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Showing papers on "Fractional Fourier transform published in 2002"


Book
06 Nov 2002
TL;DR: The Breadth and Depth of DSP Statistics, Probability and Noise ADC and DAC DSP Software Linear Systems Convolution Properties of Convolution The Discrete Fourier Transform Applications of the DFT Fourier transform Properties Fouriertransform Pairs The Fast Fouriers Transform Continuous Signal Processing Introduction to Digital Filters
Abstract: The Breadth and Depth of DSP Statistics, Probability and Noise ADC and DAC DSP Software Linear Systems Convolution Properties of Convolution The Discrete Fourier Transform Applications of the DFT Fourier Transform Properties Fourier Transform Pairs The Fast Fourier Transform Continuous Signal Processing Introduction to Digital Filters Moving Average Filters Windowed-Sinc Filters Custom Filters FFT Convolution Recursive Filters Chebyshev Filters Filter Comparison Audio Processing Image Formation and Display Linear Image Processing Special Imaging Techniques Neural Networks (and more!) Data Compression Digital Signal Processors Getting Started with DSPs Complex Numbers The Complex Fourier Transform The Laplace Transform The z-Transform Index

594 citations


Book ChapterDOI
01 Jan 2002
TL;DR: In this paper, the authors review the development and understanding of the nonlinear Fourier analysis of measured space and time series, based upon a generalization of linear Fourier Analysis referred to as the inverse scattering transform (IST) and its generalizations.
Abstract: Publisher Summary The Fourier transform has provided one of the most important mathematical tools for understanding the dynamics of linear wave trains that are presumed to be governed by linear partial differential equations with a well-defined dispersion relation. The aim of this chapter is to review the development and understanding of the nonlinear Fourier analysis of measured space and time series. The approach is based upon a generalization of linear Fourier analysis referred to as the inverse scattering transform (IST) and its generalizations. From a mathematical point of view, IST solves particular “integrable” nonlinear partial differential wave equations such as the Korteweg-de Vries (KdV), the nonlinear Schrodinger (NLS) and the Kadomtsev–Petviashvili (KP) equations. Because of the mathematical complexity of these theories of nonlinear wave propagation, one cannot expect to bridge all the physical possibilities for the analysis of nonlinear wave data in a single review. The chapter closes with a brief introduction to the application of the inverse scattering transform as a time series analysis tool.

357 citations


Journal ArticleDOI
TL;DR: In this article, the DC-FFT algorithm was used to analyze the contact stresses in an elastic body under pressure and shear tractions for high efficiency and accuracy, and a set of general formulas of the frequency response function for the elastic field was derived and verified.
Abstract: The knowledge of contact stresses is critical to the design of a tribological element. It is necessary to keep improving contact models and develop efficient numerical methods for contact studies, particularly for the analysis involving coated bodies with rough surfaces. The fast Fourier Transform technique is likely to play an important role in contact analyses. It has been shown that the accuracy in an algorithm with the fast Fourier Transform is closely related to the convolution theorem employed. The algorithm of the discrete convolution and fast Fourier Transform, named the DC-FFT algorithm includes two routes of problem solving: DC-FFT/Influence coefficients/Green's, function for the cases with known Green's functions and DC-FFT/Influence coefficient/conversion, if frequency response functions are known. This paper explores the method for the accurate conversion for influence coefficients from frequency response functions, further improves the DC- FFT algorithm, and applies this algorithm to analyze the contact stresses in an elastic body under pressure and shear tractions for high efficiency and accuracy. A set of general formulas of the frequency response function for the elastic field is derived and verified. Application examples are presented and discussed.

265 citations


Journal Article
TL;DR: Simulation results indicate that the energy of LFM signal will be collected effectively when the fractional order is matching with its modulation slope and in weak signals detection of underwater acoustic domain, the authors can get high anti-Doppler performance using the Fractional fourier transform algorithm.
Abstract: Based on the concept of the fractional fourier transform, its digital computation is given through computer simulation. In terms of linear frequency modulation (LFM) signal, the relation between fractional order and modulation slope is analyzed and the performance comparison with matched filter is given. Moreover, the separation of LFM signal and noise is realized in low signal-to-noise ratio through simulation. Simulation results indicate that the energy of LFM signal will be collected effectively when the fractional order is matching with its modulation slope. In weak signals detection of underwater acoustic domain, we can get high anti-Doppler performance using the Fractional fourier transform algorithm.

