Showing papers on "Discrete-time Fourier transform published in 2014"
TL;DR: This work proposes a fast local search method for recovering a sparse signal from measurements of its Fourier transform (or other linear transform) magnitude which it refers to as GESPAR: GrEedy Sparse PhAse Retrieval, which does not require matrix lifting, unlike previous approaches, and therefore is potentially suitable for large scale problems such as images.
Abstract: We consider the problem of phase retrieval, namely, recovery of a signal from the magnitude of its Fourier transform, or of any other linear transform. Due to the loss of Fourier phase information, this problem is ill-posed. Therefore, prior information on the signal is needed in order to enable its recovery. In this work we consider the case in which the signal is known to be sparse, i.e., it consists of a small number of nonzero elements in an appropriate basis. We propose a fast local search method for recovering a sparse signal from measurements of its Fourier transform (or other linear transform) magnitude which we refer to as GESPAR: GrEedy Sparse PhAse Retrieval. Our algorithm does not require matrix lifting, unlike previous approaches, and therefore is potentially suitable for large scale problems such as images. Simulation results indicate that GESPAR is fast and more accurate than existing techniques in a variety of settings.
337 citations
TL;DR: An approach to reconstruction is proposed that optimizes for sparsity in the continuous Fourier spectrum and can be used to reduce sampling requirements and improve reconstruction quality.
Abstract: Sparsity in the Fourier domain is an important property that enables the dense reconstruction of signals, such as 4D light fields, from a small set of samples. The sparsity of natural spectra is often derived from continuous arguments, but reconstruction algorithms typically work in the discrete Fourier domain. These algorithms usually assume that sparsity derived from continuous principles will hold under discrete sampling. This article makes the critical observation that sparsity is much greater in the continuous Fourier spectrum than in the discrete spectrum. This difference is caused by a windowing effect. When we sample a signal over a finite window, we convolve its spectrum by an infinite sinc, which destroys much of the sparsity that was in the continuous domain. Based on this observation, we propose an approach to reconstruction that optimizes for sparsity in the continuous Fourier spectrum. We describe the theory behind our approach and discuss how it can be used to reduce sampling requirements and improve reconstruction quality. Finally, we demonstrate the power of our approach by showing how it can be applied to the task of recovering non-Lambertian light fields from a small number of 1D viewpoint trajectories.
240 citations
TL;DR: The nonlinear Fourier transform (NFT), a powerful tool in soliton theory and exactly solvable models, is exploited for data transmission over integrable channels, such as optical fibers, where pulse propagation is governed by the nonlinear Schrödinger equation.
Abstract: The nonlinear Fourier transform (NFT), a powerful tool in soliton theory and exactly solvable models, is a method for solving integrable partial differential equations governing wave propagation in certain nonlinear media. The NFT decorrelates signal degrees-of-freedom in such models, in much the same way that the Fourier transform does for linear systems. In this three-part series of papers, this observation is exploited for data transmission over integrable channels, such as optical fibers, where pulse propagation is governed by the nonlinear Schrodinger equation. In this transmission scheme, which can be viewed as a nonlinear analogue of orthogonal frequency-division multiplexing commonly used in linear channels, information is encoded in the nonlinear frequencies and their spectral amplitudes. Unlike most other fiber-optic transmission schemes, this technique deals with both dispersion and nonlinearity directly and unconditionally without the need for dispersion or nonlinearity compensation methods. This paper explains the mathematical tools that underlie the method.
208 citations
TL;DR: The sparse Fourier transform (SFT) addresses the big data setting by computing a compressed Fouriertransform using only a subset of the input data, in time smaller than the data set size.
