Conclusions

A recursive algorithm was presented for digital sinusoid generation that has excellent properties for VLSI implementation. It is like the coupled form in that it offers instantaneous amplitude from its state and constant amplitude in the presence of frequency modulation. However, its implementation requires only one or two multiplies per sample instead of four.

While these properties make the new oscillator appear ideally suited for FM applications in VLSI, there are issues to be resolved regarding conversion from modulator output to carrier coefficients. Preliminary experiments indicate that FM indices less than $1$ are well behaved when the output of a modulating oscillator simply adds to the coefficient of the carrier oscillator (bypassing the exact FM formulas). Approximate amplitude normalizing coefficients have also been derived which provide a first-order approximation to the exact AM compensation at low cost. For music synthesis applications, we believe a distortion in the details of the FM instantaneous frequency trajectory and a moderate amount of incidental AM can be tolerated since they produce only second-order timbral effects in many situations.