Digital Sound & Music: Concepts, Applications, & Science, Chapter 6, last updated 6/25/2013

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gives the amplitude of the frequency component for each

kth

component in the

phase-modulated signal. These scaling functions are called Bessel functions of the first

kind. It's beyond the scope of the book to define these functions further. You can experiment for

yourself to see that the frequency components have amplitudes that depend on I. If you listen to

the sounds created, you'll find that the timbres of the sounds can also be caused to change over

time by changing I. The frequencies of the components, on the other hand, depend on the ratio

of . You can try varying the MATLAB commands above to experience the wide variety

of sounds that can be created with phase modulation. You should also consider the possibilities

of applying additive or subtractive synthesis to multiple phase-modulated waveforms.

Frequency Modulation (FM) 6.3.3.7

We have seen in the previous section that phase modulation can be applied to the digital

synthesis of a wide variety of waveforms. Frequency modulation is equally versatile and

frequently used in digital synthesizers. Frequency modulation is defined recursively as follows:

and

( ( )),

, and

,

,

,

Equation 6.5 Frequency modulation for digital synthesis

Frequency modulation can yield results identical to phase modulation, depending on how

inputs parameters are handled in the implementation. A difference between phase and frequency

modulation is the perspective from which the modulation is handled. Obviously, the former is

shaping a waveform by modulating the phase, while the latter is modulating the frequency. In

frequency modulation, the change in the frequency can be handled by a parameter d, an absolute

change in carrier signal frequency, which is defined by . The input parameters

, , , , , and yield the graphs shown in Figure

6.56 and Figure 6.57. We suggest that you try to replicate these results by writing a MATLAB

program based on Equation 6.5 defining frequency modulation.

Figure 6.56 Frequency modulation using two sinusoidals,

where and