Abstract: The discrete Fourier transform (DFT) is a fundamental component of numerous computational techniques in signal processing and scientific computing. The most popular means of computing the DFT is the fast Fourier transform (FFT). However, with the emergence of big data problems, in which the size of the processed data sets can easily exceed terabytes, the "fast" in FFT is often no longer fast enough. In addition, in many big data applications it is hard to acquire a sufficient amount of data to compute the desired Fourier transform in the first place. The sparse Fourier transform (SFT) addresses the big data setting by computing a compressed Fourier transform using only a subset of the input data, in time smaller than the data set size. The goal of this article is to survey these recent developments, explain the basic techniques with examples and applications in big data, demonstrate tradeoffs in empirical performance of the algorithms, and discuss the connection between the SFT and other techniques for massive data analysis such as streaming algorithms and compressive sensing.
154 citations
TL;DR: The proposed sparse discrete fractional Fourier transform algorithm achieves multicomponent resolution in addition to its low computational complexity and robustness against noise and applies to the synchronization of high dynamic direct-sequence spread-spectrum signals.
Abstract: The discrete fractional Fourier transform is a powerful signal processing tool with broad applications for nonstationary signals. In this paper, we propose a sparse discrete fractional Fourier transform (SDFrFT) algorithm to reduce the computational complexity when dealing with large data sets that are sparsely represented in the fractional Fourier domain. The proposed technique achieves multicomponent resolution in addition to its low computational complexity and robustness against noise. In addition, we apply the SDFrFT to the synchronization of high dynamic direct-sequence spread-spectrum signals. Furthermore, a sparse fractional cross ambiguity function (SFrCAF) is developed, and the application of SFrCAF to a passive coherent location system is presented. The experiment results confirm that the proposed approach can substantially reduce the computation complexity without degrading the precision.
122 citations
TL;DR: A novel parameter estimation method based on keystone transform and Radon-Fourier transform for space moving targets with high-speed maneuvering performance that can overcome the limitation of Doppler frequency ambiguity and correct range curvature for all targets in one processing step, which simplifies the operation procedure.
Abstract: This letter proposes a novel parameter estimation method based on keystone transform (KT) and Radon-Fourier transform (RFT) for space moving targets with high-speed maneuvering performance. In this method, second-order KT is used to correct the range curvature and part of the range walk for all targets simultaneously. Then, fractional Fourier transform is employed to estimate the targets' radial acceleration, followed by the quadric phase term compensation. Finally, RFT and Clean technique are carried out to correct the residual range walk, and the initial range and radial velocity of moving targets are further obtained. The advantage of the proposed method is that it can overcome the limitation of Doppler frequency ambiguity and correct range curvature for all targets in one processing step, which simplifies the operation procedure. Simulation results are presented to demonstrate the validity of the proposed method.
114 citations
TL;DR: In this article, two fast numerical methods for computing the nonlinear Fourier transform with respect to the Schrodinger equation (NSE) are presented, which achieves a runtime of O(D 2 ) floating point operations, where D is the number of sample points.
Abstract: The nonlinear Fourier transform, which is also known as the forward scattering transform, decomposes a periodic signal into nonlinearly interacting waves. In contrast to the common Fourier transform, these waves no longer have to be sinusoidal. Physically relevant waveforms are often available for the analysis instead. The details of the transform depend on the waveforms underlying the analysis, which in turn are specified through the implicit assumption that the signal is governed by a certain evolution equation. For example, water waves generated by the Korteweg-de Vries equation can be expressed in terms of cnoidal waves. Light waves in optical fiber governed by the nonlinear Schrodinger equation (NSE) are another example. Nonlinear analogs of classic problems such as spectral analysis and filtering arise in many applications, with information transmission in optical fiber, as proposed by Yousefi and Kschischang, being a very recent one. The nonlinear Fourier transform is eminently suited to address them -- at least from a theoretical point of view. Although numerical algorithms are available for computing the transform, a "fast" nonlinear Fourier transform that is similarly effective as the fast Fourier transform is for computing the common Fourier transform has not been available so far. The goal of this paper is to address this problem. Two fast numerical methods for computing the nonlinear Fourier transform with respect to the NSE are presented. The first method achieves a runtime of $O(D^2)$ floating point operations, where $D$ is the number of sample points. The second method applies only to the case where the NSE is defocusing, but it achieves an $O(D\log^2D)$ runtime. Extensions of the results to other evolution equations are discussed as well.
