Showing papers on "Discrete-time Fourier transform published in 2000"
TL;DR: In this article, Discrete convolution and FFT (DC-FFT) is adopted instead of the method of continuous convolutions and Fourier transform for the contact problems.
613 citations
TL;DR: This definition is based on a particular set of eigenvectors of the DFT matrix, which constitutes the discrete counterpart of the set of Hermite-Gaussian functions, and is exactly unitary, index additive, and reduces to the D FT for unit order.
Abstract: We propose and consolidate a definition of the discrete fractional Fourier transform that generalizes the discrete Fourier transform (DFT) in the same sense that the continuous fractional Fourier transform generalizes the continuous ordinary Fourier transform. This definition is based on a particular set of eigenvectors of the DFT matrix, which constitutes the discrete counterpart of the set of Hermite-Gaussian functions. The definition is exactly unitary, index additive, and reduces to the DFT for unit order. The fact that this definition satisfies all the desirable properties expected of the discrete fractional Fourier transform supports our confidence that it will be accepted as the definitive definition of this transform.
604 citations
TL;DR: In this paper, a computationally efficient numerical procedure to generate 2D correlation spectra from a set of spectral data collected at certain discrete intervals of an external physical variable, such as time, temperature, pressure, etc., is proposed.
Abstract: A computationally efficient numerical procedure to generate twodimensional (2D) correlation spectra from a set of spectral data collected at certain discrete intervals of an external physical variable, such as time, temperature, pressure, etc., is proposed. The method is based on the use of a discrete Hilbert transform algorithm which carries out the time-domain orthogonal transformation of dynamic spectra. The direct computation of a discrete Hilbert transform provides a definite computational advantage over the more traditional fast Fourier transform route, as long as the total number of discrete spectral data traces does not significantly exceed 40. Furthermore, the mathematical equivalence between the Hilbert transform approach and the original formal definition based on the Fourier transform offers an additional useful insight into the true nature of the asynchronous 2D spectrum, which may be regarded as a time-domain cross-correlation function between orthogonally transformed dynamic spectral intensity variations.
473 citations
TL;DR: A new type of DFRFT is introduced, which are unitary, reversible, and flexible, which works in performance similar to the continuous fractional Fourier transform (FRFT) and can be efficiently calculated by the FFT.
Abstract: The discrete fractional Fourier transform (DFRFT) is the generalization of discrete Fourier transform. Many types of DFRFT have been derived and are useful for signal processing applications. We introduce a new type of DFRFT, which are unitary, reversible, and flexible; in addition, the closed-form analytic expression can be obtained. It works in performance similar to the continuous fractional Fourier transform (FRFT) and can be efficiently calculated by the FFT. Since the continuous FRFT can be generalized into the continuous affine Fourier transform (AFT) (the so-called canonical transform), we also extend the DFRFT into the discrete affine Fourier transform (DAFT). We derive two types of the DFRFT and DAFT. Type 1 is similar to the continuous FRFT and AFT and can be used for computing the continuous FRFT and AFT. Type 2 is the improved form of type 1 and can be used for other applications of digital signal processing. Meanwhile, many important properties continuous FRFT and AFT are kept in the closed-form DFRFT and DAFT, and some applications, such as filter design and pattern recognition, are also discussed. The closed-form DFRFT we introduce has the lowest complexity among all current DFRFTs that is still similar to the continuous FRFT.
287 citations
TL;DR: A new optical encryption technique using the fractional Fourier transform to decrypt the data correctly, in which the input plane, encryp- tion plane, and output planes exist, in addition to the key used for encryption.
Abstract: We propose a new optical encryption technique using the fractional Fourier transform. In this method, the data are encrypted to a stationary white noise by two statistically independent random phase masks in fractional Fourier domains. To decrypt the data correctly, one needs to specify the fractional domains in which the input plane, encryp- tion plane, and output planes exist, in addition to the key used for en- cryption. The use of an anamorphic fractional Fourier transform for the encryption of two-dimensional data is also discussed. We suggest an optical implementation of the proposed idea. Results of a numerical simulation to analyze the performance of the proposed method are pre- sented. © 2000 Society of Photo-Optical Instrumentation Engineers. (S0091-3286(00)01811-0)
215 citations
TL;DR: A new image encryption algorithm based on a generalized fractional Fourier transform, to which it is referred as a multifractional Fouriers transform, is proposed.
