Classical waveforms

Up until now we have primarily taken two approaches synthesizing repetitive waveforms: wavetable synthesis (Chapter 2) and waveshaping (Chapters 5 and 6). This chapter introduces a third approach, in which waveforms are built up explicitly from line segments with controllable endpoints. This approach is historically at least as important as either of the other two, and was dominant during the analog synthesizer period, approximately 1965-1985. For lack of a better name, we'll use the term classical waveforms to denote waveforms composed of line segments.

Figure 10.1: Classical waveforms: (a) the sawtooth, (b) the triangle, and (c) the rectangle waves, shown as functions of a real parameter (not sampled).

The waveforms include the sawtooth, triangle, and rectangle waves pictured in figure reffig10.01, among many other possibilities. As we will see, the salient features of these waveforms are their discontinuities, either jumps (changes in value) or corners (changes in slope). In the figure, the sawtooth and rectangle waves have changes in value (once per cycle for the sawtooth, and twice for the rectangle), but the slope is constant (negative for the sawtooth wave, zero for the rectangle wave). The triangle wave has no discontinuous jumps, but the slope changes discontinuously.

To use classical waveforms effectively, it is usefile to understand how the shape of the waveform is reflected in its Fourier series. (To compute them we need materials from Chapter 9, which is why this chapter appears here and not earlier.) We will also need strategies for digitally synthesizing classical waveforms. These waveforms will prove to be much more susceptible to foldover problems than any we have treated before, so we will have to pay especially close attention to its control.

In general, our strategy for dealing with sampling effects will be to consider first those sampled waveforms whose period is an integer $N$. Then if we want to obtain a waveform of a non-integral period (call it $\tau$, say) we approximate $\tau$ as a quotient $N/R$ of two integers. Conceptually at least, we can then synthesize the desired waveform with period $N$, and then take only one of each $R$ samples of output. This last, down-sampling step is where the foldover is produced, and a careful analysis of it will help us control it.