Showing papers on "Fourier transform published in 2012"

Variants of Dynamic Mode Decomposition: Boundary Condition, Koopman, and Fourier Analyses

Abstract: Dynamic mode decomposition (DMD) is an Arnoldi-like method based on the Koopman operator. It analyzes empirical data, typically generated by nonlinear dynamics, and computes eigenvalues and eigenmodes of an approximate linear model. Without explicit knowledge of the dynamical operator, it extracts frequencies, growth rates, and spatial structures for each mode. We show that expansion in DMD modes is unique under certain conditions. When constructing mode-based reduced-order models of partial differential equations, subtracting a mean from the data set is typically necessary to satisfy boundary conditions. Subtracting the mean of the data exactly reduces DMD to the temporal discrete Fourier transform (DFT); this is restrictive and generally undesirable. On the other hand, subtracting an equilibrium point generally preserves the DMD spectrum and modes. Next, we introduce an “optimized” DMD that computes an arbitrary number of dynamical modes from a data set. Compared to DMD, optimized DMD is superior at calculating physically relevant frequencies, and is less numerically sensitive. We test these decomposition methods on data from a two-dimensional cylinder fluid flow at a Reynolds number of 60. Time-varying modes computed from the DMD variants yield low projection errors.

A Unit Root Test Using a Fourier Series to Approximate Smooth Breaks

Abstract: We develop a unit-root test based on a simple variant of Gallant's (1981) flexible Fourier form. The test relies on the fact that a series with several smooth structural breaks can often be approximated using the low frequency components of a Fourier expansion. Hence, it is possible to test for a unit root without having to model the precise form of the break. Our unit-root test employing Fourier approximation has good size and power for the types of breaks often used in economic analysis. The appropriate use of the test is illustrated using several interest rate spreads.

Tables of Integral Transforms

Abstract: In this chapter, we provide a set of short tables of integral transforms of the functions that are either cited in the text or are in most common use in mathematical, physical, and engineering applications. For exhaustive lists of integral transforms, the reader is referred to Erdelyi et al. (Tables of Integral Transforms, Vols. 1 and 2, 1954), Campbell and Foster (Fourier Integrals for Practical Applications, 1948), Ditkin and Prudnikov (Integral Transforms and Operational Calculus, 1965), Doetsch (Introduction to the Theory and Applications of the Laplace Transformation, 1970), Marichev (1983), Debnath (1995), Debnath and Bhatta (Integral Transforms and Their Applications, 2nd edition, 2007), Oberhettinger (Tables of Bessel Transforms, 1972).

Simple and practical algorithm for sparse Fourier transform

Abstract: We consider the sparse Fourier transform problem: given a complex vector x of length n, and a parameter k, estimate the k largest (in magnitude) coefficients of the Fourier transform of x. The problem is of key interest in several areas, including signal processing, audio/image/video compression, and learning theory.We propose a new algorithm for this problem. The algorithm leverages techniques from digital signal processing, notably Gaussian and Dolph-Chebyshev filters. Unlike the typical approach to this problem, our algorithm is not iterative. That is, instead of estimating "large" coefficients, subtracting them and recursing on the reminder, it identifies and estimates the k largest coefficients in "one shot", in a manner akin to sketching/streaming algorithms. The resulting algorithm is structurally simpler than its predecessors. As a consequence, we are able to extend considerably the range of sparsity, k, for which the algorithm is faster than FFT, both in theory and practice.

Nyström Method vs Random Fourier Features: A Theoretical and Empirical Comparison

Abstract: Both random Fourier features and the Nystrom method have been successfully applied to efficient kernel learning. In this work, we investigate the fundamental difference between these two approaches, and how the difference could affect their generalization performances. Unlike approaches based on random Fourier features where the basis functions (i.e., cosine and sine functions) are sampled from a distribution independent from the training data, basis functions used by the Nystrom method are randomly sampled from the training examples and are therefore data dependent. By exploring this difference, we show that when there is a large gap in the eigen-spectrum of the kernel matrix, approaches based on the Nystrom method can yield impressively better generalization error bound than random Fourier features based approach. We empirically verify our theoretical findings on a wide range of large data sets.

