### 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

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.

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}

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

The last expression can be interpreted as the Fourier superposition of the sinusoidal harmonics of ,

*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 times that for (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.

#### 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

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),

^{4.11}

*i.e.*, expressing a real-valued FM signal as the real part of a more analytically tractable complex-valued FM signal, we obtain

re | |||

re | |||

re | |||

re | |||

(4.9) |

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 Filters

**Previous Section:**

Sinusoidal Amplitude Modulation (AM)