On Minimizing the Look-up Table Size in Quasi Bandlimited Classical - - PDF document

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

13th International Conference on Digital Audio Effects (DAFx-10), Graz, Austria

Jussi Pekonen1, Juhan Nam2, Julius O. Smith2, Jonathan S. Abel2, and Vesa Välimäki1

1Department of Signal Processing and Acoustics

Aalto University School of Science and Technology, Helsinki/Espoo, Finland

2Center for Computer Research on Music and Acoustics

Stanford University, Stanford, California, USA

September 7, 2010

Oscillators in Subtractive Sound Synthesis

T0 2T0 −1

1

T0 2T0 −1

1

T0 2T0 −1

1 Time (s)

5 10 15 20 −60 −40 −20

Frequency (kHz) Magnitude (dB) Trivially sampled sawtooth Aliasing!

Minimizing the Look-up Table Size in the BLIT Oscillator 2/9 Pekonen, Nam, Smith, Abel, and Välimäki September 7, 2010 Aalto SPA / CCRMA DAFx-10, Graz, Austria

Bandlimited Impulse Train (BLIT) Algorithm

Continuous-Time Derivation

T0 2T0 −1

1

d dt

T0 2T0 2f0 − 2 2f0 − 1 2f0 Hlp(ω) T0 2T0 2f0 − 2 2f0 − 1 2f0

Time (s)

• T0

2T0 −1

1 Time (s) Bandlimited impulse trains (Stilson and Smith, 1996)

⇒ Ideally a sequence of sinc functions!

Minimizing the Look-up Table Size in the BLIT Oscillator 3/9 Pekonen, Nam, Smith, Abel, and Välimäki September 7, 2010 Aalto SPA / CCRMA DAFx-10, 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)

10 20 30 0.5 1

Table index Level

5 10 15 20 −100 −50

Frequency (kHz) Magnitude (dB) Play

Minimizing the Look-up Table Size in the BLIT Oscillator 4/9 Pekonen, Nam, Smith, Abel, and Välimäki September 7, 2010 Aalto SPA / CCRMA DAFx-10, Graz, Austria

Means to Improve the Performance?

5 10 15 20 −100 −50

• Magn. (dB)

BLIT using sinc Play Replace the windowed sinc function with the plain window function? Optimize: minimize table size while keeping aliasing inaudible and amplitude drop acceptable

5 10 15 20 −100 −50

Frequency (kHz)

• Magn. (dB)

BLIT using Hann window Play

0 0.5 1 2 3 4 −100 −50

Frequency (×fs)

• Magn. (dB)

sinc

Hann

Minimizing the Look-up Table Size in the BLIT Oscillator 5/9 Pekonen, Nam, Smith, Abel, and Välimäki September 7, 2010 Aalto SPA / CCRMA DAFx-10, Graz, Austria

Parametric Window Functions

Approach 1: 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

0.5 1 1.5 2 2.5 3 3.5 4 −150 −100 −50

Frequency (×fs) Magnitude (dB) 4 samples, 110 dB 4 samples, 220 dB 8 samples, 110 dB

Minimizing the Look-up Table Size in the BLIT Oscillator 6/9 Pekonen, Nam, Smith, Abel, and Välimäki September 7, 2010 Aalto SPA / CCRMA DAFx-10, Graz, Austria

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?

0.5 1 1.5 2 2.5 3 3.5 4 −150 −100 −50

Frequency (×fs) Magnitude (dB) LS, no weight 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, 2010 Aalto SPA / CCRMA DAFx-10, 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

Like fractional delay filters (Nam et al., 2010)

In this paper

• 1. 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, 2010 Aalto SPA / CCRMA DAFx-10, Graz, Austria

Further Pointers

Aside This Paper. . .

• J. Nam, V. Välimäki, J. S. Abel, and J. O. Smith. Efficient antialiasing
• scillator algorithms using low-order fractional delay filters. IEEE

Transactions on Audio, Speech, and Language Processing, 18(4): 773–785, May 2010.

• T. S. Stilson and J. O. Smith. Alias-free digital synthesis of classic analog
• waveforms. In Proceedings of the International Computer Music

Conference, pages 332–335, Hong Kong, China, August 1996.

Additional Material @ Companion Page

Look-up tables presented in the paper Sound examples URL: http://www.acoustics.hut.fi/go/dafx10-optosctables/

Minimizing the Look-up Table Size in the BLIT Oscillator 9/9 Pekonen, Nam, Smith, Abel, and Välimäki September 7, 2010 Aalto SPA / CCRMA DAFx-10, Graz, Austria