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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Autor*in:	Natalie Baddour [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2019

Schlagwörter:	Fourier Theory DFT in polar coordinates polar coordinates multidimensional DFT discrete Hankel Transform discrete Fourier Transform Orthogonality

Übergeordnetes Werk:	In: Mathematics - MDPI AG, 2013, 7(2019), 8, p 698
Übergeordnetes Werk:	volume:7 ; year:2019 ; number:8, p 698

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DOI / URN:	10.3390/math7080698

Katalog-ID:	DOAJ024544256

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allfields	10.3390/math7080698 doi (DE-627)DOAJ024544256 (DE-599)DOAJ1ff964732dc94103966118ac371d2202 DE-627 ger DE-627 rakwb eng QA1-939 Natalie Baddour verfasserin aut Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules 2019 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier 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. Fourier Theory DFT in polar coordinates polar coordinates multidimensional DFT discrete Hankel Transform discrete Fourier Transform Orthogonality Mathematics In Mathematics MDPI AG, 2013 7(2019), 8, p 698 (DE-627)737287764 (DE-600)2704244-3 22277390 nnns volume:7 year:2019 number:8, p 698 https://doi.org/10.3390/math7080698 kostenfrei https://doaj.org/article/1ff964732dc94103966118ac371d2202 kostenfrei https://www.mdpi.com/2227-7390/7/8/698 kostenfrei https://doaj.org/toc/2227-7390 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 7 2019 8, p 698
spelling	10.3390/math7080698 doi (DE-627)DOAJ024544256 (DE-599)DOAJ1ff964732dc94103966118ac371d2202 DE-627 ger DE-627 rakwb eng QA1-939 Natalie Baddour verfasserin aut Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules 2019 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier 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. Fourier Theory DFT in polar coordinates polar coordinates multidimensional DFT discrete Hankel Transform discrete Fourier Transform Orthogonality Mathematics In Mathematics MDPI AG, 2013 7(2019), 8, p 698 (DE-627)737287764 (DE-600)2704244-3 22277390 nnns volume:7 year:2019 number:8, p 698 https://doi.org/10.3390/math7080698 kostenfrei https://doaj.org/article/1ff964732dc94103966118ac371d2202 kostenfrei https://www.mdpi.com/2227-7390/7/8/698 kostenfrei https://doaj.org/toc/2227-7390 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 7 2019 8, p 698
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allfieldsGer	10.3390/math7080698 doi (DE-627)DOAJ024544256 (DE-599)DOAJ1ff964732dc94103966118ac371d2202 DE-627 ger DE-627 rakwb eng QA1-939 Natalie Baddour verfasserin aut Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules 2019 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier 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. Fourier Theory DFT in polar coordinates polar coordinates multidimensional DFT discrete Hankel Transform discrete Fourier Transform Orthogonality Mathematics In Mathematics MDPI AG, 2013 7(2019), 8, p 698 (DE-627)737287764 (DE-600)2704244-3 22277390 nnns volume:7 year:2019 number:8, p 698 https://doi.org/10.3390/math7080698 kostenfrei https://doaj.org/article/1ff964732dc94103966118ac371d2202 kostenfrei https://www.mdpi.com/2227-7390/7/8/698 kostenfrei https://doaj.org/toc/2227-7390 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 7 2019 8, p 698
allfieldsSound	10.3390/math7080698 doi (DE-627)DOAJ024544256 (DE-599)DOAJ1ff964732dc94103966118ac371d2202 DE-627 ger DE-627 rakwb eng QA1-939 Natalie Baddour verfasserin aut Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules 2019 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier 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. Fourier Theory DFT in polar coordinates polar coordinates multidimensional DFT discrete Hankel Transform discrete Fourier Transform Orthogonality Mathematics In Mathematics MDPI AG, 2013 7(2019), 8, p 698 (DE-627)737287764 (DE-600)2704244-3 22277390 nnns volume:7 year:2019 number:8, p 698 https://doi.org/10.3390/math7080698 kostenfrei https://doaj.org/article/1ff964732dc94103966118ac371d2202 kostenfrei https://www.mdpi.com/2227-7390/7/8/698 kostenfrei https://doaj.org/toc/2227-7390 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 7 2019 8, p 698
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source	In Mathematics 7(2019), 8, p 698 volume:7 year:2019 number:8, p 698
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author	Natalie Baddour
spellingShingle	Natalie Baddour misc QA1-939 misc Fourier Theory misc DFT in polar coordinates misc polar coordinates misc multidimensional DFT misc discrete Hankel Transform misc discrete Fourier Transform misc Orthogonality misc Mathematics Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules
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topic_title	QA1-939 Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules Fourier Theory DFT in polar coordinates polar coordinates multidimensional DFT discrete Hankel Transform discrete Fourier Transform Orthogonality
topic	misc QA1-939 misc Fourier Theory misc DFT in polar coordinates misc polar coordinates misc multidimensional DFT misc discrete Hankel Transform misc discrete Fourier Transform misc Orthogonality misc Mathematics
topic_unstemmed	misc QA1-939 misc Fourier Theory misc DFT in polar coordinates misc polar coordinates misc multidimensional DFT misc discrete Hankel Transform misc discrete Fourier Transform misc Orthogonality misc Mathematics
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title	Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules
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title_sort	discrete two-dimensional fourier transform in polar coordinates part i: theory and operational rules
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title_auth	Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules
abstract	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.
abstractGer	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.
abstract_unstemmed	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.
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title_short	Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules
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Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules

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