243 citations


Posted Content
TL;DR: In this article, an approximate Fourier transform on L$L$ elements is defined, which is computationally attractive in a certain setting, and which may find application to the problem of factoring integers with a quantum computer.
Abstract: We define an approximate version of the Fourier transform on $2^L$ elements, which is computationally attractive in a certain setting, and which may find application to the problem of factoring integers with a quantum computer as is currently under investigation by Peter Shor. (1994 IBM Internal Report)

193 citations


Journal ArticleDOI
TL;DR: To solve the problem whereby weak targets are shadowed by the sidelobes of strong ones, a new implementation of the CLEAN technique is proposed based on filtering in the fractional Fourier domain, and strong moving targets and weak ones can be detected iteratively.
Abstract: As a useful signal processing technique, the fractional Fourier transform (FrFT) is largely unknown to the radar signal processing community. In this correspondence, the FrFT is applied to airborne synthetic aperture radar (SAR) slow-moving target detection. For airborne SAR, the echo from a ground moving target can be regarded approximately as a chirp signal, and the FrFT is a way to concentrate the energy of a chirp signal. Therefore, the FrFT presents a potentially effective technique for ground moving target detection in airborne SAR. Compared with the common Wigner-Ville distribution (WVD) algorithm, the FrFT is a linear operator, and will not be influenced by cross-terms even if multiple moving targets exist. Moreover, to solve the problem whereby weak targets are shadowed by the sidelobes of strong ones, a new implementation of the CLEAN technique is proposed based on filtering in the fractional Fourier domain. In this way strong moving targets and weak ones can be detected iteratively. This combined method is demonstrated by using raw clutter data combined with simulated moving targets.

180 citations


Journal ArticleDOI
TL;DR: A new architecture is proposed that encodes a primary image to white noise based on iterative fractional Fourier transform that can provide additional keys for encryption to make the code more difficult to break.

174 citations


Journal ArticleDOI
TL;DR: In this article, a concept of integer fast Fourier transform (IntFFT) for approximating the discrete Fourier Transform (DFT) is introduced, where the lifting scheme is used to approximate complex multiplications appearing in the FFT lattice structures.
Abstract: A concept of integer fast Fourier transform (IntFFT) for approximating the discrete Fourier transform is introduced. Unlike the fixed-point fast Fourier transform (FxpFFT), the new transform has the properties that it is an integer-to-integer mapping, is power adaptable and is reversible. The lifting scheme is used to approximate complex multiplications appearing in the FFT lattice structures where the dynamic range of the lifting coefficients can be controlled by proper choices of lifting factorizations. Split-radix FFT is used to illustrate the approach for the case of 2/sup N/-point FFT, in which case, an upper bound of the minimal dynamic range of the internal nodes, which is required by the reversibility of the transform, is presented and confirmed by a simulation. The transform can be implemented by using only bit shifts and additions but no multiplication. A method for minimizing the number of additions required is presented. While preserving the reversibility, the IntFFT is shown experimentally to yield the same accuracy as the FxpFFT when their coefficients are quantized to a certain number of bits. Complexity of the IntFFT is shown to be much lower than that of the FxpFFT in terms of the numbers of additions and shifts. Finally, they are applied to noise reduction applications, where the IntFFT provides significantly improvement over the FxpFFT at low power and maintains similar results at high power.

