Understanding Harmonic Structures Through Instantaneous Frequency
The analysis of harmonics and non-sinusoidal waveform shape in time-series data is growing in importance. However, a precise definition of what constitutes a harmonic is lacking. In this paper, we propose a rigorous definition of when to consider signals to be in a harmonic relationship based on an...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
IEEE
2022-01-01
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| Series: | IEEE Open Journal of Signal Processing |
| Subjects: | |
| Online Access: | https://ieeexplore.ieee.org/document/9854188/ |
| _version_ | 1828148267836243968 |
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| author | Marco S. Fabus Mark W. Woolrich Catherine W. Warnaby Andrew J. Quinn |
| author_facet | Marco S. Fabus Mark W. Woolrich Catherine W. Warnaby Andrew J. Quinn |
| author_sort | Marco S. Fabus |
| collection | DOAJ |
| description | The analysis of harmonics and non-sinusoidal waveform shape in time-series data is growing in importance. However, a precise definition of what constitutes a harmonic is lacking. In this paper, we propose a rigorous definition of when to consider signals to be in a harmonic relationship based on an integer frequency ratio, constant phase, and a well-defined joint instantaneous frequency. We show this definition is linked to extrema counting and Empirical Mode Decomposition (EMD). We explore the mathematics of our definition and link it to results from analytic number theory. This naturally leads to us to define two classes of harmonic structures, termed strong and weak, with different extrema behaviour. We validate our framework using both simulations and real data. Specifically, we look at the harmonic structures in shallow water waves, the FitzHugh-Nagumo neuronal model, and the non-sinusoidal theta oscillation in rat hippocampus local field potential data. We further discuss how our definition helps to address mode splitting in nonlinear time-series decomposition methods. A clear understanding of when harmonics are present in signals will enable a deeper understanding of the functional roles of non-sinusoidal oscillations. |
| first_indexed | 2024-04-11T21:13:36Z |
| format | Article |
| id | doaj.art-166707f2a61647d9b45ecdb7e49399b2 |
| institution | Directory Open Access Journal |
| issn | 2644-1322 |
| language | English |
| last_indexed | 2024-04-11T21:13:36Z |
| publishDate | 2022-01-01 |
| publisher | IEEE |
| record_format | Article |
| series | IEEE Open Journal of Signal Processing |
| spelling | doaj.art-166707f2a61647d9b45ecdb7e49399b22022-12-22T04:02:53ZengIEEEIEEE Open Journal of Signal Processing2644-13222022-01-01332033410.1109/OJSP.2022.31980129854188Understanding Harmonic Structures Through Instantaneous FrequencyMarco S. Fabus0https://orcid.org/0000-0002-4966-4647Mark W. Woolrich1Catherine W. Warnaby2Andrew J. Quinn3https://orcid.org/0000-0003-2267-9897Nuffield Deparment of Clinical Neurosciences, University of Oxford, Oxford, U.K.Department of Psychiatry, University of Oxford, Oxford, U.K.Nuffield Deparment of Clinical Neurosciences, University of Oxford, Oxford, U.K.Department of Psychiatry, University of Oxford, Oxford, U.K.The analysis of harmonics and non-sinusoidal waveform shape in time-series data is growing in importance. However, a precise definition of what constitutes a harmonic is lacking. In this paper, we propose a rigorous definition of when to consider signals to be in a harmonic relationship based on an integer frequency ratio, constant phase, and a well-defined joint instantaneous frequency. We show this definition is linked to extrema counting and Empirical Mode Decomposition (EMD). We explore the mathematics of our definition and link it to results from analytic number theory. This naturally leads to us to define two classes of harmonic structures, termed strong and weak, with different extrema behaviour. We validate our framework using both simulations and real data. Specifically, we look at the harmonic structures in shallow water waves, the FitzHugh-Nagumo neuronal model, and the non-sinusoidal theta oscillation in rat hippocampus local field potential data. We further discuss how our definition helps to address mode splitting in nonlinear time-series decomposition methods. A clear understanding of when harmonics are present in signals will enable a deeper understanding of the functional roles of non-sinusoidal oscillations.https://ieeexplore.ieee.org/document/9854188/ElectrophysiologyEmpirical Mode DecompositionHarmonic AnalysisHilbert TransformInstantaneous Frequency |
| spellingShingle | Marco S. Fabus Mark W. Woolrich Catherine W. Warnaby Andrew J. Quinn Understanding Harmonic Structures Through Instantaneous Frequency IEEE Open Journal of Signal Processing Electrophysiology Empirical Mode Decomposition Harmonic Analysis Hilbert Transform Instantaneous Frequency |
| title | Understanding Harmonic Structures Through Instantaneous Frequency |
| title_full | Understanding Harmonic Structures Through Instantaneous Frequency |
| title_fullStr | Understanding Harmonic Structures Through Instantaneous Frequency |
| title_full_unstemmed | Understanding Harmonic Structures Through Instantaneous Frequency |
| title_short | Understanding Harmonic Structures Through Instantaneous Frequency |
| title_sort | understanding harmonic structures through instantaneous frequency |
| topic | Electrophysiology Empirical Mode Decomposition Harmonic Analysis Hilbert Transform Instantaneous Frequency |
| url | https://ieeexplore.ieee.org/document/9854188/ |
| work_keys_str_mv | AT marcosfabus understandingharmonicstructuresthroughinstantaneousfrequency AT markwwoolrich understandingharmonicstructuresthroughinstantaneousfrequency AT catherinewwarnaby understandingharmonicstructuresthroughinstantaneousfrequency AT andrewjquinn understandingharmonicstructuresthroughinstantaneousfrequency |