The minimality of mean square error in chirp approximation using fractional fourier series and fractional fourier transform
Abstract Chirps are familiar in nature, have a built-in resistance to noise and interference, and are connected to a wide range of highly oscillatory processes. Detecting chirp oscillating patterns by traditional Fourier series is challenging because the chirp frequencies constantly change over time...
| Main Authors: | , , , , , , , , , |
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| Format: | Article |
| Language: | English |
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Nature Portfolio
2022-11-01
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| Series: | Scientific Reports |
| Online Access: | https://doi.org/10.1038/s41598-022-23560-8 |
| _version_ | 1828098611028688896 |
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| author | Omar T. Bafakeeh Muhammad Yasir Ali Raza Sami Ullah Khan R. Naveen Kumar M. Ijaz Khan Deyab A. Almaleki Nidhal Ben Khedher Sayed M. Eldin Ahmed M. Galal |
| author_facet | Omar T. Bafakeeh Muhammad Yasir Ali Raza Sami Ullah Khan R. Naveen Kumar M. Ijaz Khan Deyab A. Almaleki Nidhal Ben Khedher Sayed M. Eldin Ahmed M. Galal |
| author_sort | Omar T. Bafakeeh |
| collection | DOAJ |
| description | Abstract Chirps are familiar in nature, have a built-in resistance to noise and interference, and are connected to a wide range of highly oscillatory processes. Detecting chirp oscillating patterns by traditional Fourier series is challenging because the chirp frequencies constantly change over time. Estimating such types of functions considering the partial sums of a Fourier series in Fourier analysis does not permit an approximate solution, which entails more Fourier coefficients required for signal reconstruction. The standard Fourier series, therefore, has a poor convergence rate and is an inadequate approximation. In this study, we use a parameterized orthonormal basis with an adjustable parameter to match the oscillating behavior of the chirp to approximate linear chirps using the partial sums of a generalized Fourier series known as fractional Fourier series, which gives the best approximation with only a small number of fractional Fourier coefficients. We used the fractional Fourier transform to compute the fractional Fourier coefficients at sample points. Additionally, we discover that the fractional parameter has the best value at which fractional Fourier coefficients of zero degrees have the most considerable magnitude, leading to the rapid decline of fractional Fourier coefficients of high degrees. Furthermore, fractional Fourier series approximation with optimal fractional parameters provides the minimum mean square error over the fractional Fourier parameter domain. |
| first_indexed | 2024-04-11T08:03:42Z |
| format | Article |
| id | doaj.art-731684e00a2e4add818c75dcec001e7e |
| institution | Directory Open Access Journal |
| issn | 2045-2322 |
| language | English |
| last_indexed | 2024-04-11T08:03:42Z |
| publishDate | 2022-11-01 |
| publisher | Nature Portfolio |
| record_format | Article |
| series | Scientific Reports |
| spelling | doaj.art-731684e00a2e4add818c75dcec001e7e2022-12-22T04:35:37ZengNature PortfolioScientific Reports2045-23222022-11-0112111010.1038/s41598-022-23560-8The minimality of mean square error in chirp approximation using fractional fourier series and fractional fourier transformOmar T. Bafakeeh0Muhammad Yasir1Ali Raza2Sami Ullah Khan3R. Naveen Kumar4M. Ijaz Khan5Deyab A. Almaleki6Nidhal Ben Khedher7Sayed M. Eldin8Ahmed M. Galal9Department of Industrial Engineering, Jazan UniversityDepartment of Mathematics, University of Engineering and TechnologyDepartment of Mathematics, University of Engineering and TechnologyDepartment of Mathematics, COMSATS University IslamabadDepartment of Studies and Research in Mathematics, Davangere UniversityDepartment of Mechanical Engineering, Lebanese American UniversityUmm Al-Qura UniversityDepartment of Mechanical Engineering, College of Engineering, University of HailCenter of Research, Faculty of Engineering, Future University in Egypt New CairoMechanical Engineering Department, College of Engineering, Prince Sattam Bin Abdulaziz UniversityAbstract Chirps are familiar in nature, have a built-in resistance to noise and interference, and are connected to a wide range of highly oscillatory processes. Detecting chirp oscillating patterns by traditional Fourier series is challenging because the chirp frequencies constantly change over time. Estimating such types of functions considering the partial sums of a Fourier series in Fourier analysis does not permit an approximate solution, which entails more Fourier coefficients required for signal reconstruction. The standard Fourier series, therefore, has a poor convergence rate and is an inadequate approximation. In this study, we use a parameterized orthonormal basis with an adjustable parameter to match the oscillating behavior of the chirp to approximate linear chirps using the partial sums of a generalized Fourier series known as fractional Fourier series, which gives the best approximation with only a small number of fractional Fourier coefficients. We used the fractional Fourier transform to compute the fractional Fourier coefficients at sample points. Additionally, we discover that the fractional parameter has the best value at which fractional Fourier coefficients of zero degrees have the most considerable magnitude, leading to the rapid decline of fractional Fourier coefficients of high degrees. Furthermore, fractional Fourier series approximation with optimal fractional parameters provides the minimum mean square error over the fractional Fourier parameter domain.https://doi.org/10.1038/s41598-022-23560-8 |
| spellingShingle | Omar T. Bafakeeh Muhammad Yasir Ali Raza Sami Ullah Khan R. Naveen Kumar M. Ijaz Khan Deyab A. Almaleki Nidhal Ben Khedher Sayed M. Eldin Ahmed M. Galal The minimality of mean square error in chirp approximation using fractional fourier series and fractional fourier transform Scientific Reports |
| title | The minimality of mean square error in chirp approximation using fractional fourier series and fractional fourier transform |
| title_full | The minimality of mean square error in chirp approximation using fractional fourier series and fractional fourier transform |
| title_fullStr | The minimality of mean square error in chirp approximation using fractional fourier series and fractional fourier transform |
| title_full_unstemmed | The minimality of mean square error in chirp approximation using fractional fourier series and fractional fourier transform |
| title_short | The minimality of mean square error in chirp approximation using fractional fourier series and fractional fourier transform |
| title_sort | minimality of mean square error in chirp approximation using fractional fourier series and fractional fourier transform |
| url | https://doi.org/10.1038/s41598-022-23560-8 |
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