103 citations
TL;DR: A new transform is proposed, named velocity synchronous discrete Fourier transform, which takes advantage of the instantaneous velocity to improve the quality of its result, reaching performances that can challenge the computed order tracking.
85 citations
TL;DR: In this paper, a spectral approach is developed to solve the elasto-static equations of field dislocation and g-disclination mechanics in periodic media, which is consistent with the pixel grid for the calculation of first and second order spatial derivatives, and compared to the classical discrete approximation of continuous Fourier transforms when deriving elastic fields of defects.
79 citations
TL;DR: This paper shows that Fourier extensions are actually numerically stable when implemented in finite arithmetic, and achieve a convergence rate that is at least superalgebraic, and demonstrates that they are particularly well suited for this problem.
Abstract: An effective means to approximate an analytic, nonperiodic function on a bounded interval is by using a Fourier series on a larger domain. When constructed appropriately, this so-called Fourier extension is known to converge geometrically fast in the truncation parameter. Unfortunately, computing a Fourier extension requires solving an ill-conditioned linear system, and hence one might expect such rapid convergence to be destroyed when carrying out computations in finite precision. The purpose of this paper is to show that this is not the case. Specifically, we show that Fourier extensions are actually numerically stable when implemented in finite arithmetic, and achieve a convergence rate that is at least superalgebraic. Thus, in this instance, ill-conditioning of the linear system does not prohibit a good approximation.
In the second part of this paper we consider the issue of computing Fourier extensions from equispaced data. A result of Platte et al. (SIAM Rev. 53(2):308---318, 2011) states that no method for this problem can be both numerically stable and exponentially convergent. We explain how Fourier extensions relate to this theoretical barrier, and demonstrate that they are particularly well suited for this problem: namely, they obtain at least superalgebraic convergence in a numerically stable manner.
70 citations
TL;DR: Based on a new convolution operation, convolution and correlation theorems are formulated for the offset linear canonical transform and the convolution theorem is used to investigate the sampling theorem for the band-limited signal in the OLCT domain.
Abstract: The offset linear canonical transform (OLCT), which is a time-shifted and frequency-modulated version of the linear canonical transform, has been shown to be a powerful tool for signal processing and optics. However, some basic results for this transform, such as convolution and correlation theorems, remain unknown. In this paper, based on a new convolution operation, we formulate convolution and correlation theorems for the OLCT. Moreover, we use the convolution theorem to investigate the sampling theorem for the band-limited signal in the OLCT domain. The formulas of uniform sampling and low-pass reconstruction related to the OLCT are obtained. We also discuss the design method of the multiplicative filter in the OLCT domain. Based on the model of the multiplicative filter in the OLCT domain, a practical method to achieve multiplicative filtering through convolution in the time domain is proposed.
TL;DR: A whole arithmetics of mathematical operations acting on distribution objects, comprising operators +, -, *, /, and ^ is defined, overloading operator + for convolution; based on this convolution, this default algorithm is overloaded by more specific algorithms where possible.
Abstract: Object orientation provides a flexible framework for the implementation of the convolution of arbitrary distributions of real-valued random variables. We discuss an algorithm which is based on the fast Fourier transform. It directly applies to lattice-supported distributions. In the case of continuous distributions an additional discretization to a linear lattice is necessary and the resulting lattice-supported distributions are suitably smoothed after convolution. We compare our algorithm to other approaches aiming at a similar generality as to accuracy and speed. In situations where the exact results are known, several checks confirm a high accuracy of the proposed algorithm which is also illustrated for approximations of non-central χ 2 distributions. By means of object orientation this default algorithm is overloaded by more specific algorithms where possible, in particular where explicit convolution formulae are available. Our focus is on R package distr which implements this approach, overloading operator + for convolution; based on this convolution, we define a whole arithmetics of mathematical operations acting on distribution objects, comprising operators +, -, *, /, and ^.