165 citations


Journal ArticleDOI
TL;DR: The eigenfunctions of the LCT are derived and it is shown that there are usually varieties of input functions that can cause the self-imaging phenomena in optics.
Abstract: The linear canonical transform (the LCT) is a useful tool for optical system analysis and signal processing. It is parameterized by a 2/spl times/2 matrix {a, b, c, d}. Many operations, such as the Fourier transform (FT), fractional Fourier transform (FRFT), Fresnel transform, and scaling operations are all the special cases of the LCT. We discuss the eigenfunctions of the LCT. The eigenfunctions of the FT, FRFT, Fresnel transform, and scaling operations have been known, and we derive the eigenfunctions of the LCT based on the eigenfunctions of these operations. We find, for different cases, that the eigenfunctions of the LCT also have different forms. When |a+d| 2, the eigenfunctions become the chirp multiplication and chirp convolution of self-similar functions (fractals). Besides, since many optical systems can be represented by the LCT, we can thus use the eigenfunctions of the LCT derived in this paper to discuss the self-imaging phenomena in optics. We show that there are usually varieties of input functions that can cause the self-imaging phenomena for an optical system.

162 citations


Journal ArticleDOI
TL;DR: Wave-front reconstruction with the use of the fast Fourier transform (FFT) and spatial filtering is shown to be computationally tractable and sufficiently accurate for use in large Shack-Hartmann-based adaptive optics systems.
Abstract: Wave-front reconstruction with the use of the fast Fourier transform (FFT) and spatial filtering is shown to be computationally tractable and sufficiently accurate for use in large Shack–Hartmann-based adaptive optics systems (up to at least 10,000 actuators). This method is significantly faster than, and can have noise propagation comparable with that of, traditional vector–matrix-multiply reconstructors. The boundary problem that prevented the accurate reconstruction of phase in circular apertures by means of square-grid Fourier transforms (FTs) is identified and solved. The methods are adapted for use on the Fried geometry. Detailed performance analysis of mean squared error and noise propagation for FT methods is presented with the use of both theory and simulation.

158 citations


Book
08 Mar 2002
TL;DR: In this article, the authors present a mathematical representation of a sum of sinusoidal signals and their properties, including the Fourier Transform and the Spectrum, as well as its properties and properties.
Abstract: 1. Introduction. Mathematical Representation of Signals. Mathematical Representation of Systems. Thinking about Systems. 2. Sinusoids. Tuning Fork Experiment. Review of Sine and Cosine Functions. Sinusoidal Signals. Sampling and Plotting Sinusoids. Complex Exponentials and Phasors. Phasor Addition. Physics of the Tuning Fork. Time Signals: More Than Formulas. 3. Spectrum Representation. The Spectrum of a Sum of Sinusoids. Beat Notes. Periodic Waveforms. More Periodic Signals. Fourier Series Analysis and Synthesis. Time-Frequency Spectrum. Frequency Modulation: Chirp Signals. 4. Sampling and Aliasing. Sampling. Spectrum View of Sampling and Reconstruction. Strobe Demonstration. Discrete-to-Continuous Conversion. The Sampling Theorem. 5. FIR Filters. Discrete-Time Systems. The Running Average Filter. The General FIR Filter. Implementation of FIR Filters. Linear Time-Invariant (LTI) Systems. Convolution and LTI Systems. Cascaded LTI Systems. Example of FIR Filtering. 6. Frequency Response of FIR Filters. Sinusoidal Response of FIR Systems. Superposition and the Frequency Response. Steady State and Transient Response. Properties of the Frequency Response. Graphical Representation of the Frequency Response. Cascaded LTI Systems. Running-Average Filtering. Filtering Sampled Continuous-Time Signals. 7. z-Transforms. Definition of the z-Transform. The z-Transform and Linear Systems. Properties of the z-Transform. The z-Transform as an Operator. Convolution and the z-Transform. Relationship between the z -Domain and the w-Domain. Useful Filters. Practical Bandpass Filter Design. Properties of Linear Phase Filters. 8. IIR Filters. The General IIR Difference Equation. Time-Domain Response. System Function of an IIR Filter. Poles and Zeros. Frequency Response of an IIR Filter. Three Domains. The Inverse z-Transform and Some Applications. Steady-State Response and Stability. Second-Order Filters. Frequency Response of Second-Order IIR Filter. Example of an IIR Lowpass Filter. 9. Continuous-Time Signals and LTI Systems. Continuous-Time Signals. The Unit Impulse. Continuous-Time Systems. Linear Time-Invariant Systems. Impulse Responses of Basic LTI Systems. Convolution of Impulses. Evaluating Convolution Integrals. Properties of LTI Systems. Using Convolution to Remove Multipath Distortion. 10. The Frequency Response. The Frequency Response Function for LTI Systems. Response to Real Sinusoidal Signals. Ideal Filters. Application of Ideal Filters. Time-Domain or Frequency-Domain? 11. Continuous-Time Fourier Transform. Definition of the Fourier Transform. The Fourier Transform and the Spectrum. Existence and Convergence of the Fourier Transform. Examples of Fourier Transform Pairs. Properties of Fourier Transform Pairs. The Convolution Property. Basic LTI Systems. The Multiplication Property. Table of Fourier Transform Properties and Pairs. Using the Fourier Transform for Multipath Analysis. 12. Filtering, Modulation, and Sampling. Linear Time-Invariant Systems. Sinewave Amplitude Modulation. Sampling and Reconstruction. 13. Computing the Spectrum. Finite Fourier Sum. Too Many Fourier Transforms? Time-windowing. Analysis of a Sum of Sinusoids. Discrete Fourier Transform. Spectrum Analysis of Finite-Length Signals. Spectrum Analysis of Periodic Signals. The Spectrogram. The Fast Fourier Transform (FFT). Appendix A: Complex Numbers. Notation for Complex Numbers. Euler's Formula. Algebraic Rules for Complex Numbers. Geometric Views of complex Operations. Powers and Roots. Appendix B: Programming in MATLAB. MATLAB Help. Matrix Operations and Variables. Plots and Graphics. Programming Constructs. MATLAB Scripts. Writing a MATLAB Function. Programming Tips. Appendix C: Laboratory Projects. Introduction to MATLAB. Encoding and Decoding Touch-Tone Signals. Two Convolution GUIs. Appendix D: CD-ROM Demos. Index.

