Antialiasing Oscillator Algorithms for Digital Subtractive Synthesis

1 0 −1

Rectangular waveform Level

Level

Sawtooth waveform

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Time (×T0 )

S-89.3580/S-89.4820 Audio Signal Processing Seminar, Lecture 5

Level

Department of Signal Processing and Acoustics Aalto University School of Science and Technology

October 15, 2010

1 0 −1

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pulse width Discontinuous ⇒ Aliasing!

2

Time (×T0 )

Output

Magn. (dB) Magn. (dB) Magn. (dB)

Subtractive Sound Synthesis

Filter

2

P is the duty cycle or the

Antialiasing Oscillator Algorithms Jussi Pekonen Aalto SPA

Source

P 1

Time (×T0 )

Triangle waveform

Jussi Pekonen

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3/40 October 15, 2010 S-89.3580/S-89.4820 Lecture 5

Contents of This Lecture Objectives and Outline

Operation principles of oscillators that reduce/remove aliasing 0 −30 −60 0 −30 −60 0 −30 −60

Outline 0

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1. Ideally Bandlimited Oscillators 2. Quasi-Bandlimited Oscillators Break

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3. Alias-Suppressing Oscillators 4. Special Approaches to Classical Waveform Synthesis

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Not covered: Filters (covered by Mikko in the seminar) and oscillator effects (covered by Jari next week)

Frequency (kHz) Antialiasing Oscillator Algorithms Jussi Pekonen Aalto SPA

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1 Ideally Bandlimited Oscillator Algorithms

Wavetable Synthesis

Chamberlin, 1985, Book & Burk, 2004, Book

1. Precompute single cycles of the sums of Fourier series terms (like in additive synthesis) 2. Tabulate the precomputed cycles 3. On the synthesis stage read the table computed for that fundamental frequency in a loop Computational complexity per sample Only control logic and table reads in the synthesis stage, hence constant

O(1) Memory requirements Huge! There are techniques to reduce the requirements, however, they are still large. . .

Antialiasing Oscillator Algorithms Jussi Pekonen Aalto SPA

1

Additive Synthesis

Chaudhary, 1998, AES 105th Convention

Count

fc = 20 kHz fc = 15 kHz

K

20

40

100

200

400

1000

2000

f0 (Hz)

Wavetable Synthesis II

Chamberlin, 1985, Book & Burk, 2004, Book

600 400 200 0

All harmonics up to 20 kHz at all f0 40

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100

15 The highest harmonic 10 at 15–20 kHz 5 0 40

Computational complexity per sample O(1/f0 ) Memory requirements Depends on the sinusoidal oscillator Antialiasing Oscillator Algorithms Jussi Pekonen Aalto SPA

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Number of required tables for sawtooth, minimum f0 = 27.5 Hz

Synthesize the components of the waveform’s Fourier series representation below a given cutoff frequency fc (the highest harmonic index K = �fc /f0 �) and add them up

1,000 750 500 250 0

Antialiasing Oscillator Algorithms Jussi Pekonen Aalto SPA

Count

1

5/40 October 15, 2010 S-89.3580/S-89.4820 Lecture 5

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200

400

1000

2000

200

400

1000

2000

Maximum f0 (Hz)

Antialiasing Oscillator Algorithms Jussi Pekonen Aalto SPA

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Discrete Summation Formulas

Winham and Steiglitz, 1970, JASA & Moorer, 1976, JAES

Using the identities of trigonometric functions reduce a sum of sinusoids into a “simpler” expression Example (Winham and Steiglitz, 1970): N � k=1

cos(kωn) =

sin((2N + 1)ωn/2) 1 − 2 sin(ωn/2) 2

Summary of Idelly Bandlimited Oscillators

Additive synthesis Accurate, but computationally heavy Wavetable synthesis Computationally light, memory requirements large, complicated control with time-varying phenomena Discrete Summation Formulas Computationally moderate/light, numerical issues

Numerical issues when the denominator is close to zero

Inverse FFT synthesis Computationally moderate, trade-off between temporal and spectral resolution, interpolation issues

Amplitude mismatches – requires a post-equalizing filter

Theoretical approaches useful for testing the other algorithms

Issues

Antialiasing Oscillator Algorithms Jussi Pekonen Aalto SPA

1

1

9/40 October 15, 2010 S-89.3580/S-89.4820 Lecture 5

Inverse FFT Synthesis

Deslauriers and Leider, 2009, AES 127th Convention

Compose the waveform in frequency-domain and apply inverse fast Fourier transform (IFFT) to the synthetic spectrum

