Source-linked AI summary
Alias-Free Oscillator Synchronization via Additive Synthesis
Jonas Roth, Domenic Keller, Oscar Castañeda, Christoph Studer
TL;DR
Straightforward digital oscillator synchronization causes aliasing, motivating direct synthesis of bandlimited synchronized signals. The paper uses spectral resampling with additive synthesis and implements it in HASY, whose output achieves SINAD above 41 dB under a stated frequency condition.
Problem
Naïve digital emulations of oscillator synchronization cause severe aliasing artifacts, while oversampling and sharp low-pass filtering require high computational overhead.
Method
The method linearly resamples Fourier-series coefficients of a bandlimited following waveform and synthesizes the synchronized signal additively, supporting three synchronization modes.
Results
SINAD values exceed 41 dB when the following oscillator’s frequency exceeds the leading oscillator’s frequency.
Takeaways & Limitations
HASY demonstrates practical real-time realization of alias-free oscillator synchronization with latency below five audio samples.
Takeaways & Limitations
Resampling can transform only harmonics already present in the free-running waveform’s Fourier-series coefficients.
Abstract
from arXiv · showhide
Oscillator synchronization is a widely used sound-synthesis technique, but straightforward digital implementations suffer from aliasing artifacts. This paper presents an alias-free method for digital emulation of oscillator synchronization of arbitrary periodic waveforms based on additive synthesis. Starting from a finite set of Fourier-series coefficients representing a bandlimited free-running waveform, we derive linear spectral-resampling transforms that map these coefficients to those of the bandlimited synchronized waveform. Beyond conventional hard synchronization, the proposed approach also supports two additional soft-synchronization modes. To address the high computational complexity of the proposed method, we introduce HASY, a 6 mm^2 application-specific integrated circuit (ASIC) fabricated in 65 nm CMOS technology. HASY generates one 96 kHz, 24 bit alias-free synchronized waveform with up to 512 harmonics and computes the spectral-resampling transform within only five audio-sample periods.
1. INTRODUCTION
Oscillator synchronization creates expressive harmonic content by resetting a following oscillator with a leading oscillator, but naïve digital implementations alias discontinuities. The paper develops alias-free additive-synthesis emulation for arbitrary periodic waveforms and three synchronization modes.
- Hard synchronization: Hard sync resets the following oscillator’s phase whenever the leading oscillator completes a period, potentially creating waveform discontinuities.The reset amplitude determines whether new high-frequency components arise beyond those in the free-running waveform.
- Soft synchronization: Soft sync modifies the reset procedure through waveform reversal or conditional resets, often reducing the perceived harshness of hard sync.These alternatives change the synchronized waveform’s spectral composition.
- Digital implementation: Naïve digital hard-sync emulations cause severe aliasing because discontinuous waveforms are not bandlimited when sampled directly.Extreme oversampling with sharp low-pass filtering can address the problem but incurs high computational overhead.
- Digital implementation: Additive synthesis offers bandlimited synthesis for arbitrary periodic waveforms, whereas software oscillator implementations are often considered computationally expensive.Wavetables and IFFT synthesis provide less computationally intensive alternatives for bandlimited synthesis.
- Proposed approach: The proposed method maps Fourier-series coefficients of a bandlimited following waveform to synchronized coefficients through spectral resampling, then uses additive synthesis to generate the signal.It supports hard, mirrored, and pulsar synchronization modes.
2. SPECTRAL RESAMPLING AND ADDITIVE RESYNTHESIS FOR OSCILLATOR SYNCHRONIZATION
The method transforms Fourier-series coefficients of a bandlimited following-oscillator waveform into synchronized-waveform coefficients, then uses additive synthesis to generate an alias-free signal. It supports hard, mirrored, and pulsar synchronization while imposing scope and timing constraints.
- Scope and limitations: The method emulates periodic synchronized waveforms rather than implementing phase-synchronous time-domain resets.Rapid modulation of Tlead can still induce aliasing whenever the transform is recomputed.
- Hard synchronization: The hard-sync derivation first time-shifts the following-oscillator waveform, then resamples its coefficients over the leading-oscillator period.The time shift aligns the waveform with the symmetrical Fourier integration window; the operation is equivalent to resetting the following oscillator at rate 1/P.
- Additive resynthesis: Only harmonics below the Nyquist frequency fs/2 are synthesized, producing the alias-free discrete-time synchronized output.The retained harmonic set satisfies n/Tlead < fs/2.
- Scope and limitations: The additive-resynthesis implementation restricts output to pure harmonics and leaves the hard-sync result time-shifted unless post-processing is applied.Pure harmonics are integer multiples of the fundamental, and the implementation omits post-rotation or equivalent time-shift compensation.
