Synthesis of Classic Waveforms

A common approach to the production of sounds with electronic devices (including computers) is subtractive synthesis, in which audio signals with rich spectra are passed through filters, the parameters of which are typically modulated over time. The most common audio signals used in subtractive synthesis are noise and the following “classic waveforms”: impulse trains, square, sawtooth, and triangular waveforms.

As noted in a previous section, periodic waveforms that have time-domain discontinuities in shape or slope have spectral recipes with an infinite number of sinusoidal components. All of the classic waveforms previously mentioned have time-domain discontinuities. Naive approaches to synthesize these waveforms that produce signal discontinuities should thus be expected to alias in a discrete-time signal processing context. To avoid this aliasing, 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.