Showing papers on "Fractional Fourier transform published in 2004"
TL;DR: A signal separation technique in the fractional Fourier domain is proposed which can effectively suppress the interferences on the detection of the weak components brought by the stronger components.
Abstract: This paper presents a new method for the detection and parameter estimation of multicomponent LFM signals based on the fractional Fourier transform. For the optimization in the fractional Fourier domain, an algorithm based on Quasi-Newton method is proposed which consists of two steps of searching, leading to a reduction in computation without loss of accuracy. And for multicomponent signals, we further propose a signal separation technique in the fractional Fourier domain which can effectively suppress the interferences on the detection of the weak components brought by the stronger components. The statistical analysis of the estimate errors is also performed which perfects the method theoretically, and finally, simulation results are provided to show the validity of our method.
235 citations
TL;DR: To overcome the limitation of the Fourier transform, the Gabor wavelet is introduced to analyze the phase distributions of the spatial carrier-fringe pattern and the theory of wavelet transform profilometry is presented.
Abstract: We present an analysis of a spatial carrier-fringe pattern in three-dimensional (3-D) shape measurement by using the wavelet transform, a tool excelling for its multiresolution in the time- and space-frequency domains. To overcome the limitation of the Fourier transform, we introduce the Gabor wavelet to analyze the phase distributions of the spatial carrier-fringe pattern. The theory of wavelet transform profilometry, an accuracy check by means of a simulation, and an example of 3-D shape measurement are shown.
188 citations
TL;DR: Heat fluctuations over a time tau in a nonequilibrium stationary state and in a transient state are studied for a simple system with deterministic and stochastic components and the exact Fourier transform of the distribution of these fluctuations is found.
Abstract: Heat fluctuations over a time tau in a nonequilibrium stationary state and in a transient state are studied for a simple system with deterministic and stochastic components: a Brownian particle dragged through a fluid by a harmonic potential which is moved with constant velocity. Using a Langevin equation, we find the exact Fourier transform of the distribution of these fluctuations for all tau. By a saddle-point method we obtain analytical results for the inverse Fourier transform, which, for not too small tau, agree very well with numerical results from a sampling method as well as from the fast Fourier transform algorithm. Due to the interaction of the deterministic part of the motion of the particle in the mechanical potential with the stochastic part of the motion caused by the fluid, the conventional heat fluctuation theorem is, for infinite and for finite tau, replaced by an extended fluctuation theorem that differs noticeably and measurably from it. In particular, for large fluctuations, the ratio of the probability for absorption of heat (by the particle from the fluid) to the probability to supply heat (by the particle to the fluid) is much larger here than in the conventional fluctuation theorem.
145 citations
TL;DR: This note makes a critical comparison of some matlab programs for the digital computation of the fractional Fourier transform that are freely available and describes the own implementation that lters the best out of the existing ones.
134 citations
TL;DR: In this paper, the input image to be encrypted is multiplied by a phase mask, and either its Fourier or fractional Fourier transform is obtained using interference with a wave from another random phase mask.
127 citations
TL;DR: The purpose of this paper is to show that the main results on nonlinear functions can be easily generalized to the case of arbitrary abelian groups if the Walsh-Hadamard transform is replaced by the discrete Fourier transform.
102 citations
04 Jul 2004
TL;DR: A truncated version of the classical Fast Fourier Transform that has the nice property of eliminating the "jumps" in the complexity at powers of two and gains a logarithmic factor with respect to the best previously known algorithms.
Abstract: In this paper, we present a truncated version of the classical Fast Fourier Transform. When applied to polynomial multiplication, this algorithm has the nice property of eliminating the "jumps" in the complexity at powers of two. When applied to the multiplication of multivariate polynomials or truncated multivariate power series, we gain a logarithmic factor with respect to the best previously known algorithms.
101 citations
TL;DR: The complex-window S-transform is similar to a wavelet transform, but has the fixed phase reference of the Fourier transform, leading to better time frequency localization of similar waveforms on the time series.
92 citations
TL;DR: In this article, the authors analyzed and compared two approaches to processing radio occultation data: (1) canonical transform method and (2) full spectrum inversion method and showed that these methods are closely related and can be explained from two view points: (a) both methods apply a Fourier transform like operator to the entire signal, and the derivative of the phase of the transformed signal is used for the computation of bending angles.
