Jussi Pekonen

Perceptually informed synthesis of bandlimited classical waveforms using integrated polynomial interpolation

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.

Audibility of aliasing distortion in sawtooth signals and its implications for oscillator algorithm design.

This paper investigates the audibility threshold of aliasing in computer-generated sawtooth signals. Listening tests were conducted to find out how much the aliased frequency components below and above the fundamental must be attenuated for them to be inaudible. The tested tones comprised the fundamental frequencies 415, 932, 1480, 2093, 3136, and 3951 Hz, presented at 60-dB SPL and 44.1-kHz sampling rate. The results indicate that above the fundamental the aliased components must be attenuated 0, 19, 26, 27, 32, and 41 dB for the corresponding fundamental frequencies, and below the fundamental the attenuation of 0, 3, 6, 11, 12, and 11 dB, respectively, is sufficient. The results imply that the frequency-masking phenomenon affects the perception of aliasing and that the masking effect is more prominent above the fundamental than below it. The A-weighted noise-to-mask ratio is proposed as a suitable quality measure for sawtooth signals containing aliasing. It was shown that the bandlimited impulse train, the differentiated parabolic waveform, and the fourth-order polynomial bandlimited step function synthesis algorithms are perceptually alias-free up to 1, 2, and 4 kHz, respectively. General design rules for antialiasing sawtooth oscillators are derived based on the results and on knowledge of level-dependence of masking.

Discrete-time modelling of the moog sawtooth oscillator waveform

Discrete-time modelling strategies of analogue Moog sawtooth oscillator waveforms are presented. Two alternative approaches suitable for real-time implementation are proposed, one modelling the analogue waveform in time domain using phase distortion synthesis and another matching the spectrum of an existing antialiasing sawtooth oscillator to the corresponding analogue spectrum using a first-order IIR post-equalising filter. A parameter estimation procedure for both approaches is explained and performed. Performance evaluation using polynomial fits for the estimated parameters is carried out, and good matches between the model outputs and recorded waveforms are obtained. The best match of the tested algorithms is produced by the phase distortion model and by post-equalising the fourth-order B-spline bandlimited step function sawtooth oscillator.

Filter-based alias reduction for digital classical waveform synthesis

The classical waveforms used in the subtractive sound synthesis have rich spectral content, which causes their sampled digital implementations to suffer from aliasing distortion. Several antialiasing waveform synthesis algorithms have been suggested, and they either remove the aliasing completely or reduce it greatly. A new approach to alias reduction is proposed where the remaining aliased components are suppressed by applying digital highpass and/or comb filtering to the output of an antialiasing algorithm. Applicable filter designs for this novel postprocessing approach are discussed and evaluated with respect to the alias reduction performance using noise-to-mask ratio (NMR). The NMR can be reduced by 10 dB at high fundamental frequencies with a computationally efficient highpass filter. The NMR can be further reduced by using a combination of an IIR comb filter and a DC blocking filter, which provides the best alias reduction performance at high fundamental frequencies.

Optimized Polynomial Spline Basis Function Design for Quasi-Bandlimited Classical Waveform Synthesis

Classical geometric waveforms used in virtual analog synthesis suffer from aliasing distortion when simple sampling is used. An efficient antialiasing technique is based on expressing the waveforms as a filtered sum of time-shifted approximately bandlimited polynomial-spline basis functions. It is shown that by optimizing the coefficients of the basis function so that the aliasing distortion is perceptually minimized, the alias-free bandwidth of classical waveforms can be expanded. With the best of the case examples given here, the generated impulse-train and sawtooth waveform are alias-free up to fundamental frequencies over 10 kHz when the sampling rate is 44.1 kHz.

Variable fractional delay filters in bandlimited oscillator algorithms for music synthesis

Trivially sampled geometric waveforms such as the rectangular pulse wave used in subtractive sound synthesis suffer from aliasing caused by the discontinuities in the waveform or its derivative. Several algorithms for the reduction of aliasing distortion have been suggested, providing either complete removal or great suppression of aliasing. Some antialiasing oscillators utilize variable fractional delay filters as an essential part of the algorithm. In this paper, these oscillators are reviewed with an emphasis on motivating the use of the fractional delay filters.