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...
| Váldodahkkit: | , , , |
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| Materiálatiipa: | Journal article |
| Giella: | English |
| Almmustuhtton: |
Institute of Electrical and Electronics Engineers
2022
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| _version_ | 1826308316407529472 |
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| author | Fabus, MS Woolrich, MW Warnaby, CW Quinn, AJ |
| author_facet | Fabus, MS Woolrich, MW Warnaby, CW Quinn, AJ |
| author_sort | Fabus, MS |
| collection | OXFORD |
| 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 | 2024-03-07T07:17:43Z |
| format | Journal article |
| id | oxford-uuid:0884c8dd-1604-4c33-bb84-97af4eef3e2c |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T07:17:43Z |
| publishDate | 2022 |
| publisher | Institute of Electrical and Electronics Engineers |
| record_format | dspace |
| spelling | oxford-uuid:0884c8dd-1604-4c33-bb84-97af4eef3e2c2022-09-02T13:08:44ZUnderstanding harmonic structures through instantaneous frequencyJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:0884c8dd-1604-4c33-bb84-97af4eef3e2cEnglishSymplectic ElementsInstitute of Electrical and Electronics Engineers2022Fabus, MSWoolrich, MWWarnaby, CWQuinn, AJThe 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. |
| spellingShingle | Fabus, MS Woolrich, MW Warnaby, CW Quinn, AJ Understanding harmonic structures through 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 |
| work_keys_str_mv | AT fabusms understandingharmonicstructuresthroughinstantaneousfrequency AT woolrichmw understandingharmonicstructuresthroughinstantaneousfrequency AT warnabycw understandingharmonicstructuresthroughinstantaneousfrequency AT quinnaj understandingharmonicstructuresthroughinstantaneousfrequency |