Issues

Antialiasing Oscillator Algorithms Jussi Pekonen Aalto SPA

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2 Quasi-Bandlimited Oscillator Algorithms

Trade-off between temporal and spectral resolution Data interpolation due to finite spectral resolution Noise due to errors in spectrum data Assumes linear amplitude and phase evolution within a frame Computational complexity and memory consumption Depend on the block size of the IFFT

Antialiasing Oscillator Algorithms Jussi Pekonen Aalto SPA

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Antialiasing Oscillator Algorithms Jussi Pekonen Aalto SPA

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Bandlimited Impulse Train Synthesis (BLIT) Continuous-Time Derivation (Stilson and Smith, 1996, ICMC)

2f0 2f0 − 1 2f0 − 2

d dt

2T0

T0

0

sinc function infinitely long! ⇒ Truncate to length N , window & tabulate 0

Ideally a sequence of sinc functions! 1 0 −1

2T0

T0

0

2f0 2f0 − 1 2f0 − 2

�

Time (s)

A f0

Hlp (Ω)

Computational Load For a discontinuity, the computational load is O(N). Per sample the load is O(Nf0 ) Memory Requirements The table length is NM(+1); hence the memory requirement is O(NM)

0

T0

2T0

13/40 October 15, 2010 S-89.3580/S-89.4820 Lecture 5

Amplitude Polarity Phase counter

Discontinuity detector

Level

0 0

P 1

BLIT synthesis

ˆy(n)

Triangle wave derivative

2

−2

Fractional delay

2

4f0

Time (×T0 )

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Alias Reduction Performance of BLIT Pekonen et al., 2010a, DAFx

Hann-windowed sinc N = 4, M = 8

0 −50

−100

0

−4f0

Antialiasing Oscillator Algorithms Jussi Pekonen Aalto SPA

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Stilson and Smith, 1996, ICMC & Stilson, 2006, PhD Thesis

fs

Discontinuity located between sampling instants ⇒ Oversampling by factor M required to get proper positioning, can be further improved by table interpolation

2T0

BLIT Algorithm

Rect. wave derivative Level

T0

Time (s) Antialiasing Oscillator Algorithms Jussi Pekonen Aalto SPA

2

Computational Load and Memory Requirements of BLIT

Play

0

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10 15 20

Plain Hann window N = 4, M = 8 Magnitude (dB)

1 0 −1

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Magnitude (dB)

2

0 −50

−100

Play

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Frequency (kHz)

5

10 15 20

Frequency (kHz)

The windowed sinc function is not the optimal! Antialiasing Oscillator Algorithms Jussi Pekonen Aalto SPA

16/40 October 15, 2010 S-89.3580/S-89.4820 Lecture 5

Approaches Not Using Look-Up Tables Modified FM pulses Use a modified FM synthesizer to generate bandlimited pulses (Timoney et al., 2008, DAFx) Fractional delay filters A handy approach

Fractional Delay Filters in BLIT Synthesis Pekonen et al., 2010b, ICGCS

The purpose of fractional delay (FD) filters?

⇒ To approximate ideal bandlimited interpolation!

The basis function of ideal bandlimited interpolation?

⇒ The sinc function!

⇒ Use FD filters to synthesize the bandlimited impulses!

Modify the algorithm: BLIT FD synthesis filter

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Time (samples)

1 0.5 0 −2 1 0.5 0 −2

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Time (samples)

Antialiasing Oscillator Algorithms Jussi Pekonen Aalto SPA

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Direct BLIT Synthesis Using FD Filters II Nam et al., 2010, IEEE TransASLP

Magn. (dB)

2

Left: Lagrange BLIT Right: B-spline BLIT

−1

17/40 October 15, 2010 S-89.3580/S-89.4820 Lecture 5

0 −50

Magn. (dB)

Antialiasing Oscillator Algorithms Jussi Pekonen Aalto SPA

1 0.5 0 −2 1 0.5 0 −2 1 0.5 0 −2

Level

Optimized tables Optimize the table entries according to selected criteria

Nam et al., 2010, IEEE TransASLP

0 −50

−100

−100

Play

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Play

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Left: Lagrange BLIT sawtooth Right: B-spline BLIT sawtooth

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Magn. (dB)

Parametric window function Use a controllable window function as the look-up table

Direct BLIT Synthesis Using FD Filters

Level

Look-Up Table Approaches (Pekonen et al., 2010a, DAFx)