Abstract: We propose a new image encryption algorithm based on a generalized fractional Fourier transform, to which we refer as a multifractional Fourier transform. We encrypt the input image simply by performing the multifractional Fourier transform with two keys. Numerical simulation results are given to verify the algorithm, and an optical implementation setup is also suggested.
182 citations
Book•
02 Mar 2000
TL;DR: With the presentation at an introductory level, the third edition of the book contains a comprehensive treatment of continuous-time and discrete-time signals and The facts were simultaneously known would operate through 175.
Abstract: Preface 1 FUNDAMENTAL CONCEPTS 1.1 Continuous-Time Signals 1.2 Discrete-Time Signals 1.3 Systems 1.4 Examples of Systems 1.5 Basic System Properties 1.6 Chapter Summary Problems 2 TIME-DOMAIN MODELS OF SYSTEMS 2.1 Input/Output Representation of Discrete-Time Systems 2.2 Convolution of Discrete-Time Signals 2.3 Difference Equation Models 2.4 Differential Equation Models 2.5 Solution of Differential Equations 2.6 Convolution Representation of Continuous-Time Systems 2.7 Chapter Summary Problems 3 THE FOURIER SERIES AND FOURIER TRANSFORM 3.1 Representation of Signals in Terms of Frequency Components 3.2 Trigonometric Fourier Series 3.3 Complex Exponential Series 3.4 Fourier Transform 3.5 Spectral Content of Common Signals 3.6 Properties of the Fourier Transform 3.7 Generalized Fourier Transform 3.8 Application to Signal Modulation and Demodulation 3.9 Chapter Summary Problems 4 FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS 4.1 Discrete-Time Fourier Transform 4.2 Discrete Fourier Transform 4.3 DFT of Truncated Signals 4.4 FFT Algorithm 4.5 Application to Data Analysis 4.6 Chapter Summary Problems 5 FOURIER ANALYSIS OF SYSTEMS 5.1 Fourier Analysis of Continuous-Time Systems 5.2 Response to Periodic and Nonperiodic Inputs 5.3 Analysis of Ideal Filters 5.4 Sampling 5.5 Fourier Analysis of Discrete-Time Systems 5.6 Application to Lowpass Digital Filtering 5.7 Chapter Summary Problems 6 THE LAPLACE TRANSFORM AND THE TRANSFER FUNCTION REPRESENTATION 6.1 Laplace Transform of a Signal 6.2 Properties of the Laplace Transform 6.3 Computation of the Inverse Laplace Transform 6.4 Transform of the Input/Output Differential Equation 6.5 Transform of the Input/Output Convolution Integral 6.6 Direct Construction of the Transfer Function 6.7 Chapter Summary Problems 7 THE z-TRANSFORM AND DISCRETE-TIME SYSTEMS 7.1 z-Transform of a Discrete-Time Signal 7.2 Properties of the z-Transform 7.3 Computation of the Inverse z-Transform 7.4 Transfer Function Representation 7.5 System Analysis Using the Transfer Function Representation 7.6 Chapter Summary Problems 8 ANALYSIS OF CONTINUOUS-TIME SYSTEMS USING THE TRANSFER FUNCTION REPRESENTATION 8.1 Stability and the Impulse Response 8.2 Routh-Hurwitz Stability Test 8.3 Analysis of the Step Response 8.4 Response to Sinusoids and Arbitrary Inputs 8.5 Frequency Response Function 8.6 Causal Filters 8.7 Chapter Summary Problems 9 APPLICATION TO CONTROL 9.1 Introduction to Control 9.2 Tracking Control 9.3 Root Locus 9.4 Application to Control System Design 9.5 Chapter Summary Problems 10 DESIGN OF DIGITAL FILTERS AND CONTROLLERS 10.1 Discretization 10.2 Design of IIR Filters 10.3 Design of IIR Filters Using MATLAB 10.4 Design of FIR Filters 10.5 Design of Digital Controllers 10.6 Chapter Summary Problems 11 STATE REPRESENTATION 11.1 State Model 11.2 Construction of State Models 11.3 Solution of State equations 11.4 Discrete-Time Systems 11.5 Equivalent State Representations 11.6 Discretization of State Model 11.7 Chapter Summary Problems APPENDIX A BRIEF REVIEW OF COMPLEX VARIABLES APPENDIX B BRIEF REVIEW OF MATRICES BIBLIOGRAPHY INDEX
93 citations
TL;DR: In this article, the authors define a discrete fractional Fourier transform (FT) which is essentially the time-evolution operator of the discrete harmonic oscillator, and define its energy eigenfunctions as a discrete algebraic analogue of the Hermite-Gaussian functions.
