Quadratic-Phase Wave-Packet Transform in <i><b>L</b></i><sup>2</sup>(<inline-formula><math display="inline"><semantics><mrow><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>)
Wavelet transform is a powerful tool for analysing the problems arising in harmonic analysis, signal and image processing, sampling, filtering, and so on. However, they seem to be inadequate for representing those signals whose energy is not well concentrated in the frequency domain. In pursuit of r...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-09-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/14/10/2018 |
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| author | Hari M. Srivastava Firdous A. Shah Waseem Z. Lone |
| author_facet | Hari M. Srivastava Firdous A. Shah Waseem Z. Lone |
| author_sort | Hari M. Srivastava |
| collection | DOAJ |
| description | Wavelet transform is a powerful tool for analysing the problems arising in harmonic analysis, signal and image processing, sampling, filtering, and so on. However, they seem to be inadequate for representing those signals whose energy is not well concentrated in the frequency domain. In pursuit of representations of such signals, we propose a novel time-frequency transform coined as quadratic-phase wave packet transform in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>L</mi><mn>2</mn></msup><mrow><mo>(</mo><mi mathvariant="double-struck">R</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. The proposed transform is aimed at rectifying the conventional wavelet transform by employing a quadratic-phase Fourier transform with extra degrees of freedom. Besides the formulation of all the fundamental results, including the orthogonality relation, reconstruction formula and the characterization of range, we also derive a direct relationship between the well-known Wigner-Ville distribution and the proposed transform. In addition, we study the quadratic-phase wave-packet transform in the framework of almost periodic functions. Finally, we extend the scope of the present work by investigating the composition of quadratic-phase wave packet transforms. |
| first_indexed | 2024-03-09T19:26:57Z |
| format | Article |
| id | doaj.art-8ff0900868dd46f5bb5763dcb63ef686 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-09T19:26:57Z |
| publishDate | 2022-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-8ff0900868dd46f5bb5763dcb63ef6862023-11-24T02:50:46ZengMDPI AGSymmetry2073-89942022-09-011410201810.3390/sym14102018Quadratic-Phase Wave-Packet Transform in <i><b>L</b></i><sup>2</sup>(<inline-formula><math display="inline"><semantics><mrow><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>)Hari M. Srivastava0Firdous A. Shah1Waseem Z. Lone2Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, CanadaDepartment of Mathematics, University of Kashmir, South Campus, Anantnag 192101, Jammu and Kashmir, IndiaDepartment of Mathematics, University of Kashmir, South Campus, Anantnag 192101, Jammu and Kashmir, IndiaWavelet transform is a powerful tool for analysing the problems arising in harmonic analysis, signal and image processing, sampling, filtering, and so on. However, they seem to be inadequate for representing those signals whose energy is not well concentrated in the frequency domain. In pursuit of representations of such signals, we propose a novel time-frequency transform coined as quadratic-phase wave packet transform in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>L</mi><mn>2</mn></msup><mrow><mo>(</mo><mi mathvariant="double-struck">R</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. The proposed transform is aimed at rectifying the conventional wavelet transform by employing a quadratic-phase Fourier transform with extra degrees of freedom. Besides the formulation of all the fundamental results, including the orthogonality relation, reconstruction formula and the characterization of range, we also derive a direct relationship between the well-known Wigner-Ville distribution and the proposed transform. In addition, we study the quadratic-phase wave-packet transform in the framework of almost periodic functions. Finally, we extend the scope of the present work by investigating the composition of quadratic-phase wave packet transforms.https://www.mdpi.com/2073-8994/14/10/2018quadratic-phase Fourier transformwave-packet transformWigner-Ville distributionperiodic functioncomposition operator |
| spellingShingle | Hari M. Srivastava Firdous A. Shah Waseem Z. Lone Quadratic-Phase Wave-Packet Transform in <i><b>L</b></i><sup>2</sup>(<inline-formula><math display="inline"><semantics><mrow><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>) Symmetry quadratic-phase Fourier transform wave-packet transform Wigner-Ville distribution periodic function composition operator |
| title | Quadratic-Phase Wave-Packet Transform in <i><b>L</b></i><sup>2</sup>(<inline-formula><math display="inline"><semantics><mrow><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>) |
| title_full | Quadratic-Phase Wave-Packet Transform in <i><b>L</b></i><sup>2</sup>(<inline-formula><math display="inline"><semantics><mrow><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>) |
| title_fullStr | Quadratic-Phase Wave-Packet Transform in <i><b>L</b></i><sup>2</sup>(<inline-formula><math display="inline"><semantics><mrow><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>) |
| title_full_unstemmed | Quadratic-Phase Wave-Packet Transform in <i><b>L</b></i><sup>2</sup>(<inline-formula><math display="inline"><semantics><mrow><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>) |
| title_short | Quadratic-Phase Wave-Packet Transform in <i><b>L</b></i><sup>2</sup>(<inline-formula><math display="inline"><semantics><mrow><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>) |
| title_sort | quadratic phase wave packet transform in i b l b i sup 2 sup inline formula math display inline semantics mrow mi mathvariant double struck r mi mrow semantics math inline formula |
| topic | quadratic-phase Fourier transform wave-packet transform Wigner-Ville distribution periodic function composition operator |
| url | https://www.mdpi.com/2073-8994/14/10/2018 |
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