SOUND SYNTHESIS FABIÁN ESQUEDA NATIVE INSTRUMENTS GMBH
1.3.2019
© 2003 – 2019 VESA VÄLIMÄKI AND FABIÁN ESQUEDA SOUND SYNTHESIS 1.3.2019
OUTLINE
‣ Introduction
‣ Introduction to Synthesis
‣ FM and Phase Distortion Synthesis
‣ Synthesis, Synthesis, Synthesis SOUND SYNTHESIS 1.3.2019 Introduction
‣ BEng in Electronic Engineering with Music Technology Systems (2012) from University of York.
‣ MSc in Acoustics and Music Technology in (2013) from University of Edinburgh.
‣ DSc in Acoustics and Audio Signal Processing in (2017) from Aalto University.
‣ Thesis topic: Aliasing reduction in nonlinear processing.
‣ Published on a variety of topics, including audio effects, circuit modeling and sound synthesis.
‣ My current role is at Native Instruments where I work as a Software Developer for our Synths & FX team. ABOUT NI SOUND SYNTHESIS 1.3.2019
About Native Instruments
‣ One of largest music technology companies in Europe.
‣ Founded in 1996.
‣ Headquarters in Berlin, offices in Los Angeles, London, Tokyo, Shenzhen and Paris.
‣ Team of ~600 people (~400 in Berlin), including ~100 developers. SOUND SYNTHESIS 1.3.2019 About Native Instruments - History
‣ First product was Generator – a software modular synthesizer.
‣ Generator became Reaktor, NI’s modular synthesis/processing environment and one of its core products to this day.
‣ The Pro-Five and B4 were NI’s first analog modeling synthesizers. SOUND SYNTHESIS 1.3.2019 About Native Instruments
‣ Pioneered software instruments and digital DJing.
‣ Combining software development, hardware engineering, sound design.
‣ Strong focus on integrated software/hardware systems (e.g. Maschine). SOUND SYNTHESIS 1.3.2019 Team Structure
‣ Cross-functional teams working using the Agile methodology
‣ Usual roles:
‣ Application developers
‣ DSP developers
‣ User experience designers (UX)
‣ User interface designers (UI)
‣ Sound designers
‣ Quality and assurance engineers (QA)
‣ Team direction set by Product Owner (PO)
‣ Day to day organization and people management handled by Agile Coach. SOUND SYNTHESIS 1.3.2019 The Agile Process
‣ Work using fast two week iterations called sprints.
‣ Tasks specified from a user’s perspective:
‣ As a … [user] …
‣ I want … [desire] …
‣ So that … [benefit] …
‣ Divide larger tasks into smaller ones that can be implemented and tested within one sprint.
‣ Goal is to always have a potentially “releasable” product at the end of a sprint. SOUND SYNTHESIS 1.3.2019 Audio DSP Development at NI
‣ Low-level research on DSP algorithms and technique.
‣ Turn algorithms into engaging devices for users (w/ input from UX designers and sound designers)
‣ Efficient C++ implementation and integration into larger products (working closely with application developers).
‣ Participate actively in workshops, “hackathons” and trainings. INTRODUCTION TO SOUND SYNTHESIS SOUND SYNTHESIS 1.3.2019 Applications of Sound Synthesis
MAKING MUSIC COMMUNICATION COMPUTER GAMING AND VIRTUAL REALITY TEST SIGNALS CAN YOU GUESS THE SOUND? SOUND SYNTHESIS 1.3.2019 SOUND SYNTHESIS 1.3.2019 SOUND SYNTHESIS 1.3.2019
Cool reading: 'Pop 'n Pour': This Electronic Music Pioneer Created the Sound of Coke's Beloved Bubbles: https://www.coca-colacompany.com/stories/meet-suzanne-ciani-the-legendary-creator-of-cokes-pop-n-pour SOUND SYNTHESIS 1.3.2019
THE PROMISE OF DIGITAL SOUND SYNTHESIS
▸ “I have indicated how almost any sound can be produced by treating the numbers generated by a computer as samples of the sound pressure wave.” — Max Mathews, Science, 1963
▸ If sound can be stored by numbers then it must be possible to compute those numbers… RIGHT? SOUND SYNTHESIS 1.3.2019 Problems in Sound Synthesis
▸ How to compute sound efficiently? ▸ Realistic synthesis requires a complicated system. ▸ If simplified too much, sounds artificial. ▸ Computers are getting faster.
