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Defining Fundamental Frequency for Almost Harmonic Signals

Elvander, Filip LU and Jakobsson, Andreas LU orcid (2020) In IEEE Transactions on Signal Processing p.6453-6466
Abstract

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... (More)

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.

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publication status
published
subject
keywords
Computational modeling, Frequency estimation, fundamental frequency estimation, Gaussian noise, Harmonic analysis, Inharmonicity, Maximum likelihood estimation, misspecified models, Numerical models, optimal mass transport
in
IEEE Transactions on Signal Processing
pages
6453 - 6466
publisher
IEEE - Institute of Electrical and Electronics Engineers Inc.
external identifiers
  • scopus:85096864263
ISSN
1053-587X
DOI
10.1109/TSP.2020.3035466
language
English
LU publication?
yes
id
e68fcef6-a4e2-4530-90e9-1a397b390709
date added to LUP
2020-12-14 10:06:40
date last changed
2022-04-19 02:45:45
@article{e68fcef6-a4e2-4530-90e9-1a397b390709,
  abstract     = {{<p>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&amp;#x2018; 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.</p>}},
  author       = {{Elvander, Filip and Jakobsson, Andreas}},
  issn         = {{1053-587X}},
  keywords     = {{Computational modeling; Frequency estimation; fundamental frequency estimation; Gaussian noise; Harmonic analysis; Inharmonicity; Maximum likelihood estimation; misspecified models; Numerical models; optimal mass transport}},
  language     = {{eng}},
  pages        = {{6453--6466}},
  publisher    = {{IEEE - Institute of Electrical and Electronics Engineers Inc.}},
  series       = {{IEEE Transactions on Signal Processing}},
  title        = {{Defining Fundamental Frequency for Almost Harmonic Signals}},
  url          = {{http://dx.doi.org/10.1109/TSP.2020.3035466}},
  doi          = {{10.1109/TSP.2020.3035466}},
  year         = {{2020}},
}