Digital Sinusoid Generators

In [178], three techniques were examined for generating sinusoids digitally by means of recursive algorithms.^F.10 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,^F.11which 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:

$$\begin{array}{crcll} (1) & x_1(n) &=& c_nx_1(n-1) + s_nx_2(n... ..._2(n-1) & \\ & x_2(n) &=& x_1(n-1) & \mbox{(DFR)} \end{array}$$

where $c_n\isdef \cos(2\pi f_n T)$, $s_n\isdef \sin(2\pi f_n T)$, $f_n$ is the instantaneous frequency of oscillation (Hz) at time sample $n$, and $T$ is the sampling period in seconds. The magic circle parameter is $\epsilon=2\sin(\pi f_n T)$.

The digital waveguide oscillator appears to have the best overall properties yet seen for VLSI implementation. This structure, introduced in [487], may be derived from the theory of digital waveguides (see Appendix F, particularly §F.14, and [459,491]). 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.