Journal ArticleDOI
TL;DR: This paper proposes a fast approximate algorithm for the associated Legendre transform by means of polynomial interpolation accelerated by the Fast Multipole Method (FMM), and shows that the algorithm is stable and is faster than the direct computation for N ≥ 511.
Abstract: The spectral method with discrete spherical harmonics transform plays an important role in many applications. In spite of its advantages, the spherical harmonics transform has a drawback of high computational complexity, which is determined by that of the associated Legendre transform, and the direct computation requires time of O(N3) for cut-off frequency N. In this paper, we propose a fast approximate algorithm for the associated Legendre transform. Our algorithm evaluates the transform by means of polynomial interpolation accelerated by the Fast Multipole Method (FMM). The divide-and-conquer approach with split Legendre functions gives computational complexity O(N2 log N). Experimental results show that our algorithm is stable and is faster than the direct computation for N ≥ 511.

Journal ArticleDOI
TL;DR: In this paper, a method for calculating the matrix elements of the Coulomb operator for Gaussian basis sets using an intermediate discrete Fourier transform of the density was described. But their results are different from those of the Gaussian and augmented-plane-wave method of Parrinello and co-workers.
Abstract: We describe a method for calculating the matrix elements of the Coulomb operator for Gaussian basis sets using an intermediate discrete Fourier transform of the density. Our goals are the same as those of the Gaussian and augmented-plane-wave method of Parrinello and co-workers [M. Krack and M. Parrinello, Phys. Chem. Chem. Phys. 2, 2105 (2000)], but our techniques are quite different. In particular, we aim at much higher numerical accuracy than typical programs using plane wave expansions. Our method is free of the effects of periodic images and yields full precision. Other low-scaling methods for the Coulomb operator are compared to the Fourier transform method with regard to numerical precision, asymptotic scaling with molecular size, asymptotic scaling with basis set size, onset point (the size of the calculation where the method outperforms traditional Gaussian integral techniques by a factor of 2), and the ability to calculate the Hartree–Fock exchange operator. The Fourier transform method is superior to alternatives by most criteria. In particular, for typical molecular applications it has an earlier onset point than fast multipole methods.