Abstract: Certain solutions to Harper's equation are discrete analogues of (and approximations to) the Hermite-Gaussian functions. They are the energy eigenfunctions of a discrete algebraic analogue of the harmonic oscillator, and they lead to a definition of a discrete fractional Fourier transform (FT). The discrete fractional FT is essentially the time-evolution operator of the discrete harmonic oscillator.
78 citations
Journal Article•
TL;DR: In this article, an original method is introduced which greatly improves the precision of the Fourier analysis not only in frequency and amplitude but also in time, thus minimizing the problem of the tradeoff of time versus frequency in the classic short-time Fourier transform.
Abstract: An original method is introduced which greatly improves the precision of the Fourier analysis not only in frequency and amplitude but also in time, thus minimizing the problem of the tradeoff of time versus frequency in the classic short-time Fourier transform. This method is of great interest when extracting spectral modeling parameters from existing sounds. A detailed theoretical presentation is made, and practical results obtained from implementing this method are presented.
75 citations
Patent•
15 Mar 2000TL;DR: In this article, a method for registering first and second images which are offset by an x and/or y displacement in sub-pixel locations is presented, which includes the steps of: multiplying the first image by a window function to create a first windowed image, transforming the first window image with a Fourier transform, multiplying the second image by the window function, and transforming the second windowing image with the Fourier transformation, and computing a collection of coordinate pairs from the two image Fourier transforms, such that at each coordinate pair the values of the first and the second
Abstract: Methods for registering first and second images which are offset by an x and/or y displacement in sub-pixel locations are provided. A preferred implementation of the methods includes the steps of: multiplying the first image by a window function to create a first windowed image; transforming the first windowed image with a Fourier transform to create a first image Fourier transform; multiplying the second image by the window function to create a second windowed image; transforming the second windowed image with a Fourier transform to create a second image Fourier transform; computing a collection of coordinate pairs from the first and second image Fourier transforms such that at each coordinate pair the values of the first and second image Fourier transforms are likely to have very little aliasing noise; computing an estimate of a linear Fourier phase relation between the-first and second image Fourier transforms using the Fourier phases of the first and second image Fourier transforms at the coordinate pairs in a minimum-least squares sense; and computing the displacements in the x and/or y directions from the linear Fourier phase relationship. Also provided are a computer program having computer readable program code and program storage device having a program of instructions for executing and performing the methods of the present invention, respectively.
64 citations
TL;DR: A maximum-likelihood estimation method is presented which in parallel with the system transfer function also estimates a parametric noise transfer function, leading to a consistent and efficient estimator.
TL;DR: The fractionalization of the Fourier transform (FT) is analyzed, starting from the minimal premise that repeated application of the fractional Fouriertransform a sufficient number of times should give back the FT.
Abstract: We analyze the fractionalization of the Fourier transform (FT), starting from the minimal premise that repeated application of the fractional Fourier transform (FrFT) a sufficient number of times should give back the FT. There is a qualitative increase in the richness of the solution manifold, from U(1) (the circle S1) in the one-dimensional case to U(2) (the four-parameter group of 2 × 2 unitary matrices) in the two-dimensional case [rather than simply U(1)×U(1)]. Our treatment clarifies the situation in the N-dimensional case. The parameterization of this manifold (a fiber bundle) is accomplished through two powers running over the torus T2=S1×S1 and two parameters running over the Fourier sphere S2. We detail the spectral representation of the FrFT: The eigenvalues are shown to depend only on the T2 coordinates; the eigenfunctions, only on the S2 coordinates. FrFT’s corresponding to special points on the Fourier sphere have for eigenfunctions the Hermite–Gaussian beams and the Laguerre–Gaussian beams, while those corresponding to generic points are SU(2)-coherent states of these beams. Thus the integral transform produced by every Sp(4, R) first-order system is essentially a FrFT.