TL;DR: This paper first proposes a generalized convolution theorem for the LCT and then derives a corresponding product theorem associated with the L CT, which is shown to be special cases of the derived results.
Abstract: The linear canonical transform (LCT), which is a generalized form of the classical Fourier transform (FT), the fractional Fourier transform (FRFT), and other transforms, has been shown to be a powerful tool in optics and signal processing. Many results of this transform are already known, including its convolution theorem. However, the formulation of the convolution theorem for the LCT has been developed differently and is still not having a widely accepted closed-form expression. In this paper, we first propose a generalized convolution theorem for the LCT and then derive a corresponding product theorem associated with the LCT. The ordinary convolution theorem for the FT, the fractional convolution theorem for the FRFT, and some existing convolution theorems for the LCT are shown to be special cases of the derived results. Moreover, some applications of the derived results are presented.
TL;DR: The discrete Fourier transform produces a Fourier representation for finite-duration data sequences and plays a key role in the implementation of a variety of digital signal-?processing algorithms.
Abstract: The discrete Fourier transform (DFT) produces a Fourier representation for finite-duration data sequences. In addition to its theoretical importance, the DFT plays a key role in the implementation of a variety of digital signal-?processing algorithms. Several algorithms including the fast Fourier transform (FFT) and the Goertzel algorithm have been introduced for the fast implementation of the DFT [1], [2].
TL;DR: In this paper, a Gauss-Fast Fourier Transform (FFT) algorithm was proposed for Fourier-domain forward modeling of potential fields, which converged to the space-domain solution much faster than the standard FFT method with grid expansion.
Abstract: We analyzed the numerical forward methods in the Fourier domain for potential fields. Existing Fourier-domain forward methods applied the standard fast Fourier transform (FFT) algorithm to inverse transform a conjugate symmetrical spectrum into a real field. It had significant speed advantages over space-domain forward methods but suffered from problems including aliasing, imposed periodicity, and edge effect. Usually, grid expansion was needed to reduce these errors, which was equivalent to the numerical evaluation of the oscillatory Fourier integral using the trapezoidal rule with smaller steps. We tested a high-precision Fourier-domain forward method based on a combined use of shift-sampling technique and Gaussian quadrature theory. The trapezoidal rule applied by the standard FFT algorithm to evaluate the continuous Fourier transform was modified by introducing a shift parameter ξ. By choosing optimum values of ξ as Gaussian quadrature nodes, we developed a Gauss-FFT method for Fourier forward modeling of potential fields. No grid expansion was needed, the sources can be set near the boundary of the fields or even go beyond the boundary. The Gauss-FFT method converged to the space-domain solution much faster than the standard FFT method with grid expansion. Forward modeling results almost identical to space-domain ones can be obtained in less time. Numerical examples, of both simple and complex 2D and 3D source forward modeling, revealed the reliability and adaptability of the method.
TL;DR: The observation that Fourier extensions, much like classical Fourier series, also have excellent resolution properties for representing oscillatory functions are explained and eminently suitable for problems with moderate to high degrees of oscillation.
TL;DR: In this article, the authors develop and study two conceptually new ways to define convolution products for hypercomplex Fourier transforms, which will enable the development and fast implementation of new filters for quaternionic signals and systems, as well as for their higher dimensional counterparts.
Abstract: Hypercomplex Fourier transforms are increasingly used in signal processing for the analysis of higher-dimensional signals such as color images. A main stumbling block for further applications, in particular concerning filter design in the Fourier domain, is the lack of a proper convolution theorem. The present paper develops and studies two conceptually new ways to define convolution products for such transforms. As a by-product, convolution theorems are obtained that will enable the development and fast implementation of new filters for quaternionic signals and systems, as well as for their higher dimensional counterparts.
01 Jun 2014
TL;DR: In this paper, the discrete component of the nonlinear Fourier transform is studied and multi-eigen value signal sets are presented that achieve spectral efficiencies greater than 3 bits/s/Hz.
