Wavetable synthesis is a popular sound synthesis method enabling the efficient creation of musical sounds. Using sample rate conversion techniques, arbitrary musical pitches can be generated from one wavetable or from a small set of wavetables: downsampling is used for raising the pitch and upsampling for lowering it. A challenge when changing the pitch of a sampled waveform is to avoid disturbing aliasing artifacts. Besides bandlimited resampling algorithms, the use of an integrated wavetable and a differentiation of the output signal has been proposed previously by Geiger. This paper extends Geiger’s method by using several integrator and differentiator stages to improve alias-reduction. The waveform is integrated multiple times before it is stored in a wavetable. During playback, a sample rate conversion method is first applied and the output signal is then differentiated as many times as the wavetable has been integrated. The computational cost of the proposed technique is independent of the pitch-shift ratio. It is shown that the higher-order integrated wavetable technique reduces aliasing more than the first-order technique with a minor increase in computational cost. Quantization effects are analyzed and are shown to become notable at high frequencies, when several integration and differentiation stages are used.

This paper was published in the Proceedings of the 15th International Conference on Digital Audio Effects (DAFx-12), York, UK, September 17-21, 2012, pp. 245-252.
A revised and extended version of this paper entitled "Higher-Order Integrated Wavetable and Sampling Synthesis" was published in the Journal of the Audio Engineering Society, vol. 61, no. 9, pp. 624-636, September 2013.

This video shows the spectrum of a trivial sawtooth signal, when its fundamental is swept from 440 Hz to 3.5 kHz. Aliasing is visible as extra spectral components:

• Video: aliasing in the trivial sawtooth signal sweep

The following sawtooth tones (F0 = 1245 Hz) are related to Fig. 5 of the paper. K denotes the order of integration and N is the order of the Lagrange interpolator used:

(a) Ideal bandlimited sawtooth signal
(b) Trivial table-lookup: K = 0, N = 0
(c) Geiger's scheme: K = 1, N = 0
(d) K = 2, N = 0
(e) K = 2, N = 1
(f) K = 3, N = 1
(g) K = 4, N = 1
(h) K = 4, N = 3
(i) K = 5, N = 3
(j) K = 6, N = 3

The following sequence of 2-sec long sawtooth tones (F0 = 1245 Hz) is related to Fig. 6 of the paper. When the integration order is K = 4 and the interpolation order is N = 3, the quantization noise becomes audible when the number of bits is reduced:

• No quantization, 24 bits, 20 bits, 16 bits, 12 bits

There are more audio and video examples available at:
http://www.idmt.fraunhofer.de/en/higher_order_integrated_wavetable_synthesis.html