In [169], three techniques were examined for generating sinusoids digitally by means of recursive algorithms.^{C.11} 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.12}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:
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.