01 Jan 2000
TL;DR: A formula for computing the cross-cor correlation of a pair of images (and hence the auto-correlation of a single image) is presented, based on a previously published hypercomplex Fourier transform.
Abstract: The auto-correlation and cross-correlation function have been generalized to color images, but only by explicit evaluation of a double summation over the whole image for each pixel of the correlation result. Practical use of any correlation method on images of reasonable spatial resolution requires realization using a Fourier transform. A formula for computing the cross-correlation of a pair of images (and hence the auto-correlation of a single image) is presented, based on a previously published hypercomplex Fourier transform.
08 Oct 2000
TL;DR: Efficient methods to estimate the spectral content of (noisy) periodic waveforms that are common in industrial processes based on the recursive discrete Fourier transform, which are quite immune to uncorrelated measurement noise.
Abstract: This paper presents efficient methods to estimate the spectral content of (noisy) periodic waveforms that are common in industrial processes The techniques presented, which are based on the recursive discrete Fourier transform, are especially useful in computing high-order derivatives of such waveforms Unlike conventional differentiating techniques, the methods presented differentiate in the frequency domain and thus are quite immune to uncorrelated measurement noise This paper also shows the theoretical relationship between the proposed methods and those of well-known resonant filters
TL;DR: This study introduces several types of simplified fractional Fourier transform (SFRFT) that are simpler than the original FRFT in terms of digital computation, optical implementation, implementation of gradient-index media, and implementation of radar systems.
Abstract: The fractional Fourier transform (FRFT) has been used for many years, and it is useful in many applications. Most applications of the FRFT are based on the design of fractional filters (such as removal of chirp noise and the fractional Hilbert transform) or on fractional correlation (such as scaled space-variant pattern recognition). In this study we introduce several types of simplified fractional Fourier transform (SFRFT). Such transforms are all special cases of a linear canonical transform (an affine Fourier transform or an ABCD transform). They have the same capabilities as the original FRFT for design of fractional filters or for fractional correlation. But they are simpler than the original FRFT in terms of digital computation, optical implementation, implementation of gradient-index media, and implementation of radar systems. Our goal is to search for the simplest transform that has the same capabilities as the original FRFT. Thus we discuss not only the formulas and properties of the SFRFT’s but also their implementation. Although these SFRFT’s usually have no additivity properties, they are useful for the practical applications. They have great potential for replacing the original FRFT’s in many applications.
TL;DR: In this article, an optimization method was used to select the wavenumbers k for the inverse Fourier transform in 2.5D electrical modeling, and the model tests showed that with the wavernumbers k selected in this way the inverse-fourier transform performs with satisfactory accuracy.
Abstract: An optimization method is used to select the wavenumbers k for the inverse Fourier transform in 2.5D electrical modelling. The model tests show that with the wavenumbers k selected in this way the inverse Fourier transform performs with satisfactory accuracy.
TL;DR: This integral is accurately evaluated with an improved trapezoidal rule and effectively transcribed using local Fourier basis and adaptive multiscale local Fouriers basis.
TL;DR: In this paper, the authors apply techniques from non-commutative harmonic analysis to the development of fast algorithms for the computation of convolution integrals on motion groups, in particular on the group of rigid-body motions in 3-space, denoted here as SE(3).
TL;DR: In this paper, a new phase-averaging method, denoted as Fourier averaging, is presented for the investigation of periodic flows, where the moments of velocity, as estimated from a small number of samples, show fluctuations in their phasewise development.