Two-dimensional Fourier transform electronic spectroscopy

Abstract: Two-dimensional Fourier transform electronic spectra of the cyanine dye IR144 in methanol are used to explore new aspects of optical 2D spectroscopy on a femtosecond timescale. The experiments reported here are pulse sequence and coherence pathway analogs of the two-dimensional magnetic resonance techniques known as COSY (correlated spectroscopy) and NOESY (nuclear Overhauser effect spectroscopy). Noncollinear three pulse scattering allows selection of electronic coherence pathways by choice of phase matching geometry, temporal pulse order, and Fourier transform variables. Signal fields and delays between excitation pulses are measured by spectral interferometry. Separate real (absorptive) and imaginary (dispersive) 2D spectra are generated by measuring the signal field at the sample exit, performing a 2D scan that equally weights rephasing and nonrephasing coherence pathways, and phasing the 2D spectra against spectrally resolved pump–probe signals. A 3D signal propagation function is used to correct the...

High Resolution Mass Spectrometry

Abstract: ■ CONTENTS Mass Resolution, Mass Resolving Power 708 Mass Resolution and Accuracy 708 Time-of-Flight Mass Analyzers 708 Orthogonal Acceleration (see ref 13) 709 Reflectron/Multipass TOF 709 Recent Advances in TOF Mass Analyzers 709 Selected Applications 710 Fourier Transform Mass Analyzers 710 Common Features of Fourier Transform Mass Analyzers 710 Ion Accumulation and Detection 711 Advances in Fourier Transform Mass Analyzers 711 Selected Applications 713 Author Information 715 Corresponding Author 715 Biographies 715 Acknowledgments 715 References 715

Radar Maneuvering Target Motion Estimation Based on Generalized Radon-Fourier Transform

Abstract: The slant range of a radar maneuvering target is usually modeled as a multivariate function in terms of its illumination time and multiple motion parameters. This multivariate range function includes the modulations on both the envelope and the phase of an echo of the coherent radar target and provides the foundation for radar target motion estimation. In this paper, the maximum likelihood estimators (MLE) are derived for motion estimation of a maneuvering target based on joint envelope and phase measurement, phase-only measurement and envelope-only measurement in case of high signal-to-noise ratio (SNR), respectively. It is shown that the proposed MLEs are to search the maximums of the outputs of the proposed generalized Radon-Fourier transform (GRFT), generalized Radon transform (GRT) and generalized Fourier transform (GFT), respectively. Furthermore, by approximating the slant range function by a high-order polynomial, the inherent accuracy limitations, i.e., the Cramer-Rao low bounds (CRLB), and some analysis are given for high order motion parameter estimations in different scenarios. Finally, some numerical experimental results are provided to demonstrate the effectiveness of the proposed methods.

Nearly Optimal Sparse Fourier Transform

Abstract: We consider the problem of computing the k-sparse approximation to the discrete Fourier transform of an n-dimensional signal. We show: * An O(k log n)-time randomized algorithm for the case where the input signal has at most k non-zero Fourier coefficients, and * An O(k log n log(n/k))-time randomized algorithm for general input signals. Both algorithms achieve o(n log n) time, and thus improve over the Fast Fourier Transform, for any k = o(n). They are the first known algorithms that satisfy this property. Also, if one assumes that the Fast Fourier Transform is optimal, the algorithm for the exactly k-sparse case is optimal for any k = n^{\Omega(1)}. We complement our algorithmic results by showing that any algorithm for computing the sparse Fourier transform of a general signal must use at least \Omega(k log(n/k)/ log log n) signal samples, even if it is allowed to perform adaptive sampling.