Journal ArticleDOI
TL;DR: An analytical and concise formula is derived for the cross-spectral density of partially coherent twisted anisotropic GSM beams passing through a FRT system in terms of the tensor method.
Abstract: The fractional Fourier transform (FRT) is applied to partially coherent twisted anisotropic Gaussian-Schell model (GSM) beams based directly on the cross-spectral density. An analytical and concise formula is derived for the cross-spectral density of partially coherent twisted anisotropic GSM beams passing through a FRT system in terms of the tensor method. The corresponding tensor ABCD law for performing a FRT is obtained. The connection between the FRT formula and the generalized Collins formula for partially coherent beams is discussed. The formulas derived provide a powerful tool for analyzing and calculating the FRTs of partially coherent beams.

Journal ArticleDOI
TL;DR: The paper introduces and develops the fractional discrete cosine transform (DCT) on the same lines, discussing multiplicity and computational aspects.
Abstract: The extension of the Fourier transform operator to a fractional power has received much attention in signal theory and is finding attractive applications. The paper introduces and develops the fractional discrete cosine transform (DCT) on the same lines, discussing multiplicity and computational aspects. Similarities and differences with respect to the fractional Fourier transform are pointed out.


Journal ArticleDOI
TL;DR: An important aspect consists in showing the advantage of wavelet transform over Fourier transform with respect to dual localization of a signal in both the original and the transformed domain enabling principal new application fields in comparison with Fouriertransform.
Abstract: The wavelet transform has been established with the Fourier transform as a data-processing method in analytical chemistry. The main fields of application in analytical chemistry are related to denoising, compression, variable reduction, and signal suppression. Analytical applications were selected showing prospects and limitations of the wavelet transform. An important aspect consists in showing the advantage of wavelet transform over Fourier transform with respect to dual localization of a signal in both the original and the transformed domain enabling principal new application fields in comparison with Fourier transform.

Journal ArticleDOI
TL;DR: This paper derives several new transforms that are the generalization of the cosine, sine, or Hartley transform and shows that the FRCT/FRST, CCT/CST, and SFRCT/SFRST are also useful for the one-sided signal processing.
Abstract: In previous papers, the Fourier transform (FT) has been generalized into the fractional Fourier transform (FRFT), the linear canonical transform (LCT), and the simplified fractional Fourier transform (SFRFT). Because the cosine, sine, and Hartley transforms are very similar to the FT, it is reasonable to think they can also be generalized by the similar way. We introduce several new transforms. They are all the generalization of the cosine, sine, or Hartley transform. We first derive the fractional cosine, sine, and Hartley transforms (FRCT/FRST/FRHT). They are analogous to the FRFT. Then, we derive the canonical cosine and sine transforms (CCT/CST). They are analogous to the LCT. We also derive the simplified fractional cosine, sine, and Hartley transforms (SFRCT/SFRST/SFRHT). They are analogous to the SFRFT and have the advantage of real-input-real-output. We also discuss the properties, digital implementation, and applications (e.g., the applications for filter design and space-variant pattern recognition) of these transforms. The transforms introduced in this paper are very efficient for digital implementation. We can just use one half or one fourth of the real multiplications required for the FRFT and LCT to implement them. When we want to process even, odd, or pure real/imaginary functions, we can use these transforms instead of the FRFT and LCT. Besides, we also show that the FRCT/FRST, CCT/CST, and SFRCT/SFRST are also useful for the one-sided (t /spl isin/ [0, /spl infin/]) signal processing.

Proceedings ArticleDOI
24 Jun 2002
TL;DR: A new and unique system for achieving transform coding aims of coefficient elimination and compensation is developed and demonstrated, based on iterative projection of signals between the image domain and transform domain.
Abstract: Overcomplete transforms, like the dual-tree complex wavelet transform, offer more flexible signal representations than critically-sampled transforms. Large numbers of transform coefficients can be discarded without much reconstruction quality loss by forcing compensatory changes in the remaining coefficients. We develop and demonstrate a new and unique system for achieving these transform coding aims of coefficient elimination and compensation. The system is based on iterative projection of signals between the image domain and transform domain.