Abstract: [1] We analyze and compare two approaches to processing radio occultation data: (1) canonical transform method and (2) full spectrum inversion method. We show that these methods are closely related and can be explained from two view points: (1) both methods apply a Fourier transform like operator to the entire radio occultation signal, and the derivative of the phase of the transformed signal is used for the computation of bending angles, and (2) they can be explained from a signal processing view point as the location of multiple tones constituting the complete signal. The full spectrum inversion method is a composition of phase correction and Fourier transform, which makes the numerical algorithm computationally more efficient as compared to the canonical transform method. We investigate the relative performance of the two methods in simulations using a wave optics propagator. We use simple analytical models of the atmospheric refractivity as well as radiosonde data in order to reproduce complex multipath situations. The numerical simulations as well as the analytical estimations indicate that a resolution of 60 m (or even higher) can be achieved.
79 citations
01 Jan 2004
TL;DR: In this paper, a new method for the detection and parameter estimation of multicomponent LFM signals based on the fractional Fourier transform is presented, which can effectively suppress the interferences on the detection of the weak components brought by the stronger components.
Abstract: This paper presents a new method for the detection and parameter estimation of multicomponent LFM signals based on the fractional Fourier transform. For the optimization in the fractional Fourier domain, an algorithm based on Quasi-Newton method is proposed which consists of two steps of searching, leading to a reduction in computation without loss of accuracy. And for multicomponent signals, we further propose a signal separation technique in the fractional Fourier domain which can effectively suppress the interferences on the detection of the weak components brought by the stronger components. The statistical analysis of the estimate errors is also performed which perfects the method theoretically, and finally, simulation results are provided to show the validity of our method.
78 citations
Book•
07 Jun 2004
TL;DR: In this article, the authors proposed a nonlinear analysis method for Oscillators, based on the transformation in the Fourier Domain of the Linear Differential Equation (LDE).
Abstract: Preface.Chapter 1. Nonlinear Analysis Methods.1.1 Introduction.1.2 Time-Domain Solution.1.3 Solution Through Series Expansion1.4 The Conversion Matrix.1.5 Bibliography.Chapter 2. Nonlinear Measurements.2.1 Introduction.2.2 Load/Source-Pull.2.3 The Vector Nonlinear Network Analyser.2.4 Pulsed Measurements.2.5 Bibliography.Chapter 3. Nonlinear Models.3.1 Introduction.3.2 Physical Models.3.3 Equivalent-Circuit Models.3.4 Black-Box Models.3.5 Simplified Models.3.6 Bibliography.Chapter 4. Power Amplifiers.4.1 Introduction.4.2 Classes of Operation.4.3 Simplified Class-A Fundamental-Frequency Design For High Efficiency.4.4 Multi-Harmonic Design For High Power And Efficiency.4.5 Bibliography.Chapter 5. Oscillators.5.1 Introduction.5.2 Linear Stability and Oscillation Conditions.5.3 From Linear To Nonlinear: Quasi-Large-Signal Oscillation And Stability Conditions.5.4 Design Methods.5.5 Nonlinear Analysis Methods For Oscillators.5.6 Noise.5.7 Bibliography.Chapter 6. Frequency Multipliers and Dividers.6.1 Introduction.6.2 Passive Multipliers.6.3 Active Multipliers.6.4 Frequency Dividers-The Rigenerative (Passive) Approach.6.5 Bibliography.Chapter 7. Mixers. 7.1 Introduction.7.2 Mixer Configurations.7.3 Mixer Design.7.4 Nonlinear Analysis.7.5 Noise.7.6 Bibliography.Chapter 8. Stability and Injection-locked Circuits.8.1 Introduction.8.2 Local Stability Of Nonlinear Circuits In Large-Signal Regime.8.3 Nonlinear Analysis, Stability And Bifurcations.8.4 Injection Locking.8.5 Bibliography.Appendix.A.1. Transformation in the Fourier Domain of the Linear Differential Equation.A.2. Time-Frequency Transformations.A.3 Generalized Fourier Transformation for the Volterra Series Expansion.A.4 Discrete Fourier Transform and Inverse Discrete Fourier Transform for Periodic Signals.A.5 The Harmonic Balance System of Equations for the Example Circuit with N=3.A.6 The Jacobian MatrixA.7 Multi-dimensional Discrete Fourier Transform and Inverse Discrete Fourier Transform for quasi-periodic signals.A.8 Oversampled Discrete Fourier Transform and Inverse Discrete Fourier Transform for Quasi-Periodic Signals.A.9 Derivation of Simplified Transport Equations.A.10 Determination of the Stability of a Linear Network.A.11 Determination of the Locking Range of an Injection-Locked Oscillator.Index.