Nearly optimal sparse fourier transform

Abstract: We consider the problem of computing the k-sparse approximation to the discrete Fourier transform of an n-dimensional signal. We show: An O(k log n)-time randomized algorithm for the case where the input signal has at most k non-zero Fourier coefficients, and An O(k log n log(n/k))-time randomized algorithm for general input signals. Both algorithms achieve o(n log n) time, and thus improve over the Fast Fourier Transform, for any k=o(n). They are the first known algorithms that satisfy this property. Also, if one assumes that the Fast Fourier Transform is optimal, the algorithm for the exactly k-sparse case is optimal for any k = nΩ(1). We complement our algorithmic results by showing that any algorithm for computing the sparse Fourier transform of a general signal must use at least Ω(k log (n/k) / log log n) signal samples, even if it is allowed to perform adaptive sampling.

Waveform Design for Active Sensing Systems: A Computational Approach

Abstract: Active sensing applications such as radar, sonar and medical imaging, demand proper designs of the probing waveform. A well-synthesized waveform can significantly increase the system performance in terms of signal-to-interference ratio, spectrum containment, beampattern matching, target parameter estimation and so on. The focus of this work is on designing probing waveforms using computational algorithms. We first investigate designing waveforms with good correlation properties, which are widely useful in applications including range compression, channel estimation and spread spectrum. We consider both the design of a single sequence and that of a set of sequences, the former with only auto-correlations and the latter with auto- and cross-correlations. The proposed algorithms leverage FFT (fast Fourier transform) operations and can efficiently generate long sequences that were previously difficult to synthesize. We present a new derivation of the lower bound for sequence correlations that arises from the proposed algorithm framework. We show that such a lower bound can be closely approached by the newly designed sequences. A two-dimensional extension of the time-delay correlation function is the ambiguity function (AF) that involves a Doppler frequency shift. We give an overview of AF properties and discuss how to minimize AF sidelobes in a discrete formation. Besides good correlation properties, we also consider the stopband constraint that is required in the scenario of avoiding reserved frequency bands or strong electronic jammer. We present an algorithm that accounts for both correlation and stopband constraints. We finally consider transmit beampattern synthesis, particularly in the wideband case. We establish the relationship between a desired beampattern and underlying waveforms by using the Fourier transform. We highlight the increased design freedom resulting from the waveform diversity of a MIMO (multi-input multi-output) system.

Information Transmission using the Nonlinear Fourier Transform, Part I: Mathematical Tools

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 Schr\"odinger 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 first paper explains the mathematical tools that underlie the method.

P3DFFT: A Framework for Parallel Computations of Fourier Transforms in Three Dimensions

Abstract: Fourier and related transforms are a family of algorithms widely employed in diverse areas of computational science, notoriously difficult to scale on high-performance parallel computers with a large number of processing elements (cores). This paper introduces a popular software package called P3DFFT which implements fast Fourier transforms (FFTs) in three dimensions in a highly efficient and scalable way. It overcomes a well-known scalability bottleneck of three-dimensional (3D) FFT implementations by using two-dimensional domain decomposition. Designed for portable performance, P3DFFT achieves excellent timings for a number of systems and problem sizes. On a Cray XT5 system P3DFFT attains 45% efficiency in weak scaling from 128 to 65,536 computational cores. Library features include Fourier and Chebyshev transforms, Fortran and C interfaces, in- and out-of-place transforms, uneven data grids, and single and double precision. P3DFFT is available as open source at http://code.google.com/p/p3dfft/. This pa...

