Showing papers on "Fractional Fourier transform published in 2005"
Book•
11 Mar 2005
TL;DR: In this article, the authors report on the work of I. I. Schoenberg and his students in the field of algebraic geometry, which is closely related to ours and has supplemented it in certain respects.
Abstract: Introduction. The material I am reporting on here was prepared in collaboration with I. I. Hirschman. It will presently appear in book form in the Princeton Mathematical Series. I wish also to refer at once to the researches of I. J. Schoenberg and his students. Their work has been closely related to ours and has supplemented it in certain respects. Let me call attention especially to an article of Schoenberg [5, p. 199] in this Bulletin where the whole field is outlined and the historical development is traced. In view of the existence of this paper I shall t ry to avoid any parallel development here. Let me take rather a heuristic point of view and concentrate chiefly on trying to entertain you with what seems to me a fascinating subject.
430 citations
Book•
01 May 2005TL;DR: The Fourier transform and the Busemann-Petty problem have been studied extensively in the literature, see as discussed by the authors for a detailed survey of the Fourier Transform Bibliography Index.
Abstract: Introduction Basic concepts Volume and the Fourier transform Intersection bodies The Busemann-Petty problem Intersection bodies and $L_p$-spaces Extremal sections of $\ell_q$-balls Projections and the Fourier transform Bibliography Index.
353 citations
TL;DR: This work investigates the nonorthogonality of the Fourier basis on an irregularly sampled grid and proposes a technique called “antileakage Fourier transform” to overcome the spectral leakage and demonstrates the robustness and effectiveness of this technique.
Abstract: Seismic data regularization, which spatially transforms irregularly sampled acquired data to regularly sampled data, is a long-standing problem in seismic data processing. Data regularization can be implemented using Fourier theory by using a method that estimates the spatial frequency content on an irregularly sampled grid. The data can then be reconstructed on any desired grid. Difficulties arise from the nonorthogonality of the global Fourier basis functions on an irregular grid, which results in the problem of “spectral leakage”: energy from one Fourier coefficient leaks onto others. We investigate the nonorthogonality of the Fourier basis on an irregularly sampled grid and propose a technique called “antileakage Fourier transform” to overcome the spectral leakage. In the antileakage Fourier transform, we first solve for the most energetic Fourier coefficient, assuming that it causes the most severe leakage. To attenuate all aliases and the leakage of this component onto other Fourier coefficients, the data component corresponding to this most energetic Fourier coefficient is subtracted from the original input on the irregular grid. We then use this new input to solve for the next Fourier coefficient, repeating the procedure until all Fourier coefficients are estimated. This procedure is equivalent to “reorthogonalizing” the global Fourier basis on an irregularly sampled grid. We demonstrate the robustness and effectiveness of this technique with successful applications to both synthetic and real data examples.
326 citations
TL;DR: In this article, the authors present the essential ideas underlying the fast Fourier transform (NUFFT) algorithm in simple terms, and illustrate its utility with application to problems in magnetic resonance imaging and heat flow.
180 citations
TL;DR: The implementation of the semiclassical quantum Fourier transform in a system of three beryllium ion qubits confined in a segmented multizone trap incorporates the key elements of a scalable ion-trap architecture, suggesting the future capability of applying the quantum Fouriers transform to a large number of qubits as required for a useful quantum factoring algorithm.
Abstract: We report the implementation of the semiclassical quantum Fourier transform in a system of three beryllium ion qubits (two-level quantum systems) confined in a segmented multizone trap. The quantum Fourier transform is the crucial final step in Shor's algorithm, and it acts on a register of qubits to determine the periodicity of the quantum state's amplitudes. Because only probability amplitudes are required for this task, a more efficient semiclassical version can be used, for which only single-qubit operations conditioned on measurement outcomes are required. We apply the transform to several input states of different periodicities; the results enable the location of peaks corresponding to the original periods. This demonstration incorporates the key elements of a scalable ion-trap architecture, suggesting the future capability of applying the quantum Fourier transform to a large number of qubits as required for a useful quantum factoring algorithm.
171 citations
TL;DR: The Fast Linear Canonical Transform (FLCT) as mentioned in this paper is derived from the linear canonical transform (LCT) and can be used for FFT, FRT, and FST calculations.