▸ How to play synthetic sound? ▸ Control data must be obtained from user. ▸ Solved partly by real-time control interfaces and music software (e.g., MIDI devices, sequencers). SOUND SYNTHESIS 1.3.2019
(Linear vs Nonlinear) ADDITIVE SYNTHESIS SOUND SYNTHESIS 1.3.2019
ADDITIVE SYNTHESIS
▸ Also called Fourier synthesis or sinusoidal modeling.
▸ Each partial is generated separately!
▸ Accurate control but lots of data!
▸ Extensions to the method: ▸ Sines + Noise Modeling (Serra & Smith, 1990) ▸ Sines + Noise + Transients (Verma & Meng, 2000) ▸ FFT–1 synthesis (Rodet and Depalle, 1992)
Source: http://www.cs.princeton.edu/~prc ADDITIVE SYNTHESIS DEMO: RAZOR SUBTRACTIVE SYNTHESIS – PART I SOUND SYNTHESIS 1.3.2019
SUBTRACTIVE SYNTHESIS – PART I
▸ Started by Bob Moog (1934 – 2015) in the 1960s
▸ (Most likely) started with publication of Moog’s seminal paper A Voltage- Controlled Low-Pass High-Pass Filter for Audio Signal Processing at the 1965 AES Convention.
▸ Moog’s instruments were designed around the traditional keyboard interface. SOUND SYNTHESIS 1.3.2019
FUNDAMENTALS OF SUBTRACTIVE SYNTHESIS
▸ Based on the source–filter model
▸ Start with a signal that has a rich frequency spectrum
▸ Process it using a filter
Source Filter Sound
▸ Perhaps it should be called “source–filter synthesis”? Nothing is subtracted, really… IS IT REALLY THAT SIMPLE?
NO SOUND SYNTHESIS 1.3.2019
AN ACTUAL ANALOG SUBTRACTIVE SYNTHESIZER SOUND SYNTHESIS 1.3.2019
AN ACTUAL ANALOG SUBTRACTIVE SYNTHESIZER SOUND SYNTHESIS 1.3.2019
IMPLEMENTING SUBTRACTIVE SYNTHESIS
▸ Example architecture of a subtractive synthesizer:
Envelope Envelope
Oscillator 1
Mixer Lowpass Filter Amplifier Out
Oscillator 2
▸ One or more oscillators typically used as the source
▸ Second- or fourth-order resonant lowpass filters typically used
▸ Envelope generators (ADSR) used for modulation SOUND SYNTHESIS 1.3.2019
OSCILLATORS IN SUBTRACTIVE SYNTHESIS
▸ Periodic geometric waveforms are at the heart of subtractive synthesis
▸ These waveforms are harmonically- rich
▸ Might contain all or only even harmonics
▸ Digital emulation of these waveforms must suppress aliasing
(Figure from: T. D. Rossing: The Science of Sound. Second Edition. Addison-Wesley, 1990.) SOUND SYNTHESIS 1.3.2019
IMPLEMENTING SUBTRACTIVE SYNTHESIS
Envelope Envelope
Oscillator 1
Mixer Lowpass Filter Amplifier Out
Oscillator 2 SOUND SYNTHESIS 1.3.2019
ENVELOPE GENERATORS IN SUBTRACTIVE SYNTHESIS SOUND SYNTHESIS 1.3.2019
A SECOND LOOK AT THE KORG MS-20 SOUND SYNTHESIS 1.3.2019
VIRTUAL ANALOG (VA)
Emulation of analog audio circuits in the digital domain SUBTRACTIVE SYNTHESIS DEMO REAKTOR SOUND SYNTHESIS 1.3.2019
ALIASING IN VA OSCILLATOR WAVEFORMS
▸ Rationale:
▸ Periodic geometric waveforms contain infinite harmonics.
▸ Sampling theorem tells us sampling rate must be at least twice the highest frequency component.
▸ Highest frequency component is infinite.
▸ No sample rate will ever be high enough!
▸ Synthesizing geometric waveforms trivially is equivalent to sampling a waveform with infinite frequency content. SOUND SYNTHESIS 1.3.2019
ALIASING: THE MOVIE
▸ Example: trivial sawtooth waveform (e.g. using a wrapping ramp function)
▸ Aliasing causes distinctive artifacts:
▸ Inharmonicity
▸ Beating
▸ Completely useless!
