Exploring the Potential of Mixed Fourier Series in Signal Processing Applications Using One-Dimensional Smooth Closed-Form Functions with Compact Support: A Comprehensive Tutorial
This paper studies and analyzes the approximation of one-dimensional smooth closed-form functions with compact support using a mixed Fourier series (i.e., a combination of partial Fourier series and other forms of partial series). To explore the potential of this approach, we discuss and revise its...
| Main Authors: | , , , , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-09-01
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| Series: | Mathematical and Computational Applications |
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| Online Access: | https://www.mdpi.com/2297-8747/28/5/93 |
| _version_ | 1797573087042994176 |
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| author | Carlos-Iván Páez-Rueda Arturo Fajardo Manuel Pérez German Yamhure Gabriel Perilla |
| author_facet | Carlos-Iván Páez-Rueda Arturo Fajardo Manuel Pérez German Yamhure Gabriel Perilla |
| author_sort | Carlos-Iván Páez-Rueda |
| collection | DOAJ |
| description | This paper studies and analyzes the approximation of one-dimensional smooth closed-form functions with compact support using a mixed Fourier series (i.e., a combination of partial Fourier series and other forms of partial series). To explore the potential of this approach, we discuss and revise its application in signal processing, especially because it allows us to control the decreasing rate of Fourier coefficients and avoids the Gibbs phenomenon. Therefore, this method improves the signal processing performance in a wide range of scenarios, such as function approximation, interpolation, increased convergence with quasi-spectral accuracy using the time domain or the frequency domain, numerical integration, and solutions of inverse problems such as ordinary differential equations. Moreover, the paper provides comprehensive examples of one-dimensional problems to showcase the advantages of this approach. |
| first_indexed | 2024-03-10T21:04:41Z |
| format | Article |
| id | doaj.art-6043033c2a8e464d9178f98052b4f720 |
| institution | Directory Open Access Journal |
| issn | 1300-686X 2297-8747 |
| language | English |
| last_indexed | 2024-03-10T21:04:41Z |
| publishDate | 2023-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematical and Computational Applications |
| spelling | doaj.art-6043033c2a8e464d9178f98052b4f7202023-11-19T17:15:35ZengMDPI AGMathematical and Computational Applications1300-686X2297-87472023-09-012859310.3390/mca28050093Exploring the Potential of Mixed Fourier Series in Signal Processing Applications Using One-Dimensional Smooth Closed-Form Functions with Compact Support: A Comprehensive TutorialCarlos-Iván Páez-Rueda0Arturo Fajardo1Manuel Pérez2German Yamhure3Gabriel Perilla4Department of Electronic Engineering, Pontificia Universidad Javeriana, Carrera 7 #40-62, Bogota 110311, ColombiaDepartment of Electronic Engineering, Pontificia Universidad Javeriana, Carrera 7 #40-62, Bogota 110311, ColombiaDepartment of Electronic Engineering, Pontificia Universidad Javeriana, Carrera 7 #40-62, Bogota 110311, ColombiaDepartment of Electronic Engineering, Pontificia Universidad Javeriana, Carrera 7 #40-62, Bogota 110311, ColombiaDepartment of Electronic Engineering, Pontificia Universidad Javeriana, Carrera 7 #40-62, Bogota 110311, ColombiaThis paper studies and analyzes the approximation of one-dimensional smooth closed-form functions with compact support using a mixed Fourier series (i.e., a combination of partial Fourier series and other forms of partial series). To explore the potential of this approach, we discuss and revise its application in signal processing, especially because it allows us to control the decreasing rate of Fourier coefficients and avoids the Gibbs phenomenon. Therefore, this method improves the signal processing performance in a wide range of scenarios, such as function approximation, interpolation, increased convergence with quasi-spectral accuracy using the time domain or the frequency domain, numerical integration, and solutions of inverse problems such as ordinary differential equations. Moreover, the paper provides comprehensive examples of one-dimensional problems to showcase the advantages of this approach.https://www.mdpi.com/2297-8747/28/5/93function reconstructionFourier seriesGibbs phenomenonconvergence accelerationexponential accuracy |
| spellingShingle | Carlos-Iván Páez-Rueda Arturo Fajardo Manuel Pérez German Yamhure Gabriel Perilla Exploring the Potential of Mixed Fourier Series in Signal Processing Applications Using One-Dimensional Smooth Closed-Form Functions with Compact Support: A Comprehensive Tutorial Mathematical and Computational Applications function reconstruction Fourier series Gibbs phenomenon convergence acceleration exponential accuracy |
| title | Exploring the Potential of Mixed Fourier Series in Signal Processing Applications Using One-Dimensional Smooth Closed-Form Functions with Compact Support: A Comprehensive Tutorial |
| title_full | Exploring the Potential of Mixed Fourier Series in Signal Processing Applications Using One-Dimensional Smooth Closed-Form Functions with Compact Support: A Comprehensive Tutorial |
| title_fullStr | Exploring the Potential of Mixed Fourier Series in Signal Processing Applications Using One-Dimensional Smooth Closed-Form Functions with Compact Support: A Comprehensive Tutorial |
| title_full_unstemmed | Exploring the Potential of Mixed Fourier Series in Signal Processing Applications Using One-Dimensional Smooth Closed-Form Functions with Compact Support: A Comprehensive Tutorial |
| title_short | Exploring the Potential of Mixed Fourier Series in Signal Processing Applications Using One-Dimensional Smooth Closed-Form Functions with Compact Support: A Comprehensive Tutorial |
| title_sort | exploring the potential of mixed fourier series in signal processing applications using one dimensional smooth closed form functions with compact support a comprehensive tutorial |
| topic | function reconstruction Fourier series Gibbs phenomenon convergence acceleration exponential accuracy |
| url | https://www.mdpi.com/2297-8747/28/5/93 |
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