2

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Approaches to Improve the Performance of BLIT

Level

2

0 −50

−100

Play

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Frequency (kHz)

5

10

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Frequency (kHz)

Trade-off: Alias reduction vs. Amplitude drop of higher harmonics

Antialiasing Oscillator Algorithms Jussi Pekonen Aalto SPA

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Third-order Lagrange FD Filter

Phase

1 0.5 0 −1 −2

Output

2

Example of FD-BLIT

BLIT

2

Prone to Numerical Errors

Replace the integrator with a second-order leaky integrator (Brandt, 2001, ICMC)

0

Hint,2 (z) = d = 0.732

d = 0.197

d = 0.662

d = 0.127

1

Inherent property of the algorithm, cannot be avoided

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ˆy(n) Hfd (z)

In the beginning 1 0 −1 0 10 20

z−D

30

40

Magn. (dB)

Nam et al., 2009, DAFx

+

Antialiasing Oscillator Algorithms Jussi Pekonen Aalto SPA

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Feedback Loop Oscillator

Aδ(n)

1 − z−1 (1 − cz−1 )2

Boosting of Aliasing at Low Frequencies

0 −1 Antialiasing Oscillator Algorithms Jussi Pekonen Aalto SPA

2

Issues With BLIT

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Bandlimited Step Function Synthesis (BLEP) Brandt, 2001, ICMC

Avoid integration in the synthesis stage 0 −20 −40 −60

1. Integrate the BLIT function Inharmonic! 0 5 10 15 20 Frequency (kHz)

After one second 1 0 −1 0 10 20

Antialiasing Oscillator Algorithms Jussi Pekonen Aalto SPA

30

40

22/40 October 15, 2010 S-89.3580/S-89.4820 Lecture 5

2. At each discontinuity, trigger the integral In principle: Accumulate the BLIT look-up table and reading it through and output a constant one when the table size is exceeded In practice: Compute the difference between the bandlimited step function and unit step function and add it onto the waveform around the discontinuity (Välimäki and Huovilainen, 2007, IEEE SPM & Leary and Bright, 2009, U.S. Patent) Computational load and memory requirements the same as with BLIT Antialiasing Oscillator Algorithms Jussi Pekonen Aalto SPA

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Summary of Quasi-Bandlimited Oscillators

Bandlimited Impulse Train (BLIT) Synthesize a sequence of bandlimited impulses and integrate, issues with the integration and boosting of aliasing at low frequencies

3

Chamberlin, 1985, Book & Puckette, 2007, Book

Synthesize the trivial waveform with a high sampling rate ⇒ aliased components will be at lower level

Issues

Spectral envelope of these waveforms decay gently ⇒ Very high oversampling factor L required!

Bandlimited Step Function (BLEP) Synthesize a sequence of bandlimited step functions or a sequence of correction functions

Antialiasing Oscillator Algorithms Jussi Pekonen Aalto SPA

Oversampled Trivial Approach

Highly oversampled oscillator consumes computational power ⇒ Computational load: O(L)

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Antialiasing Oscillator Algorithms Jussi Pekonen Aalto SPA

Level

Level

Alias Suppressing Oscillator Algorithms

Level

3

3

Filtering of Full-Wave Rectified Sine Wave Lane et al., 1997, CMJ

1 0 −1 1 0.5 0 1 0 −1

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1. Sinusoid with half of the target frequency 2. Full-wave rectify 3. Fixed lowpass filter

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Sample Antialiasing Oscillator Algorithms Jussi Pekonen Aalto SPA

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4. f0 -tracking highpass filter Other waveforms with approximations (see Lowenfels, 2003, AES 115th Convention, for practical approaches)

Antialiasing Oscillator Algorithms Jussi Pekonen Aalto SPA

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Differentiation of Waveforms (DPW)

Piecewise

Polynomial

3

Higher-Order DPWs

Välimäki et al., 2010a, IEEE TransASLP

Välimäki et al., 2010a, IEEE TransASLP

3rd-order dB

�

= (iω)F(f (t)) � F(f (t)) f (t)dt = +C iω

Integration decreases spectral tilt by about 6 dB per octave Sawtooth waveform linear within a period ⇒ Analytic integration possible

0.2 0 −0.2

30

40 dB

1 0.5 0

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5 0 −5

·10−2 0

f0 = 2.637 kHz

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5 0 −5

·10−3 0

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f0 = 800 Hz

20

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60

80

Sample Antialiasing Oscillator Algorithms Jussi Pekonen Aalto SPA

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5

Frequency (kHz)

