On Minimizing the Look-up Table Size in Quasi Bandlimited Classical Waveform Oscillators

Transcription

1 On Minimizing the Look-up Table Size in Quasi Bandlimited Classical Waveform Oscillators 3th International Conference on Digital Audio Effects (DAFx-), Graz, Austria Jussi Pekonen, Juhan Nam 2, Julius O. Smith 2, Jonathan S. Abel 2, and Vesa Välimäki Department of Signal Processing and Acoustics Aalto University School of Science and Technology, Helsinki/Espoo, Finland 2 Center for Computer Research on Music and Acoustics Stanford University, Stanford, California, USA September 7, 2 Oscillators in Subtractive Sound Synthesis T 2T T 2T T 2T 2 4 Trivially sampled sawtooth Aliasing! Minimizing the Look-up Table Size in the BLIT Oscillator 2/9 Pekonen, Nam, Smith, Abel, and Välimäki September 7, 2 DAFx-, Graz, Austria

2 Bandlimited Impulse Train (BLIT) Algorithm Continuous-Time Derivation T 2T Bandlimited impulse trains (Stilson and Smith, 996) d dt Ideally a sequence of sinc functions! 2f 2f 2f 2 T 2T H lp (ω) T 2T 2f 2f 2f 2 T 2T Minimizing the Look-up Table Size in the BLIT Oscillator 3/9 Pekonen, Nam, Smith, Abel, and Välimäki September 7, 2 DAFx-, Graz, Austria Problems in the BLIT Algorithm sinc function infinitely long! Truncation, windowing & tabulation High oversampling required in order to get proper positioning For good quality, long tables are required Short Table Example (Hann-Windowed sinc Function) Level Table index Minimizing the Look-up Table Size in the BLIT Oscillator 4/9 Pekonen, Nam, Smith, Abel, and Välimäki September 7, 2 DAFx-, Graz, Austria

3 Means to Improve the Performance? Magn. (db) Magn. (db) BLIT using sinc BLIT using Hann window Replace the windowed sinc function with the plain window function? Optimize: minimize table size while keeping aliasing inaudible and amplitude drop acceptable Magn. (db) sinc Hann Minimizing the Look-up Table Size in the BLIT Oscillator 5/9 Pekonen, Nam, Smith, Abel, and Välimäki September 7, 2 DAFx-, Graz, Austria Parametric Window Functions Approach : Kaiser & Dolph-Chebyshev Windows Allow control over the minimum stopband attenuation! Gain depends on the table parameters First-order IIR post-eq filter to compensate the amplitude drop Example: Kaiser Window samples, db 4 samples, 22 db 8 samples, db Minimizing the Look-up Table Size in the BLIT Oscillator 6/9 Pekonen, Nam, Smith, Abel, and Välimäki September 7, 2 DAFx-, Graz, Austria

4 Direct Optimization Strategies Approach 2: Minimax & Least-Squared Minimized Stopband Gain Objective Minimize the stopband gain using an error measure Subject to Passband gain constraints Design Issues Error measure: minimax, least-squares, other? Weighted error: how to choose the frequency dependency? LS, no weight 5 MM, no weight LS, with weight Minimizing the Look-up Table Size in the BLIT Oscillator 7/9 Pekonen, Nam, Smith, Abel, and Välimäki September 7, 2 DAFx-, Graz, Austria Conclusions Aliasing in BLIT algorithm investigated using short look-up tables The ideal windowed sinc function is not the optimal look-up table! Better alias reduction performance with alternative approaches In this paper Like fractional delay filters (Nam et al., 2). Parametric window functions Gain depends on parameters Amplitude compensation using post-eq 2. Direct optimization approaches Minimize a weighted error measure in stopband Independent control over the amplitude drop Minimizing the Look-up Table Size in the BLIT Oscillator 8/9 Pekonen, Nam, Smith, Abel, and Välimäki September 7, 2 DAFx-, Graz, Austria

5 Further Pointers Aside This Paper... J. Nam, V. Välimäki, J. S. Abel, and J. O. Smith. Efficient antialiasing oscillator algorithms using low-order fractional delay filters. IEEE Transactions on Audio, Speech, and Language Processing, 8(4): , May 2. T. S. Stilson and J. O. Smith. Alias-free digital synthesis of classic analog waveforms. In Proceedings of the International Computer Music Conference, pages , Hong Kong, China, August 996. Additional Companion Page Look-up tables presented in the paper Sound examples URL: Minimizing the Look-up Table Size in the BLIT Oscillator 9/9 Pekonen, Nam, Smith, Abel, and Välimäki September 7, 2 DAFx-, Graz, Austria