Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules
Abstract
:1. Introduction
2. Background: Continuous 2D Fourier Transforms in Polar Coordinates
3. Motivation for the Discrete 2D Fourier Transform in Polar Coordinates
3.1. Space-Limited Functions
3.1.1. Forward Transform
3.1.2. Inverse Transform
3.2. Band-Limited Functions
3.3. Summary of Above Relationships
4. Proposed Kernel for the Discrete Transform
4.1. Proposed Kernel for 2D Polar Discrete Fourier Transform
4.2. Another Choice of Kernel
4.3. Orthogonality of the Proposed Kernel
5. Proposed Transform
Forward and Inverse Transform
6. Properties of the Transform—Transform Rules
6.1. The Complex Exponential
6.2. The Delta Function
6.3. The Generalized Shift Operator
6.4. Forward Transform of the Generalized Shift
6.5. Modulation
6.6. Convolution–Multiplication
6.7. Multiplication–Convolution Rule
6.8. Rotation
7. Generalized Parseval Theorem
7.1. Parseval’s Theorem with the Symmetric Kernel
7.2. Parseval’s Theorem with the Non-Symmetric Kernel
8. Discussion: Interpretation of the Transform
8.1. Interpretation of the 2D Forward DFT in Polar Coordinates
8.2. Interpretation of the 2D Inverse DFT in Polar Coordinates
9. Conclusions
Funding
Conflicts of Interest
Appendix A
Appendix A.1. Hankel Transform
Appendix A.2. Fourier–Bessel Series
Appendix A.3. Orthogonality of the Discrete Complex Exponential
Appendix A.4. Discrete Orthogonality of the Bessel Functions
Appendix A.5. Fourier Series and Finite Fourier Transform
Appendix A.6. Sampling Points
Appendix A.7. Proofs of Orthogonality of the Proposed Kernel
Appendix A.7.1. Proof of Orthogonality of the Kernel over the Frequency Indices
Appendix A.7.2. Proof of Orthogonality of the Kernel over the Spatial Indices
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Baddour, N. Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules. Mathematics 2019, 7, 698. https://doi.org/10.3390/math7080698
Baddour N. Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules. Mathematics. 2019; 7(8):698. https://doi.org/10.3390/math7080698
Chicago/Turabian StyleBaddour, Natalie. 2019. "Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules" Mathematics 7, no. 8: 698. https://doi.org/10.3390/math7080698
APA StyleBaddour, N. (2019). Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules. Mathematics, 7(8), 698. https://doi.org/10.3390/math7080698