# Sinusoidal Frequency Estimation by Gradient Descent

arXiv: https://arxiv.org/abs/2210.14476 · HTML: https://mlsanity.com/paper/2210.14476/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Oct 2022.

## Built on by (most cited first, 15 of 15)

- [Unsupervised Harmonic Parameter Estimation Using Differentiable DSP and Spectral Optimal Transport](https://mlsanity.com/paper/2312.14507/) (2023)
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- [Toward Fully Self-Supervised Multi-Pitch Estimation](https://mlsanity.com/paper/2402.15569/) (2024)
- [Differentiable Modal Synthesis for Physical Modeling of Planar String Sound and Motion Simulation](https://mlsanity.com/paper/2407.05516/) (2024)
- [Real-time Timbre Remapping with Differentiable DSP](https://mlsanity.com/paper/2407.04547/) (2024)
- [The Inverse Drum Machine: Source Separation Through Joint Transcription and Analysis-by-Synthesis](https://mlsanity.com/paper/2505.03337/) (2025)
- [PESTO: Real-Time Pitch Estimation with Self-supervised Transposition-equivariant Objective](https://mlsanity.com/paper/2508.01488/) (2025)
- [CONMOD: Controllable Neural Frame-based Modulation Effects](https://mlsanity.com/paper/2406.13935/) (2024)
- [AaSP: Aliasing-aware Self-Supervised Pre-Training for Audio Spectrogram Transformers](https://mlsanity.com/paper/2512.03637/) (2025)
- [Gradient-based Optimisation of Modulation Effects](https://mlsanity.com/paper/2601.04867/) (2026)
- [Architecture and Affordances of PLAUD: Performative Latents and Unsupervised DDSP](https://mlsanity.com/paper/2608.13724/) (2026)
- [Evaluating Loss Functions in Differentiable Out-of-Domain Sound-Matching with Partial Parameter Distance](https://mlsanity.com/paper/2608.27698/) (2026)
- [Off-manifold robustness in synthesizer inversion with joint distribution flow matching](https://mlsanity.com/paper/2609.29320/) (2026)

## Abstract (arXiv)

Sinusoidal parameter estimation is a fundamental task in applications from spectral analysis to time-series forecasting. Estimating the sinusoidal frequency parameter by gradient descent is, however, often impossible as the error function is non-convex and densely populated with local minima. The growing family of differentiable signal processing methods has therefore been unable to tune the frequency of oscillatory components, preventing their use in a broad range of applications. This work presents a technique for joint sinusoidal frequency and amplitude estimation using the Wirtinger derivatives of a complex exponential surrogate and any first order gradient-based optimizer, enabling end to-end training of neural network controllers for unconstrained sinusoidal models.