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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.13which 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:

(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)}

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 [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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