TL;DR: This work proposes and examines two encryption schemes using the Fresnel transform (FST) based on the use of random phase screens and random jigsaw transforms (JT) for 2-D information encryption.
Abstract: A number of methods have been recently proposed in the literature for the encryption of 2-D information using optical systems based on linear transforms, i.e., the Fourier (FT), the fractional Fourier (FRT), the Fresnel (FST), and the linear canonical transform (LCT). We propose and examine two encryption schemes using the Fresnel transform (FST) based on the use of random phase screens and random jigsaw transforms (JT). The strength and robustness of the level of encryption of the various techniques are examined with respect to blind decryption. These systems are compared with similar FRT- and LCT-based methods. Optical implementations are also discussed and sampling conditions are investigated in the context of the space-bandwidth product.
TL;DR: In this paper, the quantum Fast Fourier Transform (QFFT) can be made exact for arbitrary orders (first showing it for large primes) by using the technique known as "amplitude amplification". Although unlikely to be of any practical use, this construction allows one to make Shor's discrete logarithm quantum algorithm exact.
Abstract: We show how the Quantum Fast Fourier Transform (QFFT) can be made exact for arbitrary orders (first showing it for large primes). Most quantum algorithms only need a good approximation of the quantum Fourier transform of order 2n to succeed with high probability, and this QFFT can in fact be done exactly. Kitaev1 showed how to approximate the Fourier transform for any order. Here we show how his construction can be made exact by using the technique known as "amplitude amplification". Although unlikely to be of any practical use, this construction allows one to make Shor's discrete logarithm quantum algorithm exact. Thus we have the first example of an exact non black box fast quantum algorithm, thereby giving more evidence that "quantum" need not be probabilistic. We also show that in a certain sense the family of circuits for the exact QFFT is uniform. Namely, the parameters of the gates can be approximated efficiently.
TL;DR: It is shown that it is possible to define the autocorrelation function of symbolic data, assuming only that the authors can compare any two symbols and decide if they are equal or distinct, and another interpretation of the spectrum is given, borrowing from the spectral envelope concept.
23 Aug 2004
TL;DR: A new Fourier-based descriptor is introduced for the characterization of the shapes for retrieval purposes that combines the benefits of the wavelet transform and Fourier transform and can be presented in multiple scales, which improves the shape retrieval accuracy of the commonly used Fouriers.
Abstract: The shapes occurring in the images are important in the content-based image retrieval. We introduce a new Fourier-based descriptor for the characterization of the shapes for retrieval purposes. This descriptor combines the benefits of the wavelet transform and Fourier transform. This way the Fourier descriptors can be presented in multiple scales, which improves the shape retrieval accuracy of the commonly used Fourier-descriptors. The multiscale Fourier descriptor is formed by applying the complex wavelet transforms to the boundary function of an object extracted from an image. After that, the Fourier transform is applied to the wavelet coefficients in multiple scales. This way the multiscale shape representation can be expressed in a rotation invariant form. The retrieval efficiency of this multiscale Fourier descriptor is compared to an ordinary Fourier descriptor and CSS-shape representation.
TL;DR: The Fourier analytic approach to sections of convex bodies has been developed and has led to several results, including a complete analytic solution to the Busemann-Petty problem, characterizations of intersection bodies, extremal sections of l p-balls, and a Fourier analytical solution to Shephard's problem as mentioned in this paper.