Video by Andreas Franck, 2012. SOUND SYNTHESIS 1.3.2019
ALIASING: THE MOVIE
▸ Example: Sawtooth waveform generated using additive synthesis.
▸ Generate only components below Nyquist limit.
▸ Method is computationally expensive!
Video by Andreas Franck, 2012. SOUND SYNTHESIS 1.3.2019
VA OSCILLATOR ALGORITHMS
▸ Bandlimited Synthesis Methods ▸ Additive synthesis and it’s variations. ▸ Quasi-Bandlimited Synthesis Methods ▸ BLIT: Bandlimited impulse train + filtering (Stilson & Smith, ICMC’96) ▸ MinBLEP: Minimum-phase bandlimited step (Brandt, ICMC’01) ▸ PolyBLEP: Polynomial bandlimited step (Välimäki & Huovilainen, 2007) ▸ Alias-Suppressing Synthesis Methods ▸ Oversampling ▸ DPW: Differentiated parabolic waveform (Välimäki, 2005) ▸ Post-Processing Synthesis Suppression Methods ▸ Aliasing suppression via filtering (Pekonen & Välimäki, 2008) SOUND SYNTHESIS 1.3.2019
DPW ALGORITHM
▸ An innovative way to generate sawtooth waveforms with reduced aliasing. Published by V Välimäki in IEEE Signal Processing Letters, March, 2005.
▸ Motivation: If aliasing is attenuated sufficiently, its effects can be neglected.
▸ Algorithm is extremely simple to implement and requires two input parameters only: fundamental frequency and sampling rate.
f Modulo Counter 2 FIR Differentiator Out (Trivial Sawtooth) ( . ) fs SOUND SYNTHESIS 1.3.2019
DWP ALGORITHM EXAMPLE
1
▸ Trivial Sawtooth x[n] 0
-1 0 10 20 30 40 50 1 ▸ Squared signal (Parabolic Waveform) 0.5 x2[n] 0 0 10 20 30 40 50 1 ▸ Differentiated Signal 0 c(x2[n] − x2[n − 1]) -1 0 10 20 30 40 50 Discrete time SOUND SYNTHESIS 1.3.2019
DPW ALGORITHM EXAMPLE (CONT’D)
0 ▸ Trivial Sawtooth Spectrum -20 -40 Level(dB) -60 0 5 10 15 20 0 ▸ Parabolic Waveform Spectrum -20 -40 Level(dB) -60 0 5 10 15 20 0 ▸ Differentiated Signal -20 Spectrum -40 Level(dB) -60 0 5 10 15 20 Frequency (kHz) SOUND SYNTHESIS 1.3.2019
COMPARISON OF ALGORITHMS
▸ Test signal: musical scale at high fundamental frequencies at audio rate (44.1 kHz)
▸ Trivial sawtooth
▸ DPW sawtooth
▸ Ideal sawtooth (additive synthesis) SOUND SYNTHESIS 1.3.2019
GENERATING RECTANGULAR WAVEFORMS USING DPW
▸ One possible approach (shown in diagram): ▸ Subtract two sawtooth waveforms. ▸ Second sawtooth has a 50% phase offset. ▸ Modulate phase offset for easy pulse width modulation (PWM).
▸ Second option (not shown): ▸ Use a single sawtooth and delay a copy of it using variable delay line.
f Sawtooth Waveform −+ Out f Sawtooth Waveform Phase Shift SOUND SYNTHESIS 1.3.2019
HIGHER-ORDER DPW FORMS
▸ Trivial waveform can be integrated multiple times to improve performance of the method (Välimäki et al., 2010).
▸ Integrated waveform must then be differentiated N–1 times and scaled. SOUND SYNTHESIS 1.3.2019
INTEGRATED POLYNOMIAL WAVEFORMS
N = 1 N = 2
N = 3 N = 4
N = 5 N = 6 SOUND SYNTHESIS 1.3.2019
DIFFERENCED WAVEFORMS
N = 1 N = 2
N = 3 N = 4
N = 5 N = 6 SOUND SYNTHESIS 1.3.2019
DIFFERENCED WAVEFORMS (FREQUENCY DOMAIN)
N = 1 N = 2
N = 3 N = 4
N = 5 N = 6 SOUND SYNTHESIS 1.3.2019
POLYNOMIAL TRANSITION REGIONS (PTR)
▸ PTR algorithm is an efficient implementation of the DPW method. ▸ Process only samples around discontinuities.
