Showing papers on "Non-uniform discrete Fourier transform published in 1998"
Book•
01 Jan 1998
TL;DR: In this paper, the Discrete Fourier Transform and Numerical Computations (DFT) were used for time-frequency analysis in periodic signals and periodical signals.
Abstract: Signals and Systems.- Periodic Signals.- The Discrete Fourier Transform and Numerical Computations.- The Lebesgue Integral.- Spaces.- Convolution and the Fourier Transform of Functions.- Analog Filters.- Distributions.- Convolution and the Fourier Transform of Distributions.- Filters and Distributions.- Sampling and Discrete Filters.- Current Trends: Time-Frequency Analysis.- References.
263 citations
TL;DR: Based on the (m, N, q)-regular Fourier matrix, a new algorithm is proposed for fast Fourier transform (FFT) of nonuniform (unequally spaced) data with accuracy much better than previously reported results with the same computation complexity.
Abstract: Based on the (m, N, q)-regular Fourier matrix, a new algorithm is proposed for fast Fourier transform (FFT) of nonuniform (unequally spaced) data. Numerical results show that the accuracy of this algorithm is much better than previously reported results with the same computation complexity of O(N log/sub 2/ N). Numerical examples are shown for the applications in computational electromagnetics.
251 citations
TL;DR: The generalized Hilbert transform has similar properties to those of the ordinary Hilbert transform, but it lacks the semigroup property of the fractional Fourier transform.
Abstract: The analytic part of a signal f(t) is obtained by suppressing the negative frequency content of f, or in other words, by suppressing the negative portion of the Fourier transform, f/spl circ/, of f. In the time domain, the construction of the analytic part is based on the Hilbert transform f/spl circ/ of f(t). We generalize the definition of the Hilbert transform in order to obtain the analytic part of a signal that is associated with its fractional Fourier transform, i.e., that part of the signal f(t) that is obtained by suppressing the negative frequency content of the fractional Fourier transform of f(t). We also show that the generalized Hilbert transform has similar properties to those of the ordinary Hilbert transform, but it lacks the semigroup property of the fractional Fourier transform.
108 citations
TL;DR: In this paper, the eigenvalues and eigenvectors of the discrete Fourier and Hartley transform matrices are investigated, and the results of the eigendecomposition of the transform matrix are used to define DFRHT and DFRFT.
Abstract: This paper is concerned with the definitions of the discrete fractional Hartley transform (DFRHT) and the discrete fractional Fourier transform (DFRFT). First, the eigenvalues and eigenvectors of the discrete Fourier and Hartley transform matrices are investigated. Then, the results of the eigendecompositions of the transform matrices are used to define DFRHT and DFRFT. Also, an important relationship between DFRHT and DFRFT is described, and numerical examples are illustrated to demonstrate that the proposed DFRFT is a better approximation to the continuous fractional Fourier transform than the conventional defined DFRFT. Finally, a filtering technique in the fractional Fourier transform domain is applied to remove chirp interference.
105 citations
TL;DR: This paper develops a 2D DFRFT which can preserve the rotation properties and provide similar results to continuous FRFT.
98 citations
TL;DR: This work allows the fractional Fourier transform orders to be specified independently for the two dimensions but also allow the input and output scale parameters and the residual spherical phase factors to be controlled.
Abstract: We provide a general treatment of optical two-dimensional fractional Fourier transforming systems. We not only allow the fractional Fourier transform orders to be specified independently for the two dimensions but also allow the input and output scale parameters and the residual spherical phase factors to be controlled. We further discuss systems that do not allow all these parameters to be controlled at the same time but are simpler and employ a fewer number of lenses. The variety of systems discussed and the design equations provided should be useful in practical applications for which an optical fractional Fourier transforming stage is to be employed.
87 citations
Book•
30 Nov 1998TL;DR: In this article, the NDFT was used to construct a 1-D and 2-D antenna pattern synthesis with Prescribed Nulls, and the Dual-Tone Multi-Frequency Signal Decoding (DTMSD) was proposed.
Abstract: 1. Introduction. 2. The Nonuniform Discrete Fourier Transform. 3. 1-D Fir Filter Design Using the NDFT. 4. 2-D Fir Filter Design Using the NDFT. 5. Antenna Pattern Synthesis with Prescribed Nulls. 6. Dual-Tone Multi-Frequency Signal Decoding. 7. Conclusions. References. Index.
79 citations
TL;DR: By using the Fourier representation and the fast Fourier transform, one generation of the infinite population simple genetic algorithm can be computed in time O(cl log2 3), where c is arity of the alphabet and l is the string length.
