Classical waveforms

Up until now we have primarily taken three approaches to synthesizing
repetitive waveforms: additive synthesis (Chapter 1),
wavetable synthesis (Chapter 2), and waveshaping
(Chapters 5 and 6). This chapter
introduces a fourth
approach, in which waveforms are built up explicitly from line segments with
controllable endpoints. This approach is historically at least as important
as the others, 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.

They include the
*sawtooth*, *triangle*, and *rectangle* waves pictured in Figure
10.1, among many other possibilities. The salient features of
classical waveforms are either discontinuous jumps (changes in value) or
corners (changes in slope). In the figure, the sawtooth and rectangle waves
have jumps (once per cycle for the sawtooth, and twice for the rectangle), and
constant slope elsewhere (negative for the sawtooth wave, zero for the
rectangle wave). The triangle wave has no discontinuous jumps, but the slope
changes discontinuously twice per cycle.

To use classical waveforms effectively, it is useful to understand how the shape of the waveform is reflected in its Fourier series. (To compute these we need background 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 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 predicting and controlling foldover will be to consider first those sampled waveforms whose period is an integer . Then if we want to obtain a waveform of a non-integral period (call it , say) we approximate as a quotient of two integers. Conceptually at least, we can then synthesize the desired waveform with period , and then take only one of each samples of output. This last, down-sampling step is where the foldover is produced, and careful handling will help us control it.

- Symmetries and Fourier series

- Dissecting classical waveforms
- Fourier series of the elementary waveforms
- Sawtooth wave
- Parabolic wave
- Square and symmetric triangle waves
- General (non-symmetric) triangle wave

- Predicting and controlling foldover

- Examples

- Exercises