AM and FM Synthesis

Some of the most important developments in early digital sound synthesis derived from extensions of the oscillator, through time-variation of the control parameters at audio rates.

AM, or amplitude modulation synthesis, in continuous time, and employing a sinusoidal carrier (of frequency $f_{0}$), and modulator (of frequency $f_{1}$) generates a waveform of the following form:

$$u(t) = \left(A_{0}+A_{1}\cos(2\pi f_{1}t)\right)\cos(2\pi f_{0}t)$$ (1.4)

where $A_{0}$ and $A_{1}$ are free parameters. The symbolic representation of AM synthesis is shown in Figure 1.5(a). Such an output consists of three components, as also shown in Figure 1.5(a), where the strength of the component at the carrier frequency is determined by $A_{0}$, and those of the side components, at frequencies $f_{0}\pm f_{1}$ by $A_{1}$. If $A_{0}=0$, then ring modulation results.

Frequency modulation (FM) synthesis, the result of a serendipitous discovery by John Chowning at Stanford in the late 1960s, was perhaps the greatest single breakthrough in digital sound synthesis [49]. Instantly, it became possible to generate a wide variety of spectrally rich sounds using a bare minimum of computer operations. FM synthesis requires no more computing power than a few digital oscillators, which is not surprising, considering that FM refers to the modulation of the frequency of a digital oscillator. As a result, real-time synthesis of complex sounds became possible in the late 1970s, as the technique was incorporated into various special purpose digital synthesizers--see [176] for details. In the 1980s, FM synthesis was very successfully commercialized by the Yamaha corporation, and thereafter permanently altered the synthetic soundscape.

FM synthesis, like AM, is also a direct descendant of synthesis based on sinusoids, in the sense that in its simplest manifestation, it makes use of only two sinusoidal oscillators, one behaving s a carrier, and the other as a modulator. See Figure 1.5(b). The functional form of the output, in continuous time, is usually written in terms of sine functions, and not cosines, as

$$u(t) = A_{0}(t)\sin(2\pi f_{0}t+I\sin(2\pi f_{1}t))$$ (1.5)

where here, $f_{0}$ is the carrier frequency, $f_{1}$ the modulation frequency, and $I$ the so-called modulation index. It is straightforward to show [49] that the spectrum of this signal will exhibit components at frequencies $f_{0}+ q f_{1}$, for integer $q$, as illustrated in Figure 1.5(b). The modulation index $I$ determines the strengths of the various components, which can vary in a rather complicated way, depending on the values of associated Bessel functions. Clearly, $A_{0}(t)$ can be used to control the envelope of the resulting sound.

In fact, a slightly better formulation of the output waveform (1.5) is:

$$u(t) = A_{0}(t)\sin(2\pi\int_{0}^{t} f_{0}+If_{1}\cos(2\pi f_{1}t') dt')$$ (1.6)

where the instantaneous frequency at time $t$ may be seen to be (or rather defined as) $f_{0}+If_{1}\cos(2\pi f_{1}t')$. The quantity $If_{1}$ is often referred to as the peak frequency deviation, and written as $\Delta f$ [146]. Though this is a subtle point, and not one which will be returned to in this book, the symbolic representation in Figure 1.5(b) should be viewed in this respect.

Figure 1.5: (a) Amplitude modulation: symbolic representation, and frequency domain description of output. (b) Frequency modulation: symbolic representation and frequency domain representation of output.

FM synthesis has been extensively researched, and many variations have resulted. Among the most important are feedback configurations, useful in regularizing the behaviour of the side component magnitudes and various series and parallel multiple oscillator combinations.