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The purpose of these plugins is to provide band-limited (alias-free) sawtooth,
square, variable pulse and variable-slope triangle oscillators for use in
LADSPA-aware audio applications. The harmonic content is to match that of the
ideal waveform as closely as possible over the required range of pitches
(between 0 and half of the sampling rate).


Aliasing

Aliasing happens when you sample a signal containing frequencies that are above
half of the sample rate, aka the Nyquist rate. The result is that the frequency
appears as a different frequency (an alias frequency) in the result.


Bandlimiting

The obvious way to generate a waveform is to literally 'draw' its shape as a
function of time.

A simple example is generating a sine wave: for each interval of time, t, the
amplitude, a = sin(t). This will present no problems, as a sine wave only
contains one harmonic, its fundamental. Any pitch up to Nyquist will still
sound like a sinewave, as no higher harmonics are present.

However, consider generating a new waveform with an additional harmonic at
double the fundamental (a = sin(t) + 0.5*sin(2t)) The presence of this
additional harmonic will limit the pitch the wave can be played at to 1/4 of
the sampling rate - any pitch above this, and the higher harmonic will alias.

A perfect sawtooth wave, drawn using a simple slope function, will contain all
harmonics and therefore the wave cannot be played at any pitch without some
harmonics aliasing. At low frequencies, the noise will be fairly quiet, but
pitch it up and it will become more easily perceptible, as more of the
prominent harmonics pass Nyquist and alias.


Ideal Solutions

Instead of using drawing the waveshape directly, a wave can be approximated
using a finite number of terms from a Fourier Series:

a = Sum[h = 1, max_h] (1/h) sin(ht)

will give a sawtooth wave, with max_h harmonics.

To generate the waveforms in realtime using this method would require a fairly
powerful computer, as the number of terms (sine calculations) required gets
large as pitch decreases. At 55Hz (a low A), this would require over 2 billion
such calculations per second for a sample rate of 48kHz.

An alternative is to pre-calculate the waveforms for each maximum harmonic and
store the results in a series of wavetables. For a given pitch, determine which
table should be used based on the maximum harmonic, and retrieve samples from
this table. In this case, storage space becomes an issue - a full waveform
period will require sample rate * sizeof(float) bytes, and we'll need sample
rate / 2 tables for each maximum harmonic.


Compromise

The first ideal solution cannot realistically be implemented without big
compromises on the harmonic content at lower pitches - even if this were done,
the cpu requirements would still be quite prohibitive.

The second approach can be implemented by reducing the number of tables - a
good compromise is to generate a table for each MIDI note, 0 to 127.


Refinement




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Gibbs Phenomenon




Playback of single wavetable at frequency, independent of sample count.

Fudge factors, interpolation.

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Indexing of multiple tables

Lookup, cross fading.

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Wrap up