Abstract: The linear canonical transform (LCT) describes the effect of any quadratic phase system (QPS) on an input optical wave field. Special cases of the LCT include the fractional Fourier transform (FRT), the Fourier transform (FT), and the Fresnel transform (FST) describing free-space propagation. Currently there are numerous efficient algorithms used (for purposes of numerical simulation in the area of optical signal processing) to calculate the discrete FT, FRT, and FST. All of these algorithms are based on the use of the fast Fourier transform (FFT). In this paper we develop theory for the discrete linear canonical transform (DLCT), which is to the LCT what the discrete Fourier transform (DFT) is to the FT. We then derive the fast linear canonical transform (FLCT), an NlogN algorithm for its numerical implementation by an approach similar to that used in deriving the FFT from the DFT. Our algorithm is significantly different from the FFT, is based purely on the properties of the LCT, and can be used for FFT, FRT, and FST calculations and, in the most general case, for the rapid calculation of the effect of any QPS.
167 citations
TL;DR: The Clifford Fourier transform is extended to include general elements of Clifford Algebra, called multivectors, including scalars and vectors, and the resulting convolution and derivative theorems are extensions of those for Convolution and the Fouriertransform on scalar fields.
Abstract: Image processing and computer vision have robust methods for feature extraction and the computation of derivatives of scalar fields. Furthermore, interpolation and the effects of applying a filter can be analyzed in detail and can be advantages when applying these methods to vector fields to obtain a solid theoretical basis for feature extraction. We recently introduced the Clifford convolution, which is an extension of the classical convolution on scalar fields and provides a unified notation for the convolution of scalar and vector fields. It has attractive geometric properties that allow pattern matching on vector fields. In image processing, the convolution and the Fourier transform operators are closely related by the convolution theorem and, in this paper, we extend the Fourier transform to include general elements of Clifford Algebra, called multivectors, including scalars and vectors. The resulting convolution and derivative theorems are extensions of those for convolution and the Fourier transform on scalar fields. The Clifford Fourier transform allows a frequency analysis of vector fields and the behavior of vector-valued filters. In frequency space, vectors are transformed into general multivectors of the Clifford Algebra. Many basic vector-valued patterns, such as source, sink, saddle points, and potential vortices, can be described by a few multivectors in frequency space.
157 citations
TL;DR: A new method is presented that is capable of extracting the full 2D phase distribution from a single fringe pattern and is compared with the Fourier transform and the integration methods, and an important conclusion that the phase of the optical fringe pattern is equal to thephase of its wavelet transform on the ridge of the wave let transform is theoretically clarified.
Abstract: A new method for phase retrieval of optical fringe patterns is presented This method is based on a wavelet transform and is capable of extracting the full 2D phase distribution from a single fringe pattern An important conclusion that the phase of the optical fringe pattern is equal to the phase of its wavelet transform on the ridge of the wavelet transform is theoretically clarified The method is compared with the Fourier transform and the integration methods A numerical simulation and an experimental example of phase retrieval are shown
153 citations
TL;DR: Based on the fractional wavelet transform (FWT), a novel method for the image encryption is proposed that encrypts the image by two fractional orders and a series of scaling factors and realizes the partial encryption by using this method.
106 citations
Journal Article•
TL;DR: In this article, the authors discuss discrete fractional Fourier transform (DFRFT), time-frequency distributions related to FRFT, optimal filter and beamformer in FRFT domain, filtering using window functions and other fractional transforms along with simulation results.
Abstract: The fractional Fourier transform (FRFT) is the generalization of the classical Fourier transform. It depends on a parameter ? (= a ?/2) and can be interpreted as a rotation by an angle ? in the time-frequency plane or decomposition of the signal in terms of chirps. This paper discusses discrete FRFT (DFRFT), time-frequency distributions related to FRFT, optimal filter and beamformer in FRFT domain, filtering using window functions and other fractional transforms along with simulation results.
103 citations
18 Mar 2005
TL;DR: The successive mean quantization transform reveals the organization or structure of the data and removes properties such as gain and bias and is applied in speech processing and image processing.
Abstract: This paper presents the successive mean quantization transform (SMQT). The transform reveals the organization or structure of the data and removes properties such as gain and bias. The transform is described and applied in speech processing and image processing. The SMQT is considered as an extra processing step for the mel frequency cepstral coefficients commonly used in speech recognition. In image processing the transform is applied in automatic image enhancement and dynamic range compression.
TL;DR: In this article, the authors proposed a discrete fractional random transform based on a generalization of the DFT with an intrinsic randomness, which is used for image encryption and decryption.
