Digital Sound & Music: Concepts, Applications, & Science, Chapter 6, last updated 6/25/2013
59
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
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