New Perspectives on Distortion Synthesis for Virtual Analog Oscillators

The term "virtual analog" (VA) first appeared in the 1990s with the commercial introduction of digital synthesizer instruments that were intended to emulate the earlier analog subtractive synthesizers, such as those produced by, among others, Moog or Sequential Circuits (Smith 2008). In creating a "virtual analog" digital model, two approaches are possible: the first is to build an explicit digitalmodel derived from the various electrical components that form the original analog circuit; the second is to use digital processing structures that will produce outputs that mimic those of the analog system. To date, the second approach has been used for the implementation of the elements of virtual analog subtractive synthesizers, as real-time computational constraints limit the efficacy of circuit models in usable software instruments (Civolani and Fontana 2008). Among the features of a subtractive synthesis system, one aspect that has received more attention than others is algorithms for real-time digital generation of the periodic waveforms associated with voltage-controlled oscillators (VCO), such as sawtooths, square waves, and triangle waves, also known as the classic analog waveforms (Stilson 2006).

Although the original analog waveforms theoretically had an infinite bandwidth, the digital versions must be bandlimited to half the sampling frequency (Stilson 2006). If this is not the case, the audio output can exhibit excessive aliasing distortion, which can severely corrupt the sound quality. This distortion is manifested as audio disturbances that include inharmonicity, beating, and heterodyning. It can be particularly severe when the ratio of the waveform fundamental frequency to the sampling frequency is not an integer, as the aliased components will then fall in between the signal harmonics (Välimäki and Huovilainen 2007). Furthermore, although a crude technique such as using oversampling with a trivial waveform generator followed by downsampling to the audio rate afterwards can help to ameliorate some of the distortion (Chamberlain 1987), the drawback is that this adds significantly to the computational cost of the implementation. This then impacts on other synthesizer features such as the permissible degree of polyphony. Therefore, for VA subtractive synthesizers, a key goal currently is to develop algorithms that can generate bandlimited digital versions of the classic periodic waveforms at a low computational cost.

To date, a number of algorithms have been proposed for the production of bandlimited digital periodic waveforms. However, even though the variety of approaches appears diverse (Välimäki and Huovilainen 2007), in fact they can be unified in that all can be shown to be connected to nonlinear distortion synthesis (Dodge and Jerse 1985). Perhaps one of the most important discoveries of early digital sound synthesis, distortion-based methods dominated the research landscape of computer music in the 1970s and 1980s. A framework for these techniques was developed through the pioneering work of John Chowning on frequency modulation (FM) synthesis (Chowning 1973); Godfrey Winham, Ken Steiglitz, and Andy Moorer on discrete summation formulae (DSF; Winham and Steiglitz 1970; Moorer 1976, 1977); and Daniel Arfib and Marc LeBrun on digital waveshaping (Arfib 1978; LeBrun 1979). These techniques were demonstrated to stem from the same principles and to have interchangeable interpretations. Perhaps the key aspect that made them so interesting in the early days of computer music was their low computational cost. A number of variants, especially of FM synthesis, were keenly explored (e.g., Schottstaedt 1977; Palamin, Palamin, and Ronveaux 1988; Chowning 1989). In the 1990s, interest in these techniques diminished, even though some important novel methods were still being proposed, such as that of Puckette (1995). More recently, distortion techniques have been used in new synthesis algorithms (Lazzarini, Timoney, and Lysaght 2008b) and for audio effects (Lazzarini, Timoney, and Lysaght 2007, 2008a, 2008c).

The development of research into VA oscillator models has somewhat rekindled the interest in techniques that can be interpreted from a nonlinear distortion perspective. However, the literature has so far been limited in terms of discussing these new methods from that angle. In the following discussion, we hope to address this issue and propose new nonlinear distortion synthesis algorithms for classic analog-waveform generation. Such signals are normally represented by the sawtooth wave (containing...