Perceptually informed synthesis of bandlimited classical waveforms using integrated polynomial interpolation

Abstract

Digital subtractive synthesis is a popular music synthesis method, which requires oscillators that are aliasing-free in a perceptual sense. It is a research challenge to find computationally efficient waveform generation algorithms that produce similar-sounding signals to analog music synthesizers but which are free from audible aliasing. A technique for approximately bandlimited waveform generation is considered that is based on a polynomial correction function, which is defined as the difference of a non-bandlimited step function and a polynomial approximation of the ideal bandlimited step function. It is shown that the ideal bandlimited step function is equivalent to the sine integral, and that integrated polynomial interpolation methods can successfully approximate it. Integrated Lagrange interpolation and B-spline basis functions are considered for polynomial approximation. The polynomial correction function can be added onto samples around each discontinuity in a non-bandlimited waveform to suppress aliasing. Comparison against previously known methods shows that the proposed technique yields the best tradeoff between computational cost and sound quality. The superior method amongst those considered in this study is the integrated third-order B-spline correction function, which offers perceptually aliasing-free sawtooth emulation up to the fundamental frequency of 7.8 kHz at the sample rate of 44.1 kHz.

Aliasing Reduction in Clipped Signals

“…Here, a correction function based on a bandlimited step (BLEP) can reduce aliasing in sawtooth and rectangular oscillators [1], [26], [27]. In the BLEP method, a trivially sampled waveform is modified around each discontinuity by adding to its sample values a correction term determined by the BLEP function.…”

Virtual Analog Models of the Lockhart and Serge Wavefolders

“…The performance of the proposed antialiasing method was further evaluated by computing the A-weighted noise-to-mask ratio (ANMR) for a set of sinusoidal input signals processed by both wavefolder models. The ANMR has been previously researched as a perceptually-informed measure to evaluate the audibility of aliasing distortion [41,65]. The algorithm computes the power ratio in decibels between the wanted harmonics and aliased components, but takes into account the masking effects of the former.…”

“…Figure 15a compares the measured ANMRs for a set of sinusoidal inputs with fundamental frequencies between 1 and 5 kHz processed by the Lockhart model at different sampling rates. The ideal alias-free signals required to compute these values were synthesized using Fourier analysis and additive synthesis, as suggested in [41]. All signals were downsampled back to audio rate (i.e., 44.1 kHz) prior to evaluation.…”

“…Moreover, Välimäki and Franck have applied the antialiasing principle behind the DPW algorithm to tackle aliasing in wavetable oscillators [45]. Recent work on antialiasing techniques has extended the use of the bandlimited ramp (BLAMP) method, originally proposed to antialias triangular oscillators in [25,41], to special cases of linear piecewise nonlinearities such as signal rectification, and inverse/hard clipping [17,46,47].…”

“…Well-known techniques include the bandlimited impulse train (BLIT) family of methods, which involves the use of bandlimited basis functions and their integrated forms [39][40][41], and the use of differentiated polynomial waveforms (DPW) [42][43][44]. Moreover, Välimäki and Franck have applied the antialiasing principle behind the DPW algorithm to tackle aliasing in wavetable oscillators [45].…”