▸ Trivial vs DPW sawtooth waveforms.
▸ Difference between both waveforms.
▸ Constant offset and PTR. SUBTRACTIVE SYNTHESIS – PART II “WEST COAST SYNTHESIS”
Don Buchla (1937–2016) SOUND SYNTHESIS 1.3.2019
WEST COAST SYNTHESIS
▸ Started by Don Buchla in the 1960s.
▸ First Buchla synthesizer built for the San Francisco Tape Music Center.
▸ Synthesizers aimed for experimental work, break from traditional paradigms.
▸ No keyboards. Sequencers or pressure- sensitive plates instead.
▸ Also associated with synth-maker Serge Tcherepnin. The Buchla Music Easel
▸ West Coast synthesis has seen resurgence thanks to Eurorack boom! Serge Tcherepnin (1941– ) SOUND SYNTHESIS 1.3.2019
WEST COAST SYNTHESIS: WAVESHAPING
▸ Serge proposed introducing a processing stage between the oscillator and filter stages in a traditional subtractive setup.
▸ Classic Serge circuits such as the Voltage-Controlled Multipliers (VCM) and the Serge Triple Waveshaper (TWS) were designed for this purpose.
(Image taken from an old Serge synthesizer manual) SOUND SYNTHESIS 1.3.2019
WEST COAST SYNTHESIS: WAVESHAPING (CONT’D)
▸ Start with a waveform with a poor harmonic spectrum (e.g. a sinewave or triangle)
▸ Expand its spectrum using nonlinear waveshaper or pitch modulation
▸ Go from signal with few harmonics to sound with high harmonics
Example Waveshaper: Wavefolder
Sound example: https://www.arturia.com/microbrute/details
Image reference: F. Esqueda, H. Pöntynen, J. D. Parker and S. Bilbao, ”Virtual analog model of the Lockhart wavefolder”, in Proc. SMC-17. SOUND SYNTHESIS 1.3.2019
WEST COAST SYNTHESIS: THE LOWPASS GATE
▸ The Lowpass Gate (LPG) is a staple element in West Coast synthesis
▸ The LPG is a filter/amplifier structure controlled by a light-sensitive resistor packaged with an LED (a vactrol).
▸ The LPG produces natural acoustic-like plucks
▸ No two vactrols sound the same!
Vactrol circuit symbol
Sound reference: J. D. Parker and S. D’Angelo, ”A digital model of the Buchla lowpass-gate”, in Proc. DAFx-13. WAVETABLE SYNTHESIS SOUND SYNTHESIS 1.3.2019
WAVETABLE SYNTHESIS
▸ Idea similar to that of wavetable oscillators used by Max Mathews in the 1950s. ▸ Store one period of a waveform and repeat it.
▸ Wavetable synthesis expanded this idea by storing multiple waveforms in a bidimensional structure (eg matrix).
▸ Timbre manipulation is achieved by morphing (ie interpolating) between waveforms at run- time.
▸ First wavetable synthesizer developed by Wolfgang Palm in the 1970s.
▸ General architecture of wavetable synthesizers similar to that of subtractive synths. Just replace oscillators for wavetables!
Wavetable Filter Sound POPULAR WAVETABLE SYNTHESIZERS
WAVETABLE SYNTHESIS DEMO: ABLETON WAVETABLE FM SYNTHESIS
YAMAHA DX7 SOUND SYNTHESIS 1.3.2019
FM SYNTHESIS
▸ Discovered by John Chowning (1934– ) in the late 1960s. ▸ Patent and publication in the 1970s (Chowning, 1973). ▸ First products arrived in the early 80s. ▸ Patent was owned by Yamaha until about 1994. ▸ Third most lucrative patent Stanford ever licensed.
▸ FM = Frequency Modulation ▸ Same idea as in FM radio.
▸ A computationally-cheap way to compute intriguing sounds.
▸ Still a very popular technique with new exciting products! SOUND SYNTHESIS 1.3.2019
BASICS OF FM SYNTHESIS
▸ Very fast vibrato is applied to a sine wave (“carrier”).
▸ Vibrato generator is called “modulator”.
