Eliminating aliasing caused by discontinuities using integrals of the sinc function

A study on the limits of bandlimited correction functions used to eliminate aliasing in audio signals with discontinuities is presented. Trivial sampling of signals with discontinuities in their waveform or their derivatives causes high levels of aliasing distortion due to the infinite bandwidth of these discontinuities. Geometrical oscillator waveforms used in subtractive synthesis are a common example of audio signals with these characteristics. However, discontinuities may also be introduced in arbitrary signals during operations such as signal clipping and rectification. Several existing techniques aim to increase the perceived quality of oscillators by attenuating aliasing sufficiently to be inaudible. One family of these techniques consists on using the bandlimited step (BLEP) and ramp (BLAMP) functions to quasi-bandlimit discontinuities. Recent work on antialiasing clipped audio signals has demonstrated the suitability of the BLAMP method in this context. This work evaluates the performance of the BLEP, BLAMP, and integrated BLAMP functions by testing whether they can be used to fully bandlimit aliased signals. Of particular interest are cases where discontinuities appear past the first derivative of a signal, like in hard clipping. These cases require more than one correction function to be applied at every discontinuity. Results obtained show that if sufficiently many samples are corrected at each discontinuity, aliasing can be virtually eliminated while preserving the spectral envelope of the signal. This work extends the understanding of the BLEP, BLAMP, and integrated BLAMP functions as antialiasing tools.