- Synchronization modes: The framework directly supports hard sync, mirrored sync, and pulsar sync as distinct synchronization modes.Mirrored sync reflects the following waveform at the reset instant, while pulsar sync periodically mutes instead of resetting it.
3. ASIC IMPLEMENTATION
The spectral-resampling approach is computationally demanding, motivating specialized hardware for real-time alias-free synchronization.
- O(N^2) complexity makes the coefficient transform computationally expensive.For N = 512, the hard-sync transform requires 524 288 sinc evaluations, each involving a sine evaluation and division.
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HASY combines a configurable spectral-resampling engine with additive synthesis to implement three alias-free oscillator-synchronization modes. Its parallel ASIC architecture supports real-time operation at 96 kHz, while synchronization increases area and the fabricated chip retains control-logic errors.
- ASIC architecture: HASY implements hard, mirrored, and pulsar synchronization through spectral-resampling transforms and additive synthesis.The spectral-resampling engine computes Fourier-coefficient transforms, while the additive-synthesis oscillator generates the alias-free output.
- Real-time capability: 32 parallel column processors complete the N = 512 Fourier-series coefficient transform in 8 240 clock cycles, or approximately 42 µs.The transform duration is independent of synchronization mode.
- Real-time capability: A new spectral-resampling transform can execute every 5 audio samples at 96 kHz, enabling an approximately 19.2 kHz parameter update rate.This update rate permits modulation of the leading-oscillator frequency flead.
- ASIC measurements: 250 MHz maximum measured clock frequency exceeds the 196.6 MHz required for 96 kHz audio operation, with 242 mW power consumption at 200 MHz.Measurements used a 1.2 V core supply, 300 K room temperature, P = 0.5, and flead = 3 kHz.
- Comparison and limitations: 6 mm^2 HASY generates 512 harmonics per sample and supports three synchronization modes, whereas the 3 mm^2 BFO generates 8192 partials but lacks synchronization.The spectral-resampling engine accounts for 97% of HASY’s cell area, while its additive-synthesis oscillator occupies less than 3%.
- Comparison and limitations: Control-logic errors affecting part of the configuration interface limit the functionality of the fabricated HASY ASIC.The fixed-point golden model and measurements in functioning modes nevertheless confirm the core architectural concept and real-time processing capability.
4. EVALUATION
The evaluation compares the fixed-point HASY model with floating-point and analytical references for alias-free hard-sync synthesis. Accuracy is high for integer period ratios but declines at small P because finite input harmonics limit resampling.
- Evaluation setup: The bit-true HASY MATLAB golden model generates alias-free hard-sync sine and sawtooth waveforms and supports mirrored and pulsar sync examples.The evaluation itself is restricted to hard-sync mode.
- Baselines: The float baseline performs spectral resampling and additive synthesis with up to 512 harmonics, while the analytical baseline resynthesizes ideal hard-sync coefficients with up to 512 harmonics.Both baselines are used to assess HASY output accuracy.
- Metric: 1 s of output, corresponding to L = 96 000 samples, is generated for different following-oscillator waveforms and period ratios P, then evaluated using sample-domain SINAD.The comparison uses HASY and both reference baselines.
- SINAD comparison: Integer P produces high SINAD because spectral resampling reduces to remapping the nth harmonic to the nPth harmonic.This behavior is reported for sine, triangle, and sawtooth following oscillators at flead = 94 Hz.
- SINAD comparison: For P < 1, SINAD drops because the 512-coefficient bandlimited input may lack higher-index harmonics needed by the resampling transform.The effect generally worsens for smaller P and flead.
5. CONCLUSIONS
The paper introduces an additive-synthesis method for alias-free synchronization of arbitrary bandlimited waveforms and implements it in the HASY ASIC. The implementation provides low latency and achieves SINAD above 41 dB under the reported frequency condition, while further work addresses bugs and polyphony.
- Conclusions: The method supports three synchronization modes and arbitrary bandlimited following-oscillator waveforms.The modes include hard sync, mirrored sync, and pulsar sync.
- Conclusions: Fewer than five audio samples of latency demonstrate practical real-time realization of the proposed approach.This result concerns the HASY ASIC implementation.
- Conclusions: SINAD values above 41 dB are achieved against analytical and floating-point baselines when the following oscillator’s frequency exceeds the leading oscillator’s frequency.The comparison is based on HASY output against both reported references.
- Future work: Future work includes a polyphonic HASY ASIC that fixes implementation bugs and a Eurorack module for practical demonstration.The authors also mention deriving Fourier coefficients directly for certain waveforms.