Abstract: The Fourier analytic approach to sections of convex bodies has recently been developed and has led to several results, including a complete analytic solution to the Busemann-Petty problem, characterizations of intersection bodies, extremal sections ofl
p-balls. In this article, we extend this approach to projections of convex bodies and show that the projection counterparts of the results mentioned above can be proved using similar methods. In particular, we present a Fourier analytic proof of the recent result of Barthe and Naor on extremal projections ofl
p-balls, and give a Fourier analytic solution to Shephard’s problem, originally solved by Petty and Schneider and asking whether symmetric convex bodies with smaller hyperplane projections necessarily have smaller volume. The proofs are based on a formula expressing the volume of hyperplane projections in terms of the Fourier transform of the curvature function.
01 Jun 2004-Nuclear Instruments & Methods in Physics Research Section A-accelerators Spectrometers Detectors and Associated Equipment
TL;DR: In this article, the authors investigate the prospect for designing a high-resolution parametric method for signal processing of both Lorentzian and non-Lorentzians spectra offering an accurate and robust performance with efficiency and stability comparable to the fast Fourier transform.
Abstract: We investigate the prospect for designing a high-resolution parametric method for signal processing of both Lorentzian and non-Lorentzian spectra offering an accurate and robust performance with efficiency and stability comparable to the fast Fourier transform. It is demonstrated that the standard fast Pade transform is well suited for successfully accomplishing this multifaceted task as opposed to various fitting algorithms that all yield non-unique solutions to quantification problems. This is substantiated by computations on time signals encoded in the brain of healthy volunteers and patients using Magnetic Resonance Spectroscopy.
TL;DR: The approach is to construct both the model and their derivatives in the Fourier domain, where they have simple and general expressions, and then Fourier transform them back to the spectral domain, for accurate curve-fitting of Fourier deconvoluted spectra.
TL;DR: In this article, the approximate analytical expressions for the output field distribution of a Hermite-cosh-Gaussian beam through apertured FRFT systems are derived and compared with those obtained from diffraction integral formulae by numerical simulations.
03 Feb 2004
TL;DR: This work considers the problem of locating a template as a subimage of a larger image and outlines how an algorithm to calculate the maxima of the correlation function would work.
Abstract: We consider the problem of locating a template as a subimage of a larger image. Computing the maxima of the correlation function solves this problem classically. Since the correlation can be calculated with the Fourier transform this problem is a good candidate for a superior quantum algorithmic solution. We outline how such an algorithm would work.
01 Jan 2004
TL;DR: The dilating Gabor transformation is introduced by using a changeable window of Gaussian function in a conventional Gabor transform to analyze the optical fringes of 3-D shape measurement, providing more accurate results than Fourier transform and Gabortransform.
Abstract: In order to overcome the limitations of conventional Fourier transform and Gabor transform analyzing nonstationary signals, dilating Gabor transform is applied to analyze the optical fringes of 3-D shape measurement. The dilating Gabor transformation is introduced by using a changeable window of Gaussian function in a conventional Gabor transform. This phase analysis method provides more accurate results than Fourier transform and Gabor transform. Simulation and experimental results are presented that demonstrate the validity of the principle.
Journal Article•
TL;DR: In this paper, the generalized Radon transform (or Hough transform) is used to detect the shape of a sphere in an image, where the amplitude at a point corresponds to the amount of evidence for that shape.
Abstract: The generalized Radon (or Hough) transform is a well-known tool for detecting parameterized shapes in an image. The Radon transform is a mapping between the image space and a parameter space. The coordinates of a point in the latter correspond to the parameters of a shape in the image. The amplitude at that point corresponds to the amount of evidence for that shape. In this paper we discuss three important aspects of the Radon transform. The first aspect is discretization. Using concepts from sampling theory we derive a set of sampling criteria for the generalized Radon transform. The second aspect is accuracy. For the specific case of the Radon transform for spheres, we examine how well the location of the maxima matches the true parameters. We derive a correction term to reduce the bias in the estimated radii. The third aspect concerns a projection-based algorithm to reduce memory requirements.
TL;DR: A windowed Fourier transform method is proposed, with advantages of signal localization and noise filtering, for demodulation of carrier fringes, and an example demonstrates the improved result.
Abstract: The Fourier transform method for demodulation of carrier fringes has been extensively developed and widely used in optical metrology. However, the Fourier transform being a global operation, it has poor ability to localize the signal properties and hence the result of FTM is not ideal. A windowed Fourier transform method is thus proposed, with advantages of signal localization and noise filtering. An example demonstrates the improved result compared to the traditional Fourier transform.