Abstract: A new phase-averaging method, denoted as Fourier averaging, is presented for the investigation of periodic flows. In such flows, the moments of velocity, as estimated from a small number of samples, show fluctuations in their phasewise development. In previous methods these fluctuations are reduced by calculating moments from large phase intervals. Fourier averaging, in contrast, neglects high-frequency fluctuations and assumes that they are of no physical relevance. This method supplies additional information on amplitudes and phase angles of discrete frequencies, which may then be used for visualizations of flow fields at any desired phase increment. The Fourier averaging method was verified empirically by LDA measurements and compared to other methods. It is shown that the results obtained by Fourier averaging are more accurate than for previously known methods.
05 Jun 2000
TL;DR: A procedure called pruning, analogous to truncation of the singular-value decomposition, underlies a number of potential applications, among which the fast implementation of space-variant linear systems is discussed.
Abstract: We introduce the fractional Fourier domain decomposition for continuous and discrete signals and systems. A procedure called pruning, analogous to truncation of the singular-value decomposition, underlies a number of potential applications, among which we discuss the fast implementation of space-variant linear systems.
Journal Article•
TL;DR: In this paper, the Riesz summability of Fourier series has been shown to converge with respect to (n 3)/2 convergence in the topology of distributions in a manifold of dimension n.
Abstract: Fourier coefficients f) of piecewise smooth functions are of the order of and Fourier series exp(2ninx) converge everywhere. Here we consider analogs of these results for eigenfunction expansions f (x) = where {À2} and are eigenvalues and an orthonormal complete system of eigenfunctions of a second order positive elliptic operator on a N-dimensional manifold. We prove that the norms of projections of piecewise smooth functions on subspaces generated by eigenfunctions with A h A + 1 satisfy the estimates I Then we give some sharp results on the Riesz summability of Fourier series. In particular we prove that the Riesz means of order 8 > (N 3)/2 converge. Mathematics Subject Classification (2000): 42C 15. Let M be a smooth manifold of dimension N and let A be a second order positive elliptic operator on M, with smooth real coefficients. Assume that this differential operator with suitable boundary conditions is self adjoint with respect to some positive smooth density d ¡.,¿ and admits a sequence of eigenvalues f;,21 and a system of eigenfunctions orthonormal and complete in JL2(M, Then to every function in one can associate a Fourier transform and a Fourier series, These Fourier series converge in the metric of JL2(M, dit) and more generally in the topology of distributions, but under appropriate conditions the PPrvPnmtn alla R eda7inn« il nttnhre 10QQ
TL;DR: In this paper, the use of the short-time Fourier transform and the wavelet transform implemented using the new harmonic wavelets for analyzing the time variation of the spectral contents of exponentially time-decaying signals is studied.
TL;DR: In this article, an effective deterministic method based on the Fast Fourier Transform (FFT) for the Boltzmann equation with Maxwell molecules is considered, and the global existence, uniqueness and limitedness of the...
Abstract: An effective deterministic method based on the Fast Fourier Transform (FFT) for the Boltzmann equation with Maxwell molecules is considered. The global existence, uniqueness and limitedness of the ...
TL;DR: Fast chirp transform (FCT) as discussed by the authors is an extension of the fast Fourier transform (FFT) for the detection of signals with variable frequency. And it can alleviate the requirement of generating complicated families of filter functions typically required in the conventional matched filtering process.
Abstract: The detection of signals with varying frequency is important in many areas of physics and astrophysics. The current work was motivated by a desire to detect gravitational waves from the binary inspiral of neutron stars and black holes, a topic of significant interest for the new generation of interferometric gravitational wave detectors such as LIGO. However, this work has significant generality beyond gravitational wave signal detection. We define a fast chirp transform (FCT) analogous to the fast Fourier transform. Use of the FCT provides a simple and powerful formalism for detection of signals with variable frequency just as Fourier transform techniques provide a formalism for the detection of signals of constant frequency. In particular, use of the FCT can alleviate the requirement of generating complicated families of filter functions typically required in the conventional matched filtering process. We briefly discuss the application of the FCT to several signal detection problems of current interest.
TL;DR: In this paper, it was shown that the discrete fractional Fourier transform recovers the continuum fractional decomposition via a limiting process whereby inner products are preserved, and it is shown that this limiting process can be used to preserve the inner products of the discrete Fourier transformation.