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DPW Scaling

Välimäki, 2005, IEEE SPL; Välimäki and Huovilainen, 2006, CMJ & Välimäki et al., 2010a, IEEE TransASLP

The output of the differentiator(s) needs to be scaled due to nonideal differentiation Scaling factor issues:

Piece-wise parabolic waveform

dB

Level Level Level

Välimäki, 2005, IEEE SPL & Huovilainen and Välimäki, 2005, ICMC

10

10

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Second-Order DPW

0

5

Sample

Antialiasing Oscillator Algorithms Jussi Pekonen Aalto SPA

1 0 −1

0

0 −40 −80

Frequency (kHz)

Differentiation increases spectral tilt by about 6 dB per octave

3

0 −40 −80

Level

F

F ��

d f (t) dt

Level

�

4th-order dB

Utilizes the following Fourier Transform properties:

One differentiator

Inversely proportional to the fundamental frequency!

0 −40 −80

0

5

10

15

20

0 −40 −80

0

5

10

15

20

The fundamental frequency is in the power of the order!

⇒ At low frequencies very large scaling (e.g. 200 dB) required ⇒ Numerical problems. . .

Frequency (kHz)

Sample Antialiasing Oscillator Algorithms Jussi Pekonen Aalto SPA

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Summary of Alias-Suppressing Algorithms

4

Digital Post-Suppression Algorithms Pekonen and Välimäki, 2008, ICASSP

Sample a waveform with a tilted spectrum

We have an oscillator that has aliasing, what can we do?

Oversampling Very high oversampling factor required

Below the fundamental frequency Highpass filtering

Filtered full-wave rectified sinusoid Approximations, approximations. . .

Between harmonics Comb filtering FIR comb filter to pass the harmonic components and to remove some aliasing between the harmonics, or IIR comb filter to pass mainly the harmonic components and to suppress the aliasing between the harmonics

Differentiated parabolic waveforms Sample integrals of linear function, problems with scaling

Comb filters require the highpass filter also as they will pass DC

Antialiasing Oscillator Algorithms Jussi Pekonen Aalto SPA

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4 Special Approaches to Classical Waveform Synthesis

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Distortion (Waveshaping) Synthesis Timoney et al., 2009a, AES 126th Convention

Distrort a sinusoid with a waveshaper like in the filtered full-wave rectified sinusoid Different waveshapers for different waveforms (Timoney et al., 2009a, AES 126th Convention & Kleimola, 2008, DAFx) Not necessarily aliasing-free Requires control to avoid aliasing (Timoney et al., 2009a, AES 126th Convention & Lazzarini and Timoney, 2010, CMJ) Example: Use Chebyshev polynomials with the number of polynomials controlled by the fundamental frequency (Pekonen, 2007, Master’s thesis)

Antialiasing Oscillator Algorithms Jussi Pekonen Aalto SPA

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4

5

Phase Distortion Synthesis Ishibashi, 1987, U.S. Patent

Like waveshaping, but for phase instead of amplitude

Papers Dealing with This Topic Timoney et al., 2009b, ICASSP; Timoney et al., 2009a, AES 126th Convention; Lazzarini et al., 2009b, DAFx; Kleimola et al., 2009, DAFx & Lazzarini et al., 2009a, DAFx Approaches to control aliasing discussed in Lazzarini and Timoney, 2010, CMJ

Antialiasing Oscillator Algorithms Jussi Pekonen Aalto SPA

Ideally bandlimited oscillators No aliasing at all, different issues in different algorithms, useful for testing the other approaches Quasi-bandlimited oscillators Aliasing allowed mainly at high frequencies, BLIT and BLEP approaches, integration issues in BLIT Alias-suppressing oscillators Sample a signal that has a tilted spectrum, oversampling, filtered full-wave rectified sine wave, DPW, scaling issues in DPW Special approaches Ad hoc approaches, post-suppression by filtering, wave- and phaseshaping, issues with aliasing in distortion approaches

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5 Summary of the Lecture

Summary of the Lecture

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Future of Bandlimited Oscillator Design

Oscillator with Desired Properties 1. Perceptually aliasing-free in the range of musical frequencies 2. Computationally efficient and low memory requirements 3. Does not require a division that depends on an oscillation parameter, e.g. fundamental frequency! The first two are obtainable, the last one still unsolved problem