Radon-Fourier Transform for Radar Target Detection (III): Optimality and Fast Implementations

Abstract: As a generalized Doppler filter bank processing, Radon-Fourier transform (RFT) has recently been proposed for long-time coherent integration detection of radar moving targets. The likelihood ratio test (LRT) detector is derived here for rectilinearly moving targets. It is found that the proposed LRT detector has the identical form as the existing RFT detector, which means that the RFT detector is an optimal detector for rectilinearly moving targets under the white Gaussian noise background. For the fast implementations of the RFT detector, instead of the joint 2-D trajectory searching and coherent integration in pulse-range domain, the 1-D fast Fourier transform (FFT)-based frequency bin RFT (FBRFT) method is proposed in the pulse-range frequency domain without loss of integration performance. Moreover, at the cost of a controllable performance loss, a suboptimal approach called subband RFT (SBRFT) is also proposed to reduce the storage memory. It is shown that not only the long-time coherent integration gain can be obtained via the proposed SBRFT, but also the computational complexity and memory cost can be reduced to the level of the conventional Doppler filter banks processing, e.g., moving target detection (MTD). Some numerical experiments are also provided to demonstrate the effectiveness of the proposed methods.

Pipelined Parallel FFT Architectures via Folding Transformation

Abstract: This paper presents a novel approach to develop parallel pipelined architectures for the fast Fourier transform (FFT). A formal procedure for designing FFT architectures using folding transformation and register minimization techniques is proposed. Novel parallel-pipelined architectures for the computation of complex and real valued fast Fourier transform are derived. For complex valued Fourier transform (CFFT), the proposed architecture takes advantage of under utilized hardware in the serial architecture to derive L-parallel architectures without increasing the hardware complexity by a factor of L. The operating frequency of the proposed architecture can be decreased which in turn reduces the power consumption. Further, this paper presents new parallel-pipelined architectures for the computation of real-valued fast Fourier transform (RFFT). The proposed architectures exploit redundancy in the computation of FFT samples to reduce the hardware complexity. A comparison is drawn between the proposed designs and the previous architectures. The power consumption can be reduced up to 37% and 50% in 2-parallel CFFT and RFFT architectures, respectively. The output samples are obtained in a scrambled order in the proposed architectures. Circuits to reorder these scrambled output sequences to a desired order are presented.

Exact acceleration of linear object detectors

Abstract: We describe a general and exact method to considerably speed up linear object detection systems operating in a sliding, multi-scale window fashion, such as the individual part detectors of part-based models The main bottleneck of many of those systems is the computational cost of the convolutions between the multiple rescalings of the image to process, and the linear filters We make use of properties of the Fourier transform and of clever implementation strategies to obtain a speedup factor proportional to the filters' sizes The gain in performance is demonstrated on the well known Pascal VOC benchmark, where we accelerate the speed of said convolutions by an order of magnitude

FM stars: a Fourier view of pulsating binary stars, a new technique for measuring radial velocities photometrically

Abstract: Some pulsating stars are good clocks. When they are found in binary stars, the frequencies of their luminosity variations are modulated by the Doppler effect caused by orbital motion. For each pulsation frequency this manifests itself as a multiplet separated by the orbital frequency in the Fourier transform of the light curve of the star. We derive the theoretical relations to exploit data from the Fourier transform to derive all the parameters of a binary system traditionally extracted from spectroscopic radial velocities, including the mass function which is easily derived from the amplitude ratio of the first orbital sidelobes to the central frequency for each pulsation frequency. This is a new technique that yields radial velocities from the Doppler shift of a pulsation frequency, thus eliminates the need to obtain spectra. For binary stars with pulsating components, an orbital solution can be obtained from the light curve alone. We give a complete derivation of this and demonstrate it both with artificial data, and with a case of a hierarchical eclipsing binary with Kepler mission data, KIC 4150611 (HD 181469). We show that it is possible to detect Jupiter-mass planets orbiting δ Sct and other pulsating stars with our technique. We also show how to distinguish orbital frequency multiplets from potentially similar non-radial m-mode multiplets and from oblique pulsation multiplets.