Abstract: It is shown that the discrete fractional Fourier transform recovers the continuum fractional Fourier transform via a limiting process whereby inner products are preserved.
TL;DR: Perspective projections in the space-frequency plane are analyzed, and it is shown that under certain conditions they can be approximately modeled in terms of the fractional Fourier transform.
Abstract: Perspective projections in the space-frequency plane are analyzed, and it is shown that under certain conditions they can be approximately modeled in terms of the fractional Fourier transform. The region of validity of the approximation is examined. Numerical examples are presented.
TL;DR: In this article, a quasi-Born approximation of the Lippman-Schwinger equation is proposed to handle strong scattering accurately and efficiently, which can efficiently produce good-quality images of complex structures with strong lateral perturbations of slowness.
Abstract: Summary
The Born approximation of the Lippman–Schwinger equation has recently been used to implement a recursive method for seismic migration of pressure wavefields. This Born-based method is stable only when the scattering from heterogeneities within an extrapolation depth interval is weak. To handle strong scattering accurately and efficiently, we propose a quasi-Born approximation of the Lippman–Schwinger equation to extrapolate pressure wavefields downwards recursively. We assume that the scattered wavefield is linearly related to the incident wavefield by a scalar function that varies slowly with lateral position within an extrapolation depth interval. The extrapolation is implemented as a dual-doma in procedure in the frequency–space and frequency–wavenumber domains. Fast Fourier transforms are used to transform data between these two domains. The quasi-Born-based depth-migration algorithm is termed the quasi-Born Fourier method. It can efficiently produce good-quality images of complex structures with strong lateral perturbations of slowness. It is stable for strong scattering and can accurately handle scattering and wave propagation along directions at large angles from the main propagation direction. Image quality obtained using the new method is similar to that of a dual-domain migration method that uses the Rytov approximation within each extrapolation depth interval, but the computational speed of the new method is approximately 27 per cent faster than the latter method for pre-stack migration of an industry standard data set—the Marmousi data set. Compared to the Born-based migration method, the quasi-Born Fourier method is slightly less efficient because it requires an additional multiplication and an additional division for each lateral gridpoint in each step of wavefield extrapolation. For weak scattering, the quasi-Born Fourier method converges to the Born-based method. To improve the efficiency of the quasi-Born Fourier method further without losing its accuracy, we propose a hybrid Born/quasi-Born Fourier method in which the Born-based method is used when the scattering within an extrapolation depth interval is weak, and the quasi-Born Fourier method is used for other cases. This hybrid method is approximately 32 per cent faster than the Rytov-based method for the pre-stack depth migration of the Marmousi data set, while the images obtained using both methods have almost the same quality.
TL;DR: A radix-7, decimation-in-space fast Fourier transform (FFT) for images defined on hexagonal aggregates, expressed in terms of the p-product, a generalization of matrix multiplication.
Abstract: Hexagonal aggregates are hierarchical arrangements of hexagonal cells These hexagonal cells may be efficiently addressed using a scheme known as generalized balanced ternary for dimension 2, or GBT_2 The objects of interest in this paper are digital images whose domains are hexagonal aggregates We define a discrete Fourier transform (DFT) for such images The main result of this paper is a radix-7, decimation-in-space fast Fourier transform (FFT) for images defined on hexagonal aggregates The algorithm has complexity N log_7 N It is expressed in terms of the p-product, a generalization of matrix multiplication Data reordering (also known as shuffle permutations) is generally associated with FFT algorithms However, use of the p-product makes data reordering unnecessary
TL;DR: In this article, the spatial resolution degradation due to Fourier transform is discussed through a signal processing technique and the formulation of sensitivity using signal processing and communication theory is also performed and analyzed.
Abstract: The spatial resolution and sensitivity of the Fourier transform method for fringe detection is analyzed. The spatial resolution degradation due to Fourier transform is discussed through a signal processing technique. It is found that the upper limit of spatial resolution for displacement measurement is half the carrier fringe pitch, or half the grid pitch for the grid method. The formulation of sensitivity using signal processing and communication theory is also performed and analyzed. Measures to improve the spatial resolution sensitivity are discussed.