Abstract: Information transmission using only the discrete component of the nonlinear Fourier transform is studied and multi-eigenvalue signal sets are presented that achieve spectral efficiencies greater than 3 bits/s/Hz.
TL;DR: In this article, the quadratic Fourier transform was examined by analyzing corresponding six subcases of the transform within a reproducing kernel Hilbert spaces framework, and the results showed that the transform can be expressed as a generalized quadrastic function for one order parameter in the ordinary Fourier transformation.
Abstract: In this paper we shall examine the quadratic Fourier transform which is introduced by the generalized quadratic function for one order parameter in the ordinary Fourier transform. This will be done by analyzing corresponding six subcases of the quadratic Fourier transform within a reproducing kernel Hilbert spaces framework.
Book•
10 Oct 2014
TL;DR: In this paper, the Fourier transform of an oscillating function has been studied in the context of radial functions and Oscillatory Integral Integrals and Fourier Transform in one variable.
Abstract: Foreword.- Introduction.- Chapter 1. Basic properties of the Fourier transform.- Chapter 2. Oscillatory integrals and Fourier transforms in one variable.- Chapter 3. The Fourier transform of an oscillating function.- Chapter 4. The Fourier transform of a radial function.- Chapter 5. Multivariate extensions.- Appendix.- Bibliography.
01 Jan 2014
TL;DR: This chapter begins with the general theory of wss discrete-time stochastic processes and gives the representation theory of arma processes, together with their prediction theory, and the realization problem: what models fit a given finite segment of autocorrelation function of a time series?
Abstract: Discrete-time wide-sense stationary stochastic processes, also called time series, arise from discrete-time measurements (sampling) of random functions. A particularly mathematically tractable class of such processes consists of the so-called moving averages and auto-regressive (and more generally, arma) time series. This chapter begins with the general theory of wss discrete-time stochastic processes (which essentially reproduces that of wss continuous-time stochastic processes) and then gives the representation theory of arma processes, together with their prediction theory. The last section is concerned with the realization problem: what models fit a given finite segment of autocorrelation function of a time series? The corresponding theory is the basis of parametric spectral analysis.
18 Dec 2014
TL;DR: The optimized code is evaluated for performance and compared to the reference implementation as well as the FFTW library, and the main result is that, depending on the input parameters, the optimized Sparse Fast Fourier Transform library is two to five times faster than theReference implementation.
Abstract: The Sparse Fast Fourier Transform is a recent algorithm developed by Hassanieh et al. at MIT for Discrete Fourier Transforms on signals with a sparse frequency domain. A reference implementation of the algorithm exists and proves that the Sparse Fast Fourier Transform can be faster than modern FFT libraries. However, the reference implementation does not take advantage of modern hardware features like vector instruction sets or multithreading. In this Master Thesis the reference implementation’s performance will be analyzed and evaluated. Several optimizations are proposed and implemented in a high-performance Sparse Fast Fourier Transform library. The optimized code is evaluated for performance and compared to the reference implementation as well as the FFTW library. The main result is that, depending on the input parameters, the optimized Sparse Fast Fourier Transform library is two to five times faster than the reference implementation.
TL;DR: In this article, the convergence of the NFFT was analyzed as a Fourier frame approximation, and a frame theoretic convolutional gridding algorithm was proposed to detect features (such as edges) from nonuniform Fourier samples of piecewise smooth functions.
Abstract: Nonuniform Fourier data are routinely collected in applications such as magnetic resonance imaging, synthetic aperture radar, and synthetic imaging in radio astronomy. To acquire a fast reconstruction that does not require an online inverse process, the nonuniform fast Fourier transform (NFFT), also called convolutional gridding, is frequently employed. While various investigations have led to improvements in accuracy, efficiency, and robustness of the NFFT, not much attention has been paid to the fundamental analysis of the scheme, and in particular its convergence properties. This paper analyzes the convergence of the NFFT by casting it as a Fourier frame approximation. In so doing, we are able to design parameters for the method that satisfy conditions for numerical convergence. Our so-called frame theoretic convolutional gridding algorithm can also be applied to detect features (such as edges) from nonuniform Fourier samples of piecewise smooth functions.