Abstract: This paper is the first part of a two-part series. It proves a number of direct relationships between the Fourier transform and the simple genetic algorithm. (For a binary representation, the Walsh transform is the Fourier transform.) The results are of a theoretical nature and are based on the analysis of mutation and crossover. The Fourier transform of the mixing matrix is shown to be sparse. An explicit formula is given for the spectrum of the differential of the mixing transformation. By using the Fourier representation and the fast Fourier transform, one generation of the infinite population simple genetic algorithm can be computed in time O(cl log2 3), where c is arity of the alphabet and l is the string length. This is in contrast to the time of O(c3l) for the algorithm as represented in the standard basis. There are two orthogonal decompositions of population space that are invariant under mixing. The sequel to this paper will apply the basic theoretical results obtained here to inverse problems and asymptotic behavior.
72 citations
TL;DR: In this article, the authors extend the fractional Fourier transform to different spaces of generalized functions using two different techniques, one analytic and the other algebraic, which makes the transform of a convolution of two functions almost equal to the product of their transform.
Abstract: In recent years the fractional Fourier transform (FRFT), which is a generalization of the Fourier transform, has been the focus of many research papers because of its application in several areas, including signal processing and optics. In this paper, we extend the fractional Fourier transform to different spaces of generalized functions using two different techniques, one analytic and the other algebraic. The algebraic approach requires the introduction of a new convolution operation for the fractional Fourier transform that makes the transform of a convolution of two functions almost equal to the product of their transform.
67 citations
TL;DR: In this paper, the authors prove orthogonality relations for some analogs of trigonometric functions on a g-quadratic grid and introduce the corresponding g-Fourier series.
Abstract: We prove orthogonality relations for some analogs of trigonometric functions on a g-quadratic grid and introduce the corresponding g-Fourier series. We also discuss several other properties of this basic trigonometric system and the g-Fourier series.
61 citations
TL;DR: The fundamentals of Fourier analysis are reviewed with emphasis on the analysis of transient signals, and the human saccade is considered to illustrate the pitfalls and advantages of various Fourier analyses.
TL;DR: In this paper, a numerical algorithm based on a single fast Fourier transform is proposed, which shows better precision and calculation efficiency than those of previously published algorithms, and if specific conditions are met, the numerical calculations of two successive fractional Fourier transforms produce results that are similar to the analytical solution.
Abstract: A numerical algorithm based on a single fast Fourier transform is proposed. Its precision and calculation efficiency show better performance than those of previously published algorithms. It is also shown that if specific conditions are met, the numerical calculations of two successive fractional Fourier transforms produce results that are similar to the analytical solution.
TL;DR: A recursive algorithm to implement phase retrieval from two intensities in the fractional Fourier transform domain is proposed that can significantly simplify computational manipulations and does not need an initial phase estimate compared with conventional iterative algorithms.
Abstract: We first discuss the discrete fractional Fourier transform and present some essential properties. We then propose a recursive algorithm to implement phase retrieval from two intensities in the fractional Fourier transform domain. This approach can significantly simplify computational manipulations and does not need an initial phase estimate compared with conventional iterative algorithms. Simulation results show that this approach can successfully recover the phase from two intensities.
TL;DR: A new, efficient method for 3-D radon inversion, i.e., reconstruction of the image from the radial derivative of the 3- D radon transform, called direct Fourier inversion (DFI) is presented, based directly on the 3D Fourier slice theorem.
Abstract: The radial derivative of the three-dimensional (3-D) radon transform of an object is an important intermediate result in many analytically exact cone-beam reconstruction algorithms. The authors briefly review Grangeat's (1991) approach for calculating radon derivative data from cone-beam projections and then present a new, efficient method for 3-D radon inversion, i.e., reconstruction of the image from the radial derivative of the 3-D radon transform, called direct Fourier inversion (DFI). The method is based directly on the 3-D Fourier slice theorem. From the 3-D radon derivative data, which is assumed to be sampled on a spherical grid, the 3-D Fourier transform of the object is calculated by performing fast Fourier transforms (FFTs) along radial lines in the radon space. Then, an interpolation is performed from the spherical to a Cartesian grid using a 3-D gridding step in the frequency domain. Finally, this 3-D Fourier transform is transformed back to the spatial domain via 3-D inverse FFT. The algorithm is computationally efficient with complexity in the order of N/sup 3/ log N. The authors have done reconstructions of simulated 3-D radon derivative data assuming sampling conditions and image quality requirements similar to those in medical computed tomography (CT).
