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.

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.