Numerical Computation of the Discrete 2D Fourier Transform in Polar Coordinates

2018 ◽  
Author(s):  
Xueyang Yao ◽  
Natalie Baddour
Keyword(s):  
Fourier Transform ◽  
Numerical Simulations ◽  
Discrete Fourier Transform ◽  
Good Accuracy ◽  
Hankel Transform ◽  
Polar Coordinates ◽  
Good Precision ◽  
Numerical Implementation ◽  
Inverse Transform ◽  
Accuracy And Precision

The discrete Fourier transform in Cartesian coordinates has proven to be invaluable in many disciplines. However, in application such as photoacoustics and tomography, a discrete 2D-Fourier transform in polar coordinates is needed. In this paper, a discrete 2D-Fourier transform in polar coordinates is presented. It is shown that numerical implementation is best achieved by interpreting the transform as a 1D-discrete Fourier transform (DFT), a 1D-discrete Hankel transform (DHT) and a 1D-discrete inverse transform (IDFT) in sequence. The transform is tested by numerical simulations with respect to accuracy and precision for computation of the continuous 2D transform at specific discrete points. It was found that both the forward and inverse transform showed good accuracy to approximate the continuous Fourier transform. Moreover, good precision results were obtained, which indicate that the proposed transform itself does not add much error.

scholarly journals Discrete Two Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules

2019 ◽  
Author(s):  
Natalie Baddour
Keyword(s):  
Fourier Transform ◽  
Discrete Fourier Transform ◽  
Hankel Transform ◽  
Polar Coordinates ◽  
Two Dimensional ◽  
Discretization Schemes ◽  
Discrete Counterpart ◽  
Standard Set ◽  
Discrete Theory

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.

scholarly journals Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 698 ◽  
Author(s):  
Baddour
Keyword(s):  
Fourier Transform ◽  
Discrete Fourier Transform ◽  
Hankel Transform ◽  
Polar Coordinates ◽  
Two Dimensional ◽  
Discretization Schemes ◽  
Discrete Counterpart ◽  
Standard Set ◽  
Discrete Theory

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.

scholarly journals Discrete two dimensional Fourier transform in polar coordinates part II: numerical computation and approximation of the continuous transform

2020 ◽  
Vol 6 ◽  
pp. e257
Author(s):  
Xueyang Yao ◽  
Natalie Baddour
Keyword(s):  
Fourier Transform ◽  
Numerical Computation ◽  
Discrete Fourier Transform ◽  
Inverse Fourier Transform ◽  
Hankel Transform ◽  
Polar Coordinates ◽  
Two Dimensional ◽  
Computational Aspects ◽  
Discrete Counterpart ◽  
Standard Set

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 the first part of this two-paper series, we proposed and evaluated the theory of the 2D Discrete Fourier Transform (DFT) in polar coordinates. The theory of the actual manipulated quantities was shown, including the standard set of shift, modulation, multiplication, and convolution rules. In this second part of the series, we address the computational aspects of the 2D DFT in polar coordinates. Specifically, we demonstrate how the decomposition of the 2D DFT as a DFT, Discrete Hankel Transform and inverse DFT sequence can be exploited for coding. We also demonstrate how the proposed 2D DFT can be used to approximate the continuous forward and inverse Fourier transform in polar coordinates in the same manner that the 1D DFT can be used to approximate its continuous counterpart.

scholarly journals The Solvability of a Class of Convolution Equations Associated with 2D FRFT

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1928
Author(s):  
Zhen-Wei Li ◽  
Wen-Biao Gao ◽  
Bing-Zhao Li
Keyword(s):  
Fourier Transform ◽  
Convolution Operator ◽  
Fractional Fourier Transform ◽  
Hankel Transform ◽  
Polar Coordinates ◽  
Two Dimensional ◽  
Spatial Shift ◽  
Fractional Hankel Transform ◽  
The Relationship ◽  
Convolution Equations

In this paper, the solvability of a class of convolution equations is discussed by using two-dimensional (2D) fractional Fourier transform (FRFT) in polar coordinates. Firstly, we generalize the 2D FRFT to the polar coordinates setting. The relationship between 2D FRFT and fractional Hankel transform (FRHT) is derived. Secondly, the spatial shift and multiplication theorems for 2D FRFT are proposed by using this relationship. Thirdly, in order to analyze the solvability of the convolution equations, a novel convolution operator for 2D FRFT is proposed, and the corresponding convolution theorem is investigated. Finally, based on the proposed theorems, the solvability of the convolution equations is studied.

scholarly journals Calculation of Lightning Transient Responses on Wind Turbine Towers

