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Digital Sinusoid
Generators
In [169], three techniques were examined for
generating sinusoids digitally by means of recursive
algorithms.C.16 The recursions can be
interpreted as implementations of second-order digital resonators in
which the damping is set to zero. The three methods considered were
(1) the
2D rotation (2DR), or
complex multiply (also called the ``coupled form''), (2) the
modified coupled form (MCF), or ``magic circle'' algorithm,C.17which is similar to (1) but with better numerical behavior, and (3)
the direct-form, second-order, digital resonator (DFR) with its poles
set to the unit circle.
These three recursions may be defined as follows:
where
,
,
is the instantaneous frequency of oscillation (Hz) at time sample
,
and
is the sampling period in seconds. The magic circle parameter
is
.
The digital waveguide oscillator appears to have the best overall
properties yet seen for VLSI implementation. This structure,
introduced in [462], may be derived from the theory of
digital waveguides (see Appendix C, particularly §C.9, and
[437,466]). Any second-order digital filter structure can
be used as a starting point for developing a corresponding sinusoidal
signal generator, so in this case we begin with the second-order
waveguide filter.
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