Patent•
01 Oct 2004TL;DR: In this paper, a first divider divides the pilot signal contained in the frequency domain OFDM signal by a specified pilot signal, and a zero insertion means then inserts zero signals in the first dividers output.
Abstract: An OFDM signal receiver reduces frequency response estimation error, and reduces the circuit scale needed for a hardware implementation and the number of operations performed in a software implementation. A first Fourier transform circuit converts an OFDM signal to the frequency domain by a Fourier transform. A first divider divides the pilot signal contained in the frequency domain OFDM signal by a specified pilot signal. A zero insertion means then inserts zero signals in the first divider output. A window function multiplying means multiplies the zero insertion means output by a window function, and an inverse Fourier transform means applies an inverse Fourier transform to the multiplier output. A coring means then cores the inverse Fourier transform output, and truncation means truncates the coring means output at a specified data length. A second Fourier transform circuit applies another Fourier transform to the truncated result. A window function dividing means then divides the Fourier transform result by the window function, and a second dividing means divides the output of the first Fourier transform means by the output of the window function dividing means.
TL;DR: A lithium niobate crystal has been used as a phase contrast filter to reconstruct the decrypted phase image, alleviating the need of alignment in the Fourier plane making the system rugged.
TL;DR: It is shown that the same methods can be applied to process a single closed-fringe pattern in either noise reduction or phase approximation, which adds to the versatility of the windowed Fourier-transform method for fringe pattern analysis.
Abstract: Novel approaches based on windowed Fourier transform for demodulation of fringe patterns were previously presented [Appl. Opt. 43, 2695-2702 (2004)], where extraction of phase and phase derivatives from either phase-shifted fringe patterns or a single-carrier fringe pattern was the main focus. I show that the same methods can be applied to process a single closed-fringe pattern in either noise reduction or phase approximation, which adds to the versatility of the windowed Fourier-transform method for fringe pattern analysis.
TL;DR: An improved fast algorithm for numerical calculation of the fractional Fourier transforms of a rectangular function and a Gaussian function is proposed and the results show that the calculation accuracy of the improved algorithm is much higher.
Abstract: Through the optimization of the main interval of the fractional order, an improved fast algorithm for numerical calculation of the fractional Fourier transforms is proposed. With this improved algorithm, the fractional Fourier transforms of a rectangular function and a Gaussian function are calculated. Its calculation errors are compared with those calculated with the previously published algorithm, and the results show that the calculation accuracy of the improved algorithm is much higher.
TL;DR: The quadratic phase function is fundamental in describing and computing wave-propagation-related phenomena under the Fresnel approximation; it is also frequently used in many signal processing algorithms and has interesting properties and Fourier transform relations.
Abstract: The quadratic phase function is fundamental in describing and computing wave-propagation-related phenomena under the Fresnel approximation; it is also frequently used in many signal processing algorithms. This function has interesting properties and Fourier transform relations. For example, the Fourier transform of the sampled chirp is also a sampled chirp for some sampling rates. These properties are essential in interpreting the aliasing and its effects as a consequence of sampling of the quadratic phase function, and lead to interesting and efficient algorithms to simulate Fresnel diffraction. For example, it is possible to construct discrete Fourier transform (DFT)-based algorithms to compute exact continuous Fresnel diffraction patterns of continuous, not necessarily bandlimited, periodic masks at some specific distances.
Patent•
17 Dec 2004TL;DR: In this article, a 2D transform and its inverse have been implemented as a sequence of lifting steps arranged for reduced computational complexity (i.e., reducing a number of non-trivial operations).
Abstract: A 2D transform and its inverse have an implementation as a sequence of lifting steps arranged for reduced computational complexity (i.e., reducing a number of non-trivial operations). This transform pair has energy compaction properties similar to the discrete cosine transform (DCT), and is also lossless and scale-free. As compared to a separable DCT transform implemented as 1D DCT transforms applied separably to rows and columns of a 2D data block, the transforms operations are re-arranged into a cascade of elementary transforms, including the 2×2 Hadamard transform, and 2×2 transforms incorporating lifting rotations. These elementary transforms have implementations as a sequence of lifting operations.