Recovery of sparse 1-D signals from the magnitudes of their Fourier transform

Abstract: The problem of signal recovery from the autocorrelation, or equivalently, the magnitudes of the Fourier transform, is of paramount importance in various fields of engineering. In this work, for one-dimensional signals, we give conditions, which when satisfied, allow unique recovery from the autocorrelation with very high probability. In particular, for sparse signals, we develop two non-iterative recovery algorithms. One of them is based on combinatorial analysis, which we prove can recover signals upto sparsity o(n1/3) with very high probability, and the other is developed using a convex optimization based framework, which numerical simulations suggest can recover signals upto sparsity o(n1/2) with very high probability.

Computation of multi-region relaxed magnetohydrodynamic equilibria

Abstract: We describe the construction of stepped-pressure equilibria as extrema of a multi-region, relaxed magnetohydrodynamic (MHD) energy functional that combines elements of ideal MHD and Taylor relaxation, and which we call MRXMHD. The model is compatible with Hamiltonian chaos theory and allows the three-dimensional MHD equilibrium problem to be formulated in a well-posed manner suitable for computation. The energy-functional is discretized using a mixed finite-element, Fourier representation for the magnetic vector potential and the equilibrium geometry; and numerical solutions are constructed using the stepped-pressure equilibrium code, SPEC. Convergence studies with respect to radial and Fourier resolution are presented.

Stable and robust sampling strategies for compressive imaging

Abstract: In many signal processing applications, one wishes to acquire images that are sparse in transform domains such as spatial finite differences or wavelets using frequency domain samples. For such applications, overwhelming empirical evidence suggests that superior image reconstruction can be obtained through variable density sampling strategies that concentrate on lower frequencies. The wavelet and Fourier transform domains are not incoherent because low-order wavelets and low-order frequencies are correlated, so compressive sensing theory does not immediately imply sampling strategies and reconstruction guarantees. In this paper we turn to a more refined notion of coherence -- the so-called local coherence -- measuring for each sensing vector separately how correlated it is to the sparsity basis. For Fourier measurements and Haar wavelet sparsity, the local coherence can be controlled and bounded explicitly, so for matrices comprised of frequencies sampled from a suitable inverse square power-law density, we can prove the restricted isometry property with near-optimal embedding dimensions. Consequently, the variable-density sampling strategy we provide allows for image reconstructions that are stable to sparsity defects and robust to measurement noise. Our results cover both reconstruction by $\ell_1$-minimization and by total variation minimization. The local coherence framework developed in this paper should be of independent interest in sparse recovery problems more generally, as it implies that for optimal sparse recovery results, it suffices to have bounded \emph{average} coherence from sensing basis to sparsity basis -- as opposed to bounded maximal coherence -- as long as the sampling strategy is adapted accordingly.

A windowed graph Fourier transform

Abstract: The prevalence of signals on weighted graphs is increasing; however, because of the irregular structure of weighted graphs, classical signal processing techniques cannot be directly applied to signals on graphs. In this paper, we define generalized translation and modulation operators for signals on graphs, and use these operators to adapt the classical windowed Fourier transform to the graph setting, enabling vertex-frequency analysis. When we apply this transform to a signal with frequency components that vary along a path graph, the resulting spectrogram matches our intuition from classical discrete-time signal processing. Yet, our construction is fully generalized and can be applied to analyze signals on any undirected, connected, weighted graph.

The fractional Fourier transform and its applications

Abstract: The main objective of this paper is to study the fractional Fourier transform (FrFT) and its some basic properties. Applications of the FrFT in solving nth order linear non-homogeneous ordinary generalized differential equations and a generalized wave equation are also given.