Journal ArticleDOI
TL;DR: In this paper, the authors describe improvements in the implementation of the mixed Fourier transform, which make the method more robust and efficient and avoid potential numerical instabilities, which occasionally caused problems in the previous implementation.
Abstract: [1] A standard method for modeling electromagnetic propagation in the troposphere is the Fourier split-step algorithm for solving the parabolic wave equation. An important advance in this technique was the introduction of the mixed Fourier transform, which permitted the extension of the method from propagation over only smooth perfectly conducting surfaces to quite general surfaces with impedance boundary conditions. This paper describes improvements in the implementation of the mixed Fourier transform, which make the method more robust and efficient and avoid potential numerical instabilities, which occasionally caused problems in the previous implementation. Some examples are also presented.

Proceedings ArticleDOI
11 Aug 2002
TL;DR: A new method to generate feature vectors to be used in symbol recognition and a new exploitation of the Radon transform to generate relevant-features is proposed.
Abstract: We introduce a new method to generate feature vectors to be used in symbol recognition. We propose a new exploitation of the Radon transform to generate relevant-features. The Radon transform is essentially a transformation of an image into a transform plane (/spl rho/,/spl theta/) represented by an accumulator in the discrete case. From the accumulator array we extract a signature (R-signature) which provides global information of a binary shape whatever its type and its form. The signature allows to keep fundamental geometrical transformation like scale, translation and rotation.

Book
01 Aug 2002
TL;DR: The z Transform and Its Properties are compared to Solving Linear Differential Equation and Eigenvalues in Digital Signal Processing, which is a very simple and straightforward way to model the dynamic response of a discrete-time system.
Abstract: Preface. 1. Introduction to Linear Systems. 1.1 Continuous and Discrete Linear Systems and Signals. 1.2 System Linearity and Time Invariance. 1.3 Mathematical Modeling of Systems. 1.4 System Classification. 1.5 MATLAB System Computer Analysis and Design. 1.6 Book Organization. 1.7 Chapter One Summary. 1.8 References. 1.9 Problems. 2. Introduction to Signals. 2.1 Common Signals in Linear Systems. 2.2 Signal Operations. 2.3 Signal Classification. 2.4 MATLAB Laboratory Experiment on Signals. 2.5 Chapter Two Summary. 2.6 References. 2.7 Problems. I. FREQUENCY DOMAIN TECHNIQUES. 3. Fourier Series and Fourier Transform. 3.1 Fourier Series. 3.2 Fourier Transform and Its Properties. 3.3 Fourier Transform in System Analysis. 3.4 Fourier Series in Systems Analysis. 3.5 From Fourier Transform to Laplace Transform. 3.6 Fourier Analysis MATLAB Laboratory Experiment. 3.7 Chapter Three Summary. 3.8 References. 3.9 Problems. 4. Laplace Transform. 4.1 Laplace Transform and Its Properties. 4.2 Inverse Laplace Transform. 4.3 Laplace Transform in Linear System Analysis. 4.4 Block Diagrams. 4.5 From Laplace to the z-Transform. 