### Digital Sinusoid Generators

In [168], three techniques were examined for
generating sinusoids digitally by means of recursive
algorithms.^{C.12} 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.13}which 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:

The digital waveguide oscillator appears to have the best overall properties yet seen for VLSI implementation. This structure, introduced in [460], may be derived from the theory of digital waveguides (see Appendix C, particularly §C.9, and [433,464]). 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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The Second-Order Waveguide Filter

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Additive Synthesis