ShearLab: A Rational Design of a Digital Parabolic Scaling Algorithm

Abstract: Multivariate problems are typically governed by anisotropic features such as edges in images. A common bracket of most of the various directional representation systems which have been proposed to deliver sparse approximations of such features is the utilization of parabolic scaling. One prominent example is the shearlet system. Our objective in this paper is threefold: We first develop a digital shearlet theory which is rationally designed in the sense that it is the digitization of the existing shearlet theory for continuous data. This implies that shearlet theory provides a unified treatment of both the continuum and digital realms. Second, we analyze the utilization of pseudo-polar grids and the pseudo-polar Fourier transform for digital implementations of parabolic scaling algorithms. We derive an isometric pseudo-polar Fourier transform by careful weighting of the pseudo-polar grid, allowing exploitation of its adjoint for the inverse transform. This leads to a digital implementation of the shearlet...

Spline-Kernelled Chirplet Transform for the Analysis of Signals With Time-Varying Frequency and Its Application

Abstract: The conventional time-frequency analysis (TFA) methods, including continuous wavelet transform, short-time Fourier transform, and Wigner-Ville distribution, have played important roles in analyzing nonstationary signals. However, they often show less capability in dealing with nonstationary signals with time-varying frequency due to the bad energy concentration in the time-frequency plane. On the other hand, by introducing an extra transform kernel that matches the instantaneous frequency of the signal, parameterized TFA methods show powerful ability in characterizing time-frequency patterns of nonstationary signals with time-varying frequency. In this paper, a novel time-frequency transform, called spline-kernelled chirplet transform (SCT), is proposed. By introducing a frequency-rotate operator and a frequency-shift operator constructed with spline kernel function, the SCT is particularly powerful for the strongly nonlinear frequency-modulated signals. In addition, an effective algorithm is developed to estimate the parameters of transform kernel in the SCT. The capabilities of the SCT and parameter estimation algorithm are validated by their applications for numerical signals and a set of vibration signal collected from a rotor test rig.

Seismic data interpolation and denoising in the frequency-wavenumber domain

Abstract: I introduce a unified approach for denoising and interpolation of seismic data in the frequency-wavenumber (f-k) domain. First, an angular search in the f-k domain is carried out to identify a sparse number of dominant dips, not only using low frequencies but over the whole frequency range. Then, an angular mask function is designed based on the identified dominant dips. The mask function is utilized with the least-squares fitting principle for optimal denoising or interpolation of data. The least-squares fit is directly applied in the time-space domain. The proposed method can be used to interpolate regularly sampled data as well as randomly sampled data on a regular grid. Synthetic and real data examples are provided to examine the performance of the proposed method.

Image encryption based on interference that uses fractional Fourier domain asymmetric keys

Abstract: We propose an image encryption technique based on the interference principle and phase-truncation approach in the fractional Fourier domain. The proposed scheme offers multiple levels of security with asymmetric keys and is free from the silhouette problem. Multiple input images bonded with random phase masks are independently fractional Fourier transformed. Amplitude truncation of obtained spectrum helps generate individual and universal keys while phase truncation generates two phase-only masks analytically. For decryption, these two phase-only masks optically interfere, and this results in the phase-truncated function in the output. After using the correct random phase mask, universal key, individual key, and fractional orders, the original image is retrieved successfully. Computer simulation results with four gray-scale images validate the proposed method. To measure the effectiveness of the proposed method, we calculated the mean square error between the original and the decrypted images. In this scheme, the encryption process and decryption keys formation are complicated and should be realized digitally. For decryption, an optoelectronic scheme has been suggested.

Digital Filters for Fast Harmonic Sequence Component Separation of Unbalanced and Distorted Three-Phase Signals

Abstract: In this paper, two methods for determining the fundamental frequency and harmonic positive- and negative-sequence components of three-phase signals are investigated. Many aspects of the space-vector discrete Fourier transform and generalized delayed signal cancellation (GDSC) such as response time for different possible implementations, frequency adaptation schemes, stability of recursive implementation, and rounding error effects are discussed. A new design procedure for GDSC transformations is presented. New indices for characterizing three-phase unbalanced and distorted signals are proposed. Simulations and experiments are included in order to verify the performances and illustrate the theoretical conclusions.