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Xiaoqing Zhang ◽  
Yongzheng Zhang
Keyword(s):  
Fourier Transform ◽  
Wind Turbine ◽  
Frequency Domain ◽  
Equivalent Circuit ◽  
Discrete Fourier Transform ◽  
Efficient Method ◽  
Transient Responses ◽  
Numerical Example ◽  
Inverse Transform ◽  
Wind Turbine Towers

An efficient method is proposed in this paper for calculating lightning transient responses on wind turbine towers. In the proposed method, the actual tower body is simplified as a multiconductor grid in the shape of cylinder. A set of formulas are given for evaluating the circuit parameters of the branches in the multiconductor grid. On the basis of the circuit parameters, the multiconductor grid is further converted into an equivalent circuit. The circuit equation is built in frequency-domain to take into account the effect of the frequency-dependent characteristic of the resistances and inductances on lightning transients. The lightning transient responses can be obtained by using the discrete Fourier transform with exponential sampling to take the inverse transform of the frequency-domain solution of the circuit equation. A numerical example has been given for examining the applicability of the proposed method.

scholarly journals Discrete Two Dimensional Fourier Transform in Polar Coordinates Part II: Numerical Computation and Approximation of the Continuous Transform

2019 ◽  
Author(s):  
Xueyang Yao ◽  
Natalie Baddour
Keyword(s):  
Fourier Transform ◽  
Numerical Computation ◽  
Inverse Fourier Transform ◽  
Hankel Transform ◽  
Polar Coordinates ◽  
Two Dimensional ◽  
Computational Aspects ◽  
Discrete Counterpart ◽  
Efficient Code ◽  
Standard Set

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 the first part of this two-paper series, we proposed and evaluated the theory of the 2D discrete Fourier Transform (DFT) in polar coordinates. The theory of the actual manipulated quantities was shown, including the standard set of shift, modulation, multiplication, and convolution rules. In this second part of the series, we address the computational aspects of the 2D DFT in polar coordinates. Specifically, we demonstrate how the decomposition of the 2D DFT as a DFT, Discrete Hankel Transform (DHT) and inverse DFT sequence can be exploited for efficient code. We also demonstrate how the proposed 2D DFT can be used to approximate the continuous forward and inverse Fourier transform in polar coordinates in the same manner that the 1D DFT can be used to approximate its continuous counterpart.

scholarly journals Droplet activation parameterization: the population splitting concept revisited

2014 ◽  
Vol 7 (3) ◽  
pp. 2903-2932 ◽  
Author(s):  
R. Morales Betancourt ◽  
A. Nenes
Keyword(s):  
Numerical Simulations ◽  
Number Concentration ◽  
Activation Process ◽  
Numerical Implementation ◽  
First Order ◽  
Water Condensation ◽  
Accuracy And Precision ◽  
Droplet Activation ◽  
Input Variables ◽  
Improved Accuracy

Abstract. In this work we postulate, implement and evaluate modifications to the "population splitting" concept introduced by Nenes and Seinfeld (2003) for calculation of water condensation rates in droplet activation parameterizations. The modifications introduced here lead to an improved accuracy and precision of the parameterization-derived maximum supersaturation, smax, and droplet number concentration, Nd, as determined by comparing against those of detailed numerical simulations of the activation process. A numerical computation of the first-order derivatives ∂ Nd/∂ χj of the parameterized Nd to input variables χj was performed, and compared against the corresponding parcel model derived sensitivities, providing a thorough evaluation of the impacts of the introduced modifications in the parameterization ability to respond to aerosol characteristics. The proposed modifications require only minor changes for their numerical implementation in existing codes based on the population splitting concept.

scholarly journals Estimation and Correction of Gain Mismatch and Timing Error in Time-Interleaved ADCs Based on DFT

2014 ◽  
Vol 21 (3) ◽  
pp. 535-544 ◽  
Author(s):  
Lianping Guo ◽  
Shulin Tian ◽  
Zhigang Wang
Keyword(s):  
Fourier Transform ◽  
Numerical Simulations ◽  
Discrete Fourier Transform ◽  
Analog To Digital Converter ◽  
Digital Converter ◽  
Correction Algorithm ◽  
Timing Error ◽  
Analog To Digital ◽  
Time Interleaved ◽  
Estimation And Correction

Abstract Time-interleaved analog-to-digital converter (ADC) architecture is crucial to increase the maximum sample rate. However, offset mismatch, gain mismatch, and timing error between time-interleaved channels degrade the performance of time-interleaved ADCs. This paper focuses on the gain mismatch and timing error. Techniques based on Discrete Fourier Transform (DFT) for estimating and correcting gain mismatch and timing error in an M-channel ADC are depicted. Numerical simulations are used to verify the proposed estimation and correction algorithm.

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