Generalized Sampling Theory in the Quaternion Domain: A Fractional Fourier Approach
The field of quaternions has made a substantial impact on signal processing research, with numerous studies exploring their applications. Building on this foundation, this article extends the study of sampling theory using the quaternion fractional Fourier Transform (QFRFT). We first propose a gener...
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MDPI AG
2024-12-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/8/12/748 |
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| author | Muhammad Adnan Samad Yuanqing Xia Nader Al-Rashidi Saima Siddiqui Muhammad Younus Bhat Huda M. Alshanbari |
| author_facet | Muhammad Adnan Samad Yuanqing Xia Nader Al-Rashidi Saima Siddiqui Muhammad Younus Bhat Huda M. Alshanbari |
| author_sort | Muhammad Adnan Samad |
| collection | DOAJ |
| description | The field of quaternions has made a substantial impact on signal processing research, with numerous studies exploring their applications. Building on this foundation, this article extends the study of sampling theory using the quaternion fractional Fourier Transform (QFRFT). We first propose a generalized sampling expansion (GSE) for fractional bandlimited signals via the QFRFT, extending the classical Papoulis expansion. Next, we design fractional quaternion Fourier filters to reconstruct both the signals and their derivatives, based on the GSE and QFRFT properties. We illustrate the practical utility of the QFRFT-based GSE framework with a case study on signal denoising, demonstrating its effectiveness in noise reduction with the Mean Squared Error (MSE), highlighting the improvement in signal restoration. |
| format | Article |
| id | doaj-art-4c070f01667a4f1ba5f2da1070ef6d60 |
| institution | DOAJ |
| issn | 2504-3110 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj-art-4c070f01667a4f1ba5f2da1070ef6d602025-08-20T02:53:37ZengMDPI AGFractal and Fractional2504-31102024-12-0181274810.3390/fractalfract8120748Generalized Sampling Theory in the Quaternion Domain: A Fractional Fourier ApproachMuhammad Adnan Samad0Yuanqing Xia1Nader Al-Rashidi2Saima Siddiqui3Muhammad Younus Bhat4Huda M. Alshanbari5School of Automation, Beijing Institute of Technology, Beijing 100081, ChinaSchool of Automation, Beijing Institute of Technology, Beijing 100081, ChinaDepartment of Mathematics, College of Science and Humanities, Shaqra University, P.O. Box 1390, Dwadmy 11911, Saudi ArabiaDepartment of Mathematics, Fergana Polytechnic Institute, Fergana 150100, UzbekistanDepartment of Mathematical Sciences, Islamic University of Science and Technology, Awantipora 192122, IndiaDepartment of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi ArabiaThe field of quaternions has made a substantial impact on signal processing research, with numerous studies exploring their applications. Building on this foundation, this article extends the study of sampling theory using the quaternion fractional Fourier Transform (QFRFT). We first propose a generalized sampling expansion (GSE) for fractional bandlimited signals via the QFRFT, extending the classical Papoulis expansion. Next, we design fractional quaternion Fourier filters to reconstruct both the signals and their derivatives, based on the GSE and QFRFT properties. We illustrate the practical utility of the QFRFT-based GSE framework with a case study on signal denoising, demonstrating its effectiveness in noise reduction with the Mean Squared Error (MSE), highlighting the improvement in signal restoration.https://www.mdpi.com/2504-3110/8/12/748quaternion algebrafractional bandlimited signalquaternion fractional Fourier transformquaternion fractional Fourier filtersgeneralized sampling expansion |
| spellingShingle | Muhammad Adnan Samad Yuanqing Xia Nader Al-Rashidi Saima Siddiqui Muhammad Younus Bhat Huda M. Alshanbari Generalized Sampling Theory in the Quaternion Domain: A Fractional Fourier Approach Fractal and Fractional quaternion algebra fractional bandlimited signal quaternion fractional Fourier transform quaternion fractional Fourier filters generalized sampling expansion |
| title | Generalized Sampling Theory in the Quaternion Domain: A Fractional Fourier Approach |
| title_full | Generalized Sampling Theory in the Quaternion Domain: A Fractional Fourier Approach |
| title_fullStr | Generalized Sampling Theory in the Quaternion Domain: A Fractional Fourier Approach |
| title_full_unstemmed | Generalized Sampling Theory in the Quaternion Domain: A Fractional Fourier Approach |
| title_short | Generalized Sampling Theory in the Quaternion Domain: A Fractional Fourier Approach |
| title_sort | generalized sampling theory in the quaternion domain a fractional fourier approach |
| topic | quaternion algebra fractional bandlimited signal quaternion fractional Fourier transform quaternion fractional Fourier filters generalized sampling expansion |
| url | https://www.mdpi.com/2504-3110/8/12/748 |
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