TL;DR: This novel fractional Fourier transform has removed the restriction on the dimension of transform order and highly enhances the security of image encryption scheme proposed in this paper without increasing the computational complexity and hardware cost.
TL;DR: A novel method for color image enhancement based on the discrete quaternion Fourier transform that not only provides true color fidelity for poor quality images but also averages the color components to gray value for balancing colors.
Abstract: This paper presents a novel method for color image enhancement based on the discrete quaternion Fourier transform. We choose the quaternion Fourier transform, because it well-suited for color image processing applications, it processes all 3 color components (R,G,B) simultaneously, it capture the inherent correlation between the components, it does not generate color artifacts or blending , finally it does not need an additional color restoration process. Also we introduce a new CEME measure to evaluate the quality of the enhanced color images. Preliminary results show that the α-rooting based on the quaternion Fourier transform enhancement method out-performs other enhancement methods such as the Fourier transform based α-rooting algorithm and the Multi scale Retinex. On top, the new method not only provides true color fidelity for poor quality images but also averages the color components to gray value for balancing colors. It can be used to enhance edge information and sharp features in images, as well as for enhancing even low contrast images. The proposed algorithms are simple to apply and design, which makes them very practical in image enhancement.
TL;DR: In this article, a novel method is proposed based on feedback corrected adaptive notch filters and Hilbert transformation for CMF signal processing and aim to resolve the problems of precision sharp-decline occurrence when using adaptive notch filter (ANF) for tracking the signal frequency for a long time.
Abstract: In this paper, we focus on CMF signal processing and aim to resolve the problems of precision sharp-decline occurrence when using adaptive notch filters (ANFs) for tracking the signal frequency for a long time and phase difference calculation depending on frequency by the sliding Goertzel algorithm (SGA) or the recursive DTFT algorithm with negative frequency contribution. A novel method is proposed based on feedback corrected ANF and Hilbert transformation. We design an index to evaluate whether the ANF loses the signal frequency or not, according to the correlation between the output and input signals. If the signal frequency is lost, the ANF parameters will be adjusted duly. At the same time, singular value decomposition (SVD) algorithm is introduced to reduce noise. And then, phase difference between the two signals is detected through trigonometry and Hilbert transformation. With the frequency and phase difference obtained, time interval of the two signals is calculated. Accordingly, the mass flow rate is derived. Simulation and experimental results show that the proposed method always preserves a constant high precision of frequency tracking and a better performance of phase difference measurement compared with the SGA or the recursive DTFT algorithm with negative frequency contribution.
TL;DR: It is observed through the simulation results that the use of taking the transformation before the decomposition step improves the quality of fused image, and the results of using the fractional Fourier transform and discrete cosine transform before the decay step are encouraging.
Abstract: Image fusion has been receiving increasing attention in the research community in a wide spectrum of applications. Several algorithms in spatial and frequency domains have been developed for this purpose. In this paper we propose a novel algorithm which involves the use of fractional Fourier domains which are intermediate between spatial and frequency domains. The proposed image fusion scheme is based on decomposition of source images (or its transformed version) into self-fractional Fourier functions. The decomposed images are then fused by maximum absolute value selection rule. The selected images are combined and inverse transformation is taken to obtain the final fused image. The proposed decomposition scheme and the use of some transformation before the decomposition step offer additional degrees of freedom in the image fusion scheme. Simulation results of the proposed scheme for different transformation of the source images for two different sets of images are also presented. It is observed through the simulation results that the use of taking the transformation before the decomposition step improves the quality of fused image. In particular the results of using the fractional Fourier transform and discrete cosine transform before the decomposition step are encouraging.