TL;DR: An efficient realization of discrete Legendre function transforms based on a modified and stabilized version of the Driscoll-Healy algorithm for the stable and efficient computation of Fourier expansions of square integrable functions on the unit sphere S ⊂ R 3.
TL;DR: An algorithm called the quick Fourier transform (QFT) is developed that reduces the number of floating-point operations necessary to compute the DFT by a factor of two or four over direct methods or Goertzel's method for prime lengths.
Abstract: This paper looks at an approach that uses symmetric properties of the basis function to remove redundancies in the calculation of the discrete Fourier transform (DFT). We develop an algorithm called the quick Fourier transform (QFT) that reduces the number of floating-point operations necessary to compute the DFT by a factor of two or four over direct methods or Goertzel's method for prime lengths. By further application of the idea to the calculation of a DFT of length-2/sup M/, we construct a new O(NlogN) algorithm, with computational complexities comparable to the Cooley-Tukey algorithm. We show that the power-of-two QFT can be implemented in terms of discrete sine and cosine transforms. The algorithm can be easily modified to compute the DFT with only a subset of either input or output points and reduces by nearly half the number of operations when the data are real.
TL;DR: A nonseparable definition for the two-dimensional fractional Fourier transform that includes the separable definition as a special case is presented and its digital and optical implementations are presented.
Abstract: Previous generalizations of the fractional Fourier transform to two dimensions assumed separable kernels. We present a nonseparable definition for the two-dimensional fractional Fourier transform that includes the separable definition as a special case. Its digital and optical implementations are presented. The usefulness of the nonseparable transform is justified with an image-restoration example.
TL;DR: A nonuniform inverse fast Fourier transform (NU-IFFT) for non ununiformly sampled data is realised by combining the conjugate-gradient fast Fouriers transform (CG-FFT) method with the newly developed NUFFT algorithms.
Abstract: A nonuniform inverse fast Fourier transform (NU-IFFT) for nonuniformly sampled data is realised by combining the conjugate-gradient fast Fourier transform (CG-FFT) method with the newly developed nonuniform fast Fourier transform (NUFFT) algorithms. An example application of the algorithm in computational electromagnetics is presented.
06 Jul 1998
TL;DR: In this paper, the authors proposed an accurate algorithm for the non-uniform forward FFT (NUFFT) based on a new class of matrices, the regular Fourier matrices.
Abstract: Regular fast Fourier transform (FFT) algorithms require uniformly sampled data. In many practical situations, however, the input data is nonuniform, and hence the regular FFT does not apply. To overcome this difficulty the authors have proposed an accurate algorithm for the nonuniform forward FFT (NUFFT) based on a new class of matrices, the regular Fourier matrices. For the nonuniform inverse FFT (NU-IFFT) algorithm, the conjugate-gradient method and the regular FFT algorithm are combined to speed up a matrix inversion. Numerical results show that these algorithms are more than one order of magnitude more accurate than existing algorithms.
01 Jan 1998
TL;DR: This chapter discusses the FFT algorithm as a means of computing the DFT much more rapidly than can be achieved with the D FT algorithm directly, and presents some practical guidelines for using the F FT.
Abstract: The preceding chapters have made extensive mention of the Fourier transform (FT), the discrete Fourier transform (DFT), and the fast Fourier transform (FFT). This chapter examines the relationship between the FT and the DFT, discusses the FFT algorithm as a means of computing the DFT much more rapidly than can be achieved with the DFT algorithm directly, and presents some practical guidelines for using the FFT.
01 Jan 1998
TL;DR: In this paper, the Fourier transform was used to detect irregular heart beat from EKG signals in magnetic resonance imaging (MRI) images, and the results showed that it can be used to identify irregular heartbeat in MRI images.
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.
01 Nov 1998
TL;DR: A new transformation for discrete signals with time-varying spectra is proposed, which provides the energy density of the signal in time-frequency and a representation for the signal as well as its time- frequencies energy density.
Abstract: We propose a new transformation for discrete signals with time-varying spectra. The kernel of this transformation provides the energy density of the signal in time-frequency. With this discrete evolutionary transform we obtain a representation for the signal as well as its time-frequency energy density. To obtain the kernel of the transformation we use either the Gabor or the Malvar discrete signal representations. Signal adaptive analysis can be done using modulated or chirped bases, and implemented with either masking or image segmentation on the time-frequency plane. Different examples illustrate the implementation of the discrete evolutionary transform.
Book•
01 Jul 1998TL;DR: This book discusses the Steady State Response of Analogue Networks to Sinusoids and to the complex exponential EJWT, and an Introduction to Digital Networks and the Z-Transform.
