Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules
The theory of the continuous two-dimensional (2D) Fourier transform in polar coordinates has been recently developed but no discrete counterpart exists to date. In this paper, we propose and evaluate the theory of the 2D discrete Fourier transform (DFT) in polar coordinates. This discrete theory is...
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| Format: | Article |
| Language: | English |
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MDPI AG
2019-08-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/7/8/698 |
| _version_ | 1811287918686240768 |
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| author | Natalie Baddour |
| author_facet | Natalie Baddour |
| author_sort | Natalie Baddour |
| collection | DOAJ |
| description | The theory of the continuous two-dimensional (2D) Fourier transform in polar coordinates has been recently developed but no discrete counterpart exists to date. In this paper, we propose and evaluate the theory of the 2D discrete Fourier transform (DFT) in polar coordinates. This discrete theory is shown to arise from discretization schemes that have been previously employed with the 1D DFT and the discrete Hankel transform (DHT). The proposed transform possesses orthogonality properties, which leads to invertibility of the transform. In the first part of this two-part paper, the theory of the actual manipulated quantities is shown, including the standard set of shift, modulation, multiplication, and convolution rules. Parseval and modified Parseval relationships are shown, depending on which choice of kernel is used. Similar to its continuous counterpart, the 2D DFT in polar coordinates is shown to consist of a 1D DFT, DHT and 1D inverse DFT. |
| first_indexed | 2024-04-13T03:26:56Z |
| format | Article |
| id | doaj.art-1ff964732dc94103966118ac371d2202 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-04-13T03:26:56Z |
| publishDate | 2019-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-1ff964732dc94103966118ac371d22022022-12-22T03:04:37ZengMDPI AGMathematics2227-73902019-08-017869810.3390/math7080698math7080698Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational RulesNatalie Baddour0Department of Mechanical Engineering, University of Ottawa, 161 Louis Pasteur, Ottawa, ON K1N 6N5, CanadaThe theory of the continuous two-dimensional (2D) Fourier transform in polar coordinates has been recently developed but no discrete counterpart exists to date. In this paper, we propose and evaluate the theory of the 2D discrete Fourier transform (DFT) in polar coordinates. This discrete theory is shown to arise from discretization schemes that have been previously employed with the 1D DFT and the discrete Hankel transform (DHT). The proposed transform possesses orthogonality properties, which leads to invertibility of the transform. In the first part of this two-part paper, the theory of the actual manipulated quantities is shown, including the standard set of shift, modulation, multiplication, and convolution rules. Parseval and modified Parseval relationships are shown, depending on which choice of kernel is used. Similar to its continuous counterpart, the 2D DFT in polar coordinates is shown to consist of a 1D DFT, DHT and 1D inverse DFT.https://www.mdpi.com/2227-7390/7/8/698Fourier TheoryDFT in polar coordinatespolar coordinatesmultidimensional DFTdiscrete Hankel Transformdiscrete Fourier TransformOrthogonality |
| spellingShingle | Natalie Baddour Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules Mathematics Fourier Theory DFT in polar coordinates polar coordinates multidimensional DFT discrete Hankel Transform discrete Fourier Transform Orthogonality |
| title | Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules |
| title_full | Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules |
| title_fullStr | Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules |
| title_full_unstemmed | Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules |
| title_short | Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules |
| title_sort | discrete two dimensional fourier transform in polar coordinates part i theory and operational rules |
| topic | Fourier Theory DFT in polar coordinates polar coordinates multidimensional DFT discrete Hankel Transform discrete Fourier Transform Orthogonality |
| url | https://www.mdpi.com/2227-7390/7/8/698 |
| work_keys_str_mv | AT nataliebaddour discretetwodimensionalfouriertransforminpolarcoordinatespartitheoryandoperationalrules |