▸ Both signals are in the audio range.
x(t) = A sin(wct + I sin(wmt))
VCO = Voltage-Controlled Oscillator VCA = Voltage-Controlled Amplifier
Figure: D. Rossing: The Science of Sound SOUND SYNTHESIS 1.3.2019
FM SYNTHESIS: MODULATION INDEX
▸ Number of harmonics is proportional to modulation index I
x(t) = A sin(wct + I sin(wmt))
Example: Effect increasing I from 1 to 15
Figure: K. Steiglitz, A Digital Signal Processing Primer with Applications to Digital Audio and Computer Music SOUND SYNTHESIS 1.3.2019
FM SYNTHESIS – THEORY
▸ Frequency components get mirrored, if they occur above the Nyquist frequency (aliasing) or below 0 Hz.
▸ Inharmonic sounds when the frequency ratio between carrier and modulator is not an integer.
▸ Note: The form we discussed is technically known as phase modulation (PM).
▸ PM and FM are equivalent in some cases, but PM is easier to control.
▸ The DX-7 used PM.
PM FM
x(t) = A sin(wct + I sin(wmt)) x(t) = A sin(wc(1 + I sin(wmt))t)
Figure: K. Steiglitz, A Digital Signal Processing Primer with Applications to Digital Audio and Computer Music FM (PM) SYNTHESIS DEMO PHASE DISTORTION SYNTHESIS
SOUND SYNTHESIS 1.3.2019
PHASE DISTORTION SYNTHESIS
▸ Synthesis method introduced by Casio in 1984 for its CZ line of synthesizers.
▸ Very similar to Yamaha’s phase/frequency modulation.
▸ Popularized by artists such as Vince Clarke (Depeche Mode/Erasure). SOUND SYNTHESIS 1.3.2019
PHASE DISTORTION SYNTHESIS
▸ Works by manipulating the rate at which a sine lookup table is read out.
▸ Advertised as simpler than Yamaha’s FM synthesis!
(Image from Sound Synthesis and Sampling by Martin Russ) SOUND SYNTHESIS 1.3.2019
PHASE DISTORTION SYNTHESIS
▸ Consider the sinusoidal signal with modulo phase counter Φ ( t ) x(t) = sin(2πΦ(t)) ▸ We can produce complex harmonic spectra by shaping the phase with a nonlinear function f ( x ) x(t) = sin(2πf(Φ(t)))
▸ Depending on the shape of f ( x ) different spectra can be produced.
▸ Waveforms available on CZ series: PHASE DISTORTION DEMO TO CONCLUDE… SOUND SYNTHESIS 1.3.2019
SYNTHESIS METHODS WE DIDN’T TALK ABOUT (BUT PROBABLY SHOULD HAVE)
▸ Sampling synthesis: ▸ Samplers & Romplers
▸ Drum Synthesis: ▸ Roland TR line (subtractive synthesis)
▸ Linear Arithmetic Synthesis: ▸ Subtractive & Sampling Hybrid ▸ Example: Roland D-50 SOUND SYNTHESIS 1.3.2019
CLASSIFICATION OF SYNTHESIS METHODS (SMITH, 1991)
▸ Abstract Algorithms: Simple & smart algorithms that produce interesting sounds. ▸ Not based on any physical mechanisms. ▸ Examples: PRETTY MUCH EVERYTHING WE DISCUSSED TODAY.
▸ Processed Recordings: Digital recording, manipulation and playback. ▸ Examples: Musique concrète, sampling, wavetable synthesis.
▸ Spectral Modeling: Algorithms model the spectrum of sound. ▸ Examples: Additive synthesis.
