Non-uniform discrete Fourier transform

About: Non-uniform discrete Fourier transform is a research topic. Over the lifetime, 4067 publications have been published within this topic receiving 123952 citations.

A Digraph Fourier Transform With Spread Frequency Components

Abstract: We study the problem of constructing a graph Fourier transform (GFT) for directed graphs (digraphs), which decomposes graph signals into different modes of variation with respect to the underlying network Accordingly, to capture low, medium and high frequencies we seek a digraph (D)GFT such that the orthonormal frequency components are as spread as possible in the graph spectral domain This specification gives rise to a challenging nonconvex optimization problem, so we resort to a simple yet efficient heuristic to construct the DGFT basis from Laplacian eigenvectors of an undirected version of the digraph To select frequency components which are as spread as possible, we define a spectral dispersion function and show that it is supermodular Moreover, we show that orthonormality can be enforced via a matroid basis constraint, which motivates adopting a scalable greedy algorithm to obtain an approximate solution with provable performance guarantee The effectiveness of the novel DGFT is illustrated through numerical tests on synthetic and real-world graphs

Use of the fast Fourier transform method for analyzing linear and equispaced Fizeau fringes

Abstract: The Fourier transform method is applied to analyze the initial phase of linear and equispaced Fizeau fringes We develop an algorithm for high-precision phase measurement by using the Fourier coefficient that corresponds to the spatial frequency of the Fizeau fringes, and we describe methods for determining the fringe carrier frequency Errors caused by carrier frequency fluctuation and data truncation are studied theoretically and by computer simulation To demonstrate the method we apply it to the real-time calibration of a piezoelectric transducer mirror in a Twyman–Green interferometer

The Fourier transform in biomedical engineering

Abstract: 1 Introduction to the Fourier Transform.- 1.1 Introduction.- 1.2 Basic Functions.- 1.3 Sines, Cosines and Composite waves.- 1.4 Orthogonality.- 1.5 Waves in time and space.- 1.6 Complex numbers. A Mathematical Tool.- 1.7 The Fourier transform.- 1.8 Fourier transforms in the physical world: The Lens as an FT computer.- 1.9 Blurring and convolution.- 1.9.1 Blurring.- 1.9.2 Convolution.- 1.10 The "Point" or "Impulse" response function..- 1.11 Band-limited functions.- 1.12 Summary.- 1.13 Bibliography.- 2 The 1-D Fourier Transform.- 2.1 Introduction.- 2.2 Re-visiting the Fourier transform.- 2.3 The Sampling Theorem.- 2.4 Aliasing.- 2.5 Convolution.- 2.6 Digital Filtering.- 2.7 The Power Spectrum.- 2.8 Deconvolution.- 2.9 System Identification.- 2.10 Summary.- 2.11 Bibliography.- 3 The 2-D Fourier Transform.- 3.1 Introduction.- 3.2 Linear space-invariant systems in two dimensions.- 3.3 Ideal systems.- 3.4 A simple X-ray imaging system.- 3.5 Modulation Transfer Function (MTF).- 3.6 Image processing.- 3.7 Tomography.- 3.8 Computed Tomography.- 3.9 Summary.- 3.10 Bibliography.- 4 The Fourier Transform in Magnetic Resonance Imaging.- 4.1 Introduction.- 4.2 The 2-D Fourier transform.- 4.3 Magnetic Resonance Imaging.- 4.3.1 Nuclear Magnetic Resonance.- 4.3.2 Excitation, Evolution, and Detection.- 4.3.3 The Received Signal: FIDs and Echos.- 4.4 MRI.- 4.4.1 Localization: Magnetic Field Gradients.- 4.4.2 The MRI Signal Equation.- 4.4.3 2-D Spin-Warp Imaging.- 4.4.4 Fourier Sampling: Resolution, Field-of-View, and Aliasing.- 4.4.5 2-D Multi-slice and 3-D Spin Warp Imaging.- 4.4.6 Alternate k -space Sampling Strategies.- 4.5 Magnetic Resonance Spectroscopic Imaging.- 4.5.1 Nuclear Magnetic Resonance Spectroscopy: 1-D.- 4.5.2 Magnetic Resonance Spectroscopic Imaging: 2-D, 3-D, and 4-D.- 4.6 Motion in MRI.- 4.6.1 Phase Contrast Velocity Imaging.- 4.6.2 Phase Contrast Angiography.- 4.7 Conclusion.- 4.8 Bibliography.- 5 The Wavelet Transform.- 5.1 Introduction.- 5.1.1 Frequency analysis: Fourier transform.- 5.2 Time-Frequency analysis.- 5.2.1 Generalities.- 5.2.2. How does time-frequency analysis work?.- 5.2.3 Windowed Fourier transform.- 5.2.4 Wavelet transform.- 5.3 Multiresolution Analysis.- 5.3.1 Scaling Functions.- 5.3.2 Definition.- 5.3.3 Scaling Relation.- 5.3.4 Relationship of multiresolution analysis to wavelets.- 5.3.5 Multiresolution signal decomposition.- 5.3.6 Digital filter interpretation.- 5.3.7 Fast Wavelet Transform Algorithm.- 5.3.8 Multidimensional Wavelet Transforms.- 5.3.9 Fourier vs. Wavelet Digital Signal Processing.- 5.4 Applications.- 5.4.1 Image Compression.- 5.4.2 Irregular heart beat detection from EKG signals.- 5.5 Summary.- 5.6 Bibliography.- 6 The Discrete Fourier Transform and Fast Fourier Transform.- 6.1 Introduction.- 6.2 From Continuous to Discrete.- 6.2.1 The comb function.- 6.2.2 Sampling.- 6.2.3 Interpreting DFT data in a cyclic buffer.- 6.3 The Discrete Fourier Transform.- 6.4 The Fast Fourier Transform.- 6.4.1 The DFT as a matrix equation.- 6.4.2 Simplifying the transition matrix.- 6.4.3 Signal-flow-graph notation.- 6.4.4 The DFT expressed as a signal flow graph.- 6.4.5 Speed advantages of the FFT.- 6.5 Caveats to using the DFT/FFT.- 6.6 Conclusion.- 6.7 Bibliography.

Three-dimensional Green's function for an anisotropic multi-layered half-space

Abstract: A new numerical approach is presented to calculate the Green's function for an anisotropic multi-layered half space The formulation is explicit and unconditionally stable It imposes no limit to the thickness of the layered medium and the magnitude of the frequency In the analysis, the Fourier transform and the precise integration method (PIM) are employed Here, the Fourier transform is employed to transform the wave motion equation from the spatial domain to the wavenumber domain A second order ordinary differential equation (ODE) is observed Then, the dual vector representation of the wave motion equation is used to reduce the second order ODE to first order It is solved by the PIM Finally, the Green's function in the wavenumber domain is obtained For the evaluation of the Green's function in the spatial domain, the double inverse Fourier transform over the wavenumber is employed to derive the solutions Especially, for the transversely isotropic medium, the double inverse Fourier transform can be further reduced to a single integral by the cylindrical polar coordinate transform Numerical examples are provided Comparisons with other methods are done Very promising results are obtained