Modeling of Analog Oscillator Outputs First attempts done by De Sanctis and Sarti, 2010, IEEE TransASLP, and by Kleimola et al., 2010, SMC Antialiasing Oscillator Algorithms Jussi Pekonen Aalto SPA

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References II

Appendix

Papers Dealing with Oscillator Algorithms G. De Sanctis and A. Sarti. Virtual analog modeling in the wave-digital domain. IEEE Transactions on Audio, Speech, and Language Processing, 18(4): 715–727, May 2010. G. Deslauriers and C. Leider. A bandlimited oscillator by frequency-domain synthesis for virtual analog applications. In Proceedings of the 127th Audio Engineering Society Convention, New York, NY, October 2009. Preprint number 7923.

References

A. Huovilainen and V. Välimäki. New approaches to digital subtractive synthesis. In Proceedings of the International Computer Music Conference, pages 399–402, Barcelona, Spain, September 2005. M. Ishibashi. Electronic musical instrument. U.S. Patent 4,658,691, 1987.

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References

3/11 October 15, 2010 S-89.3580/S-89.4820 Lecture 5

References III

Papers Dealing with Oscillator Algorithms

Papers Dealing with Oscillator Algorithms

E. Brandt. Hard sync without aliasing. In Proceedings of the International Computer Music Conference, pages 365–368, Havana, Cuba, September 2001. P. Burk. Band limited oscillators using wave table synthesis. In K. Greenebaum and R. Barzel, editors, Audio Anecdotes II – Tools, Tips, and Techniques for Digital Audio, pages 37–53. A. K. Peters, Ltd, Wellesley, MA, 2004. H. Chamberlin. Musical Applications of Microprocessors, chapter 13, pages 418–480. Hayden Book Company, Hasbrouck Heights, NJ, 2nd edition, 1985. A. Chaudhary. Bandlimited simulation of analog synthesizer modules by additive synthesis. In Proceedings of the 105th Audio Engineering Society Convention, San Francisco, CA, September 1998. Preprint number 4779.

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Antialiasing Oscillator Algorithms Jussi Pekonen Aalto SPA

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J. Kleimola. Audio synthesis by bitwise logical modulation. In Proceedings of the 11th International Conference on Digital Audio Effects (DAFx-08), pages 67–70, Espoo, Finland, September 2008. J. Kleimola, J. Pekonen, H. Penttinen, V. Välimäki, and J. S. Abel. Sound synthesis using an allpass filter chain with audio-rate coefficient modulation. In Proceedings of the 12th International Conference on Digital Audio Effects (DAFx-09), pages 305–312, Como, Italy, September 2009. J. Kleimola, V. Lazzarini, J. Timoney, and V. Välimäki. Phaseshaping oscillator algorithms for musical sound synthesis. In Proceedings of the 7th Sound and Music Computing Conference, pages 94–101, Barcelona, Spain, July 2010. J. Lane, D. Hoory, E. Martinez, and P. Wang. Modeling analog synthesis with DSPs. Computer Music Journal, 21(4):23–41, Winter 1997.

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References IV

References VI

Papers Dealing with Oscillator Algorithms

Papers Dealing with Oscillator Algorithms

V. Lazzarini and J. Timoney. New perspectives on distortion synthesis for virtual analog oscillators. Computer Music Journal, 34(1):28–40, Spring 2010. V. Lazzarini, J. Timoney, J. Kleimola, and V. Välimäki. Five variations on a feedback theme. In Proceedings of the 12th International Conference on Digital Audio Effects (DAFx-09), pages 139–145, Como, Italy, September 2009a. V. Lazzarini, J. Timoney, J. Pekonen, and V. Välimäki. Adaptive phase distortion synthesis. In Proceedings of the 12th International Conference on Digital Audio Effects (DAFx-09), pages 28–35, Como, Italy, September 2009b. A. B. Leary and C. T. Bright. Bandlimited digital synthesis of analog waveforms. U.S. Patent 7,589,272, September 2009.

Antialiasing Oscillator Algorithms Jussi Pekonen Aalto SPA

J. Pekonen. Computationally efficient music synthesis – methods and sound design. Master’s thesis, TKK Helsinki University of Technology, Espoo, Finland, June 2007. Available online

http://www.acoustics.hut.fi/publications/files/theses/pekonen_mst/. J. Pekonen and V. Välimäki. Filter-based alias reduction in classical waveform synthesis. In Proceedings of the 2008 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP’08), pages 133–136, Las Vegas, NV, April 2008. J. Pekonen, J. Nam, J. O. Smith, J. S. Abel, and V. Välimäki. On minimizing the look-up table size in quasi bandlimited classical waveform synthesis. In Proceedings of the 13th International Conference on Digital Audio Effects (DAFx-10), pages 57–64, Graz, Austria, September 2010a.

