beta/2)/z]$](http://www.dsprelated.com/josimages_new/mdft/img530.png)
(see footnote
pertaining to Eq.

(
4.6)).
Figure
4.15 illustrates the first eleven Bessel functions of the first
kind for arguments up to

. It can be seen in the figure
that when the FM index

is zero,

and

for
all

. Since

is the amplitude of the carrier
frequency, there are no side bands when

. As the FM index
increases, the sidebands begin to grow while the carrier term
diminishes. This is how
FM synthesis produces an expanded, brighter
bandwidth as the FM index is increased.
Figure 4.15:
Bessel functions of the first kind
for a range of orders
and argument
.
![\includegraphics[width=\twidth]{eps/bessel}](http://www.dsprelated.com/josimages_new/mdft/img537.png) |
FM Spectra
Using the expansion in Eq.

(
4.7), it is now easy to determine
the
spectrum of
sinusoidal FM. Eliminating scaling and
phase offsets for simplicity in Eq.

(
4.5) yields
![$\displaystyle x(t) = \cos[\omega_c t + \beta\sin(\omega_m t)], \protect$](http://www.dsprelated.com/josimages_new/mdft/img538.png) |
(4.8) |
where we have changed the modulator amplitude

to the more
traditional symbol

, called the
FM index in FM sound
synthesis contexts. Using
phasor analysis (where
phasors
are defined below in §
4.3.11),
4.11i.e., expressing a real-valued FM
signal as the real part of a more
analytically tractable complex-valued FM signal, we obtain
where we used the fact that

is real when

is real.
We can now see clearly that the sinusoidal FM spectrum consists of an
infinite number of side-bands about the carrier frequency

(when

). The side bands occur at multiples of the
modulating frequency

away from the carrier frequency

.
Next Section: Analytic Signals and Hilbert Transform FiltersPrevious Section: Sinusoidal Amplitude Modulation (AM)