Discontinuous Fast Fourier Transform with Triangle Mesh for Two-dimensional Discontinuous Functions

An Accurate Conformal Fourier Transform Method for 2d Discontinuous Functions

“…2 shows an example of the Gaussian quadrature points before and after the curvilinear coordinate transformation when P = 4 and M g = 8. Note that the 2D DFFT method proposed in [12] is actually equivalent to the case with P = 1.…”

“…Similar to [12], 2D NUFFT [20] is used to reduce the computational complexity. Equation (5) can be rewritten into…”

“…N 1 and N 2 are the numbers of sampling points in the frequency domain in each dimension, µ = 2 and Q max(N 1 , N 2 ) is a constant. Numerical results illustrate the advantages of the developed algorithm over the traditional 2D FFT algorithm and the algorithm in [12].…”

“…In practice, however, many functions to be transformed are discontinuous across the boundary of an irregular area. For example, in volume integral equation solvers in electromagnetics, some components of the unknown electric current density fields to be transformed are discontinuous across the material interfaces, which in general have arbitrary shapes; another example is the analysis of radiation patterns of reflector antennas and planar near-field to far-field transformation, where the Fourier transform integral of spatially limited functions with discontinuities at the boundary has to be evaluated [11,12]. For this kind of functions, however, there usually exist significant stair-casing errors due to the uniform Cartesian orthogonal grid required by the traditional 2D FFT algorithm, and the accuracy is limited since FFT is based on the trapezoidal quadrature scheme.…”

“…A fast algorithm for evaluating the Fourier transform of 2D discontinuous functions has been proposed in [12] as an extension of [11] to allow an arbitrary boundary shape. This method adopts a triangular mesh to conform with the arbitrary shape of the discontinuity boundary, the Gaussian quadrature to increase the accuracy, and 2D nonuniform fast Fourier transform (NUFFT) [18][19][20][21] to provide a computational complexity comparable with the traditional FFT algorithm.…”