### 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:

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