Bandlimited Synthesis of Classic Waveforms

As noted in our earlier discussion of additive synthesis and Fourier theory, any complex waveform can be decomposed into a (possibly infinite) sum of sinusoids, each with its own amplitude, frequency, and phase parameters. In general, waveforms with time-domain discontinuities in shape or slope require an infinite number of sinusoids to be perfectly reconstructed. There are a number of periodic signals commonly used in sound synthesis that exhibit such discontinuities, including impulse trains, square, sawtooth, and triangular waveforms. To avoid aliasing in a digital synthesis context, it is necessary to determine bandlimited approximations for these signals so that their spectral content does not exceed half the sample rate. A more theoretical analysis of this topic is found in Alias-Free Digital Synthesis of Classic Analog Waveforms by Tim Stilson and Julius Smith. Another approach (not discussed here) to this problem is found in Hard Sync Without Aliasing by Eli Brandt.

Finally, a more recent variant of the BLIT approach described below that includes control of lowpass filter cutoff frequency and stop-band roll-off is described in LP-BLIT: Bandlimited Impulse Train Synthesis of Lowpass-Filtered Waveforms by Sebastian Kraft and Udo Zölzer.