4.6 MATLAB Laboratory Experiment. 4.7 Chapter Four Summary. 4.8 References. 4.9 Problems. 5. The z Transform. 5.1 The z Transform and Its Properties. 5.2 Inverse of the z Transform. 5.3 The z Transform in Linear System Analysis. 5.4 Block Diagram. 5.5 Discrete-Time Frequency Spectra. 5.6 MATLAB Laboratory Experiment. 5.7 Chapter Five Summary. 5.8 References. 5.9 Problems. II. TIME DOMAIN TECHNIQUES. 6. Convolution. 6.1 Convolution of Continuous-Time Signals. 6.2 Convolution for Linear Continuous-Time Systems. 6.3 Convolution of Discrete-Time Signals. 6.4 Convolution for Linear Discrete-Time Systems. 6.5 Numerical Convolution Using MATLAB. 6.6 MATLAB Laboratory Experiments on Convolution. 6.7 Chapter Six Summary. 6.8 References. 6.9 Problems. 7. System Response in Time Domain. 7.1 Solving Linear Differential Equations. 7.2 Solving Linear Difference Equations. 7.3 Discrete-Time System Impulse Response. 7.4 Continuous-Time System Impulse Response. 7.5 Complete Continuous-Time System Response. 7.6 Complete Discrete-Time System Response. 7.7 Stability of Continuous-Time Linear Systems. 7.8 Stability of Discrete-Time Linear Systems. 7.9 MATLAB Experiment on Continuous-Time Systems. 7.10 MATLAB Experiment on Discrete-Time Systems. 7.11 Chapter Seven Summary. 7.12 References. 7.13 Problems. 8. State Space Approach. 8.1 State Space Models. 8.2 Time Response from the State Equation. 8.3 Discrete-Time Models. 8.4 System Characteristic Equation and Eigenvalues. 8.5 Cayley-Hamilton Theorem. 8.6 Linearization of Nonlinear System. 8.7 State Space MATLAB Laboratory Experiments. 8.8 Chapter Eight Summary. 8.9 References. 8.10 Problems. III. SYSTEMS IN ELECTRICAL ENGINEERING. 9. Signals in Digital Signal Processing. 9.1 Sampling Theorem. 9.2 Discrete-Time Fourier Transform (DFDT). 9.3 Double Sided z-Transform. 9.4 Discrete Fourier Transform. 9.5 Discrete-Time Fourier Series. 9.6 Correlation of Discrete-Time Signals. 9.7 FIR and IIR Filters. 9.8 Laboratory Experiment on Digital Signal Processing. 9.9 Chapter Nine Summary. 9.10 References. 9.11 Problems. 10. Signals in Communication Systems. 10.1 Signal Transmission in Communications. 10.2 Signal Correlation, Energy and Power Spectra. 10.3 Hilbert Transform. 10.4 Ideal Filter. 10.5 Modulation and Demodulation. 10.6 Digital Communication System. 10.7 Communication Systems Laboratory Experiment. 10.8 Chapter Ten Summary. 10.9 References. 10.10 Problems. 11. Linear Electric Circuits. 11.1 Basic Relations. 11.2 First-Order Linear Electrical Circuits. 11.3 Second-Order Linear Electrical Circuits. 11.4 Higher-Order Linear Electrical Circuits. 11.5 Chapter Eleven Summary. 11.6 References. 11.7 MATLAB Laboratory Experiment. 11.8 Problems. 12. Linear Controls Systems. 12.1 The Essence of Feedback. 12.2 Transient Response of Second-Order Systems. 12.3 Feedback System Steady State Errors. 12.4 Feedback System Frequency Characteristics. 12.5 Bode Diagrams. 12.6 Common Dynamic Controllers: PD, PI, PID. 12.7 Laboratory Experiment on Control Systems. 12.8 Chapter Twelve Summary. 12.9 References. 12.10 Problems. Appendices. A. Linear Algebra. B. Some Results from Calculus. C. Introduction to MATLAB. D. Introduction to SIMULINK. Index.