Abstract: 1. Getting Started in Matlab and an Introduction to Systems and Signal Processing. 2. Impulse Functions, Impulse Responses, and Convolution. 3. The Steady State Response of Analogue Networks to Sinusoids and to the complex exponential EJWT. 4. Phasors. 5. Line Spectra and the Fourier Series. 6. Spectral Density Functions and the Fourier Transform. 7 The Sampling and Digitization of Signals. 8. The Discrete Fourier Transform. 9. The Fast Fourier Transform and Some Applications. 10. The Steady State Response of Analogue Systems By Consideration of the 11. Natural Responses, Transients and Stability. 12. The Laplace Transform. 13. Synthesis of Analogue Filters. 14. An Introduction to Digital Networks and the Z-Transform. 15. Synthesis of Digital Filters. 16. Correlation. 17. Processing Techniques for Bandpass Signals. Index.
TL;DR: In this paper, a finite spectral method for pointwise or cellwise local spectral schemes based on non-periodic Fourier transform and two finite spectral schemes are presented. But this method is not suitable for wave propagation problems.
Patent•
12 Nov 1998TL;DR: In this paper, the amplitude replacement section replaces the amplitude distribution with a predetermined function using the distance from a center of a frequency plane as a parameter, and the inverse fast Fourier transform section forms an image corresponding to the original image.
Abstract: An image processing apparatus capable of extracting widely viewed features of an entire image and speedily performing image processing, and a method and an information recording medium for such processing. Image data of an original image is obtained by imaging the entire original image with an image pickup unit at a time. An amplitude distribution of a signal is obtained from the image data by fast Fourier transform performed by a fast Fourier transform section of a signal processing unit. An amplitude replacement section replaces the amplitude distribution with a predetermined function using the distance from a center of a frequency plane as a parameter. An inverse fast Fourier transform section forms an image corresponding to the original image by inverse fast Fourier transform from a phase distribution of points obtained by the fast Fourier transform and from an amplitude distribution obtained by the above-described replacement.
Patent•
05 Nov 1998
TL;DR: In this paper, a pixel addressable spatial modulator is used to adjust the phase of the light of each pixel, which can be either a reflective or transmissive type device.
Abstract: A system that optically performs complex transforms, such as Fourier transforms. The system includes a pixel addressable spatial modulator that, in parallel, adjusts the phase of the light of each pixel. The modulator can be a reflective or transmissive type device. A transform lens, such as a Fourier lens, performs a two dimensional transform of the pixels outputs. This operation is repeated for the characteristic function (real and imaginary) of the function. The transformed outputs of the characteristic functions are sampled by a light detector and processed by a computer using simple fast operations, such as addition, into the final transform.
TL;DR: In this article, a method to eliminate the effect of these errors on multicomponent analysis is introduced, and a practical example of the use of the correction method is presented, where the nature of the errors has been experimentally verified.
Abstract: In Fourier transform absorbance spectroscopy, several types of baseline errors may be caused by the variations in the optical system, especially in the interferometer. The nature and causes of these errors have been studied. Furthermore, a method to eliminate the effect of these errors on multicomponent analysis is introduced. The nature of the errors has been experimentally verified, and a practical example of the use of the correction method is presented.
04 Oct 1998
TL;DR: A discrete two-dimensional Fourier transform based on quaternion (or hypercomplex) numbers allows colour images to be transformed as a whole, rather than as colour-separated components.
Abstract: A discrete two-dimensional Fourier transform based on quaternion (or hypercomplex) numbers allows colour images to be transformed as a whole, rather than as colour-separated components. The transform is reviewed and its basis functions presented with example images.
Patent•
30 Jun 1998TL;DR: In this article, a high-speed discrete Fourier transform (DFT) is applied to a network analyzer which obtains a transfer function of a device in a frequency domain and converts the transfer function to a time domain response to a simulated test signal.
Abstract: A high-speed discrete Fourier transform (DFT) apparatus utilizes a processor operating in parallel with data acquisition to calculate terms of a Fourier transform corresponding to the incoming data Since the processor calculates the Fourier terms in real-time, overall transformation time is substantially reduced and is limited by only the data acquisition time In another aspect, substantial reduction of the number of computations are achieved by transforming the plurality of terms in Fourier equations at the same time In a further aspect, the high-speed DFT is advantageously applied to a network analyzer which obtains a transfer function of a device in a frequency domain and converts the transfer function to a time domain response to a simulated test signal
TL;DR: In this article, a spectral analysis of gravity anomalies due to slab like structures with linearly varying density using the Hartley transform, a real valued replacement for the well known complex Fourier transform which is conventionally used in such an analysis, is presented.