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References V

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References VII

Papers Dealing with Oscillator Algorithms

Papers Dealing with Oscillator Algorithms

D. Lowenfels. Virtual analog synthesis with a time-varying comb filter. In Proceedings of the 115th Audio Engineering Society Convention, New York, NY, October 2003. Preprint number 5960. J. A. Moorer. The synthesis of complex audio spectra by means of discrete summation formulas. Journal of the Audio Engineering Society, 24(9): 717–727, November 1976.

J. Pekonen, V. Välimäki, J. Nam, J. S. Abel, and J. O. Smith. Variable fractional delay filters in bandlimited oscillator algorithms for music synthesis. In Proceedings of the 2010 International Conference on Green Circuits and Systems (ICGCS2010), pages 148–153, Shanghai, China, June 2010b. M. Puckette. The Theory and Technique of Electronic Music, pages 301–322. World Scientific Publishing Co., Hackensack, NJ, 2007.

J. Nam, V. Välimäki, J. S. Abel, and J. O. Smith. Alias-free oscillators using feedback delay loops. In Proceedings of the 12th International Conference on Digital Audio Effects (DAFx-09), pages 347–352, Como, Italy, September 2009.

T. Stilson. Efficiently-Variable Non-Oversampling Algorithms in Virtual-Analog Music Synthesis – A Root-Locus Perspective. PhD thesis, Stanford University, Stanford, CA, June 2006. Available online http://ccrma.stanford.edu/~stilti/papers/.

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, 18(4):773–785, May 2010.

T. 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.

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References VIII

References X

Papers Dealing with Oscillator Algorithms

Papers Dealing with Oscillator Algorithms

J. Timoney, V. Lazzarini, and T. Lysaght. A modified FM synthesis approach to bandlimited signal generation. In Proceedings of the 11th International Conference on Digital Audio Effects (DAFx-08), pages 27–33, Espoo, Finland, September 2008.

G. Winham and K. Steiglitz. Input generators for digital sound synthesis. Journal of the Acoustical Society of America, 47(2):665–666, February 1970.

J. Timoney, V. Lazzarini, B. Carty, and J. Pekonen. Phase and amplitude distortion methods for digital synthesis of classic analogue waveforms. In Proceedings of the 126th Audio Engineering Society Convention, Munich, Germany, May 2009a. Preprint number 7792. J. Timoney, V. Lazzarini, J. Pekonen, and V. Välimäki. Spectrally rich phase distortion sound synthesis using an allpass filter. In Proceedings of the 2009 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP’09), pages 293–296, Taipei, Taiwan, April 2009b.

Antialiasing Oscillator Algorithms Jussi Pekonen Aalto SPA

9/11 October 15, 2010 S-89.3580/S-89.4820 Lecture 5

References IX

Papers Dealing with Oscillator Algorithms V. Välimäki. Discrete-time synthesis of the sawtooth waveform with reduced aliasing. IEEE Signal Processing Letters, 12(3):214–217, March 2005. V. Välimäki and A. Huovilainen. Oscillator and filter algorithms for virtual analog synthesis. Computer Music Journal, 30(2):19–31, Summer 2006. V. Välimäki and A. Huovilainen. Antialiasing oscillators in subtractive synthesis. IEEE Signal Processing Magazine, 24(2):116–125, March 2007. V. Välimäki, J. Nam, J. O. Smith, and J. S. Abel. Alias-suppressed oscillators based on differentiated polynomial waveforms. IEEE Transactions on Audio, Speech, and Language Processing, 18(4):786–798, May 2010a. V. Välimäki, J. Pekonen, and J. Nam. Synthesis of bandlimited classical waveforms using integrated polynomial interpolation. Journal of the Acoustical Society of America, 2010b. Submitted for publication (accepted conditionally).

Antialiasing Oscillator Algorithms Jussi Pekonen Aalto SPA

10/11 October 15, 2010 S-89.3580/S-89.4820 Lecture 5

Antialiasing Oscillator Algorithms Jussi Pekonen Aalto SPA

11/11 October 15, 2010 S-89.3580/S-89.4820 Lecture 5

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