▸ Physical Modelling: Physics-based algorithms. ▸ Next week’s topic. I think… QUESTIONS? SOUND SYNTHESIS 1.3.2019 References
▸ H.-M. Lehtonen, V. Välimäki, and T. I. Laakso, “Canceling and selecting partials from musical tones using fractional-delay filters,” Computer Music Journal, vol. 32, no. 2, pp.43–56, Summer 2008
▸ V. Välimäki, J. S. Abel, and J. O. Smith, “Spectral delay filters,” Journal of the Audio Engineering Society, vol. 57, no. 7/8, pp. 521–531, July/Aug. 2009
▸ Max Mathews and Julius O. Smith III “Methods for synthesizing very high Q parametrically well behaved two pole filters” in Proc. Stockholm Musical Acoustic Conference(SMAC), 2003
▸ A. Degani, M. Dalai, R. Leonradi and P. Migliorati, “Time-frequency analysis of musical signals using the phase coherence” in Proc. 16th Int. Conference on Digital Audio Effects, (DAFx-14), Maynooth, Ireland, September 2-5, 2013
▸ F. Eichas and U. Zölzer, “Black-box modeling of distortion circuits with block-oriented models”, in Proc. 19th Int. Conference on Digital Audio Effects, (DAFx-16), Brno, Czech Republic, September 5-9, 2016
▸ A. Franck and V. Välimäki, “Higher-order integrated wavetable synthesis” in Proc. 15th Int. Conf. Digital Audio Effects (DAFx-12), York, Sept. 2012 SOUND SYNTHESIS 1.3.2019 References
▸ J. Chowning, “The synthesis of complex audio spectra by means of frequency modulation,” J. Audio Eng. Soc., vol. 21, no. 7, pp. 526–534, Sept. 1973.
▸ M. Mathews, “The digital computer as a musical instrument,” Science, vol. 142, no. 3592, pp. 533–557, Nov. 1, 1963.
▸ J. Kleimola, V. Lazzarini, J. Timoney, and V. Välimäki, “Vector phaseshaping synthesis,” in Proc. Int. Conf. Digital Audio Effects (DAFx-11), pp. 233–240, Paris, France, Sept. 2011.
▸ X. Serra, “Musical sound modeling with sinusoids plus noise,” in C. Roads et al. (eds.), Musical Signal Processing. Swets & Zeitlinger, 1997. Available on-line: http://www.iua.upf.es/~xserra/articles/msm/
▸ J. O. Smith, “Viewpoints on the history of digital synthesis,” in Proc. Int. Computer Music Conf. (ICMC’91), pp. 1–10 (keynote talk), Montreal, Canada, Oct. 1991. A revised version is available on-line at:
▸ http://www-ccrma.stanford.edu/~jos/kna/
▸ T. Tolonen, V. Välimäki, and M. Karjalainen, Evaluation of Modern Sound Synthesis Methods. Report no. 48, Lab. of Acoustics and Audio Signal Processing, Helsinki Univ. of Tech., Espoo, 1998. Available at: http://www.acoustics.hut.fi/ ~ttolonen/sound_synth_report.html
▸ T. Verma and T. Meng, ”Extending spectral modeling synthesis with transient modeling synthesis,” Computer Music J. vol. 24, no. 2, pp. 47–59, Summer 2000. SOUND SYNTHESIS 1.3.2019 VA Papers from Aalto University
▸ A. Huovilainen, “Non-linear Digital Implementation of the Moog Ladder Filter,” in Proc. 7th Int. Conf. Digital Audio Effects (DAFx'04), pp. 61-64, Naples, Italy, October 5-8, 2004. Available online at: http://dafx04.na.infn.it/WebProc/Proc/P_061.pdf
▸ J. Kleimola and V. Välimäki, “Reducing aliasing from synthetic audio signals using polynomial transition regions,” IEEE Signal Processing Letters, vol. 19, no. 2, pp. 67–70, Feb. 2012.
▸ Pekonen, J., and Välimäki, V., ''Filter-Based Alias Reduction in Classical Waveform Synthesis,'' in Proc. ICASSP'08, pp. 133-136, Las Vegas, Nevada, March 2008.
▸ V. Välimäki, “Discrete-Time Synthesis of the Sawtooth Waveform with Reduced Aliasing,” IEEE Signal Processing Letters, vol. 12, no. 3, pp. 214-217, March 2005.
▸ A. Huovilainen and V. Välimäki, “New Approaches to Digital Subtractive Synthesis,” in Proc. Int. Computer Music Conf. (ICMC’05), Barcelona, Spain, pp. 399-402, Sept. 2005.
▸ V. Välimäki and A. Huovilainen, “Oscillator and Filter Algorithms for Virtual Analog Synthesis,” Computer Music J., vol. 30, no. 2, pp. 19-31, summer 2006.
▸ V. Välimäki & A. Huovilainen, “Antialiasing Oscillators in Subtractive Synthesis,” IEEE Signal Processing Magazine, vol. 24, no. 2, pp. 116–125, Mar. 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, May 2010.