Journal ArticleDOI
TL;DR: Raw amplitude and time-of-flight patterns acquired from a real sonar system are processed, demonstrating reduced error in both recognition and position estimation of objects.

Journal ArticleDOI
Ahmed I. Zayed1
TL;DR: A new class of fractional integral transforms is introduced that includes the fractional Fourier and Hankel transforms and the fractionsal integration and differentiation operators as special cases.
Abstract: The paper presents a systematic and unified approach to fractional integral transforms. We introduce a new class of fractional integral transforms that includes the fractional Fourier and Hankel transforms and the fractional integration and differentiation operators as special cases. These fractional transforms may also be viewed as angular transforms, indexed by an angular parameter /spl alpha/, since their kernels are obtained by taking the limits of analytic functions in the unit disc along a radius making an angle /spl alpha/ with the x-axis.

01 Jan 2002
TL;DR: In this paper, some of the recent work on he “separation of variables” approach to computing a Fourier transform on an arbitrary finite group is surveyed, a natural generalization of the Cooley–Tukey algorithm.
Abstract: In 1965 J. Cooley and J. Tukey published an article detailing an efficient algorithm to compute the Discrete Fourier Transform, necessary for processing the newly available reams of digital time series produced by recently invented analog-to-digital converters. Since then, the Cooley– Tukey Fast Fourier Transform and its variants has been a staple of digital signal processing. Among the many casts of the algorithm, a natural one is as an efficient algorithm for computing the Fourier expansion of a function on a finite abelian group. In this paper we survey some of our recent work on he “separation of variables” approach to computing a Fourier transform on an arbitrary finite group. This is a natural generalization of the Cooley–Tukey algorithm. In addition we touch on extensions of this idea to compact and noncompact groups. Pure and Applied Mathematics: Two Sides of a Coin The Bulletin of the AMS for November 1979 had a paper by L. Auslander and R. Tolimieri [3] with the delightful title “Is computing with the Finite Fourier Transform pure or applied mathematics?” This rhetorical question was answered by showing that in fact, the finite Fourier transform, and the family of efficient algorithms used to compute it, the Fast Fourier Transform (FFT), a pillar of the world of digital signal processing, were of interest to both pure and applied mathematicians. Mathematics Subject Classification: 20C15; Secondary 65T10.

Journal ArticleDOI
TL;DR: The concept of higher order short time Fourier transform phase derivatives is introduced, which may be used to estimate signal trajectories instantaneously in both time and frequency and to determine convergence of the remapped time–frequency surface.

Patent
18 Oct 2002
TL;DR: In this article, a method for coding in frequency, module and phase a digital representation, in the space field, of a ring-shaped element, including the steps of: applying to any point of the element a polar conversion at constant angle, whereby the element is unfolded in rectangular form; transferring, to the frequency field, any points of the converted rectangular shape by means of a Fourier transform; filtering the discrete data resulting from the transfer by at least one real, bidimensional, band-pass filter, oriented along the phase axis; applying a Hilbert transform to the filtering results
Abstract: A method for coding in frequency, module and phase a digital representation, in the space field, of a ring-shaped element, including the steps of: applying to any point of the element a polar conversion at constant angle, whereby the element is unfolded in rectangular form; transferring, to the frequency field, any point of the converted rectangular shape by means of a Fourier transform; filtering the discrete data resulting from the transfer by means of at least one real, bidimensional, band-pass filter, oriented along the phase axis; applying a Hilbert transform to the filtering results; applying an inverse Fourier transform to the results of the Hilbert transform; and extracting phase and module information in the space field.

Journal ArticleDOI
TL;DR: In this paper, the Fourier transform is considered as a Henstock/Kurzweil integral and sufficient conditions for its existence are given for continuous Fourier transformation. But the Riemann-Lebesgue lemma fails.
Abstract: The Fourier transform is considered as a Henstock/Kurzweil integral. Sufficient conditions are given for the existence of the Fourier transform and necessary and sufficient conditions are given for it to be continuous. The Riemann-Lebesgue lemma fails: Henstock/Kurzweil Fourier transforms can have arbitrarily large point-wise growth. Convolution and inversion theorems are established. An appendix gives sufficient conditions for interchanging repeated Henstock/Kurzweil integrals and gives an estimate on the integral of a product. AMS (MOS) subject classification: 42A38, 26A39

Journal ArticleDOI
TL;DR: In this article, an implementation of the quantum fast Fourier transform algorithm in an entangled system of multilevel atoms is presented, where wave-packet control of the internal states of the ions in the linear ion-trap scheme for quantum computing is used.
Abstract: We propose an implementation of the quantum fast Fourier transform algorithm in an entangled system of multilevel atoms. The Fourier transform occurs naturally in the unitary time evolution of energy eigenstates and is used to define an alternative wave-packet basis for quantum information in the atom. A change of basis from energy levels to wave packets amounts to a discrete quantum Fourier transform within each atom. The algorithm then reduces to a series of conditional phase transforms between two entangled atoms in mixed energy and wave-packet bases. We show how to implement such transforms using wave-packet control of the internal states of the ions in the linear ion-trap scheme for quantum computing.

Journal ArticleDOI
TL;DR: WAVEWAT is a new processing algorithm to suppress the on-resonance water signal in NMR spectra based on a multiresolution analysis of the free induction decay using a dyadic discrete wavelet transform (DWT).