# Defining Fundamental Frequency for Almost Harmonic Signals

arXiv: https://arxiv.org/abs/2003.10767 · HTML: https://mlsanity.com/paper/2003.10767/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Mar 2020.

## Built on by (most cited first, 4 of 4)

- [Fredformer: Frequency Debiased Transformer for Time Series Forecasting](https://mlsanity.com/paper/2406.09009/) (2024)
- [Tokenizing Single-Channel EEG with Time-Frequency Motif Learning](https://mlsanity.com/paper/2502.16060/) (2025)
- [FredNormer: Frequency Domain Normalization for Non-stationary Time Series Forecasting](https://mlsanity.com/paper/2410.01860/) (2024)
- [AdaMamba: Adaptive Frequency-Gated Mamba for Long-Term Time Series Forecasting](https://mlsanity.com/paper/2604.23239/) (2026)

## Abstract (arXiv)

In this work, we consider the modeling of signals that are almost, but not quite, harmonic, i.e., composed of sinusoids whose frequencies are close to being integer multiples of a common frequency. Typically, in applications, such signals are treated as perfectly harmonic, allowing for the estimation of their fundamental frequency, despite the signals not actually being periodic. Herein, we provide three different definitions of a concept of fundamental frequency for such inharmonic signals and study the implications of the different choices for modeling and estimation. We show that one of the definitions corresponds to a misspecified modeling scenario, and provides a theoretical benchmark for analyzing the behavior of estimators derived under a perfectly harmonic assumption. The second definition stems from optimal mass transport theory and yields a robust and easily interpretable concept of fundamental frequency based on the signals' spectral properties. The third definition interprets the inharmonic signal as an observation of a randomly perturbed harmonic signal. This allows for computing a hybrid information theoretical bound on estimation performance, as well as for finding an estimator attaining the bound. The theoretical findings are illustrated using numerical examples.