TL;DR: Comparing with single FRT encryption, the security using this method for optical image encryption is greatly improved due to the introduction of the pixel scrambling technique.
TL;DR: This letter defines a DFRFT based on a centered version of the DFT (CDFRFT) using eigenvectors derived from the Gru/spl uml/nbaum tridiagonal commutor that serve as excellent discrete approximations to the Hermite-Gauss functions.
Abstract: Existing versions of the discrete fractional Fourier transform (DFRFT) are based on the discrete Fourier transform (DFT). These approaches need a full basis of DFT eigenvectors that serve as discrete versions of Hermite-Gauss functions. In this letter, we define a DFRFT based on a centered version of the DFT (CDFRFT) using eigenvectors derived from the Gru/spl uml/nbaum tridiagonal commutor that serve as excellent discrete approximations to the Hermite-Gauss functions. We develop a fast and efficient way to compute the multiangle version of the CDFRFT for a discrete set of angles using the FFT algorithm. We then show that the associated chirp-frequency representation is a useful analysis tool for multicomponent chirp signals.
TL;DR: Simulated data have been used to analyze the artefacts produced by the Lagrange interpolation and the proposed non-linear sampling scheme is simple and highly robust since no parameters need to be adjusted by the user.
Abstract: Rapid acquisition of high-resolution 2D and 3D NMR spectra is essential for studying biological macromolecules. In order to minimize the experimental time, a non-linear sampling scheme is proposed for the indirect dimensions of multidimensional experiments. These data can be processed using the algorithm proposed by Dutt and Rokhlin (Appl. Comp. Harm. Anal. 1995, 2, 85–100) for fast Fourier transforms of non equispaced data. Examples of 1H−15N HSQC spectra are shown, where crowded correlation peaks can be resolved using non-linear acquisition. Simulated data have been used to analyze the artefacts produced by the Lagrange interpolation. As compared to non-linear processing methods, this algorithm is simple and highly robust since no parameters need to be adjusted by the user.
TL;DR: The angular difference function (ADF) is presented and its applicability to rotation estimation is demonstrated and it is efficiently computed using the pseudopolar Fourier transform, which computes the discrete Fouriertransform of an image on a near spherical grid.
Abstract: The estimation of large motions without prior knowledge is an important problem in image registration. In this paper, we present the angular difference function (ADF) and demonstrate its applicability to rotation estimation. The ADF of two functions is defined as the integral of their spectral difference along the radial direction. It is efficiently computed using the pseudopolar Fourier transform, which computes the discrete Fourier transform of an image on a near spherical grid. Unlike other Fourier-based registration schemes, the suggested approach does not require any interpolation. Thus, it is more accurate and significantly faster.
TL;DR: The fractional Fourier transform and its applications for phase retrieval, beam characterization, space-variant pattern recognition, adaptive filter design, encryption, watermarking, and so forth is discussed in detail.
Abstract: We review the progress achieved in optical information processing during the last decade by applying fractional linear integral transforms. The fractional Fourier transform and its applications for phase retrieval, beam characterization, space-variant pattern recognition, adaptive filter design, encryption, watermarking, and so forth is discussed in detail. A general algorithm for the fractionalization of linear cyclic integral transforms is introduced and it is shown that they can be fractionalized in an infinite number of ways. Basic properties of fractional cyclic transforms are considered. The implementation of some fractional transforms in optics, such as fractional Hankel, sine, cosine, Hartley, and Hilbert transforms, is discussed. New horizons of the application of fractional transforms for optical information processing are underlined.
TL;DR: A new method for image encryption and decryption in which the image is broken up into bit planes and each bit plane undergoes a jigsaw transform, which increases the robustness of the encryption system by several orders of magnitude.
Abstract: We propose a new method for image encryption and decryption in which the image is broken up into bit planes. Each bit plane undergoes a jigsaw transform. The transformed bit planes are combined together and then encrypted using random phase masks and fractional Fourier transforms. The different fractional parameters, the random phase codes, and the jigsaw transform index form the key to the encrypted data. This increases the robustness of the encryption system by several orders of magnitude. Different variations of the juxtaposition of the pieces of the image are also considered. These include the rotated version of the jigsaw pieces. The computational complexity of the bit-plane-based jigsaw algorithm is further improved using the third dimension (i.e., along different bit planes) for scrambling as well. The results of computer simulation are presented to verify the proposed idea and analyze the performance of the proposed techniques.
TL;DR: It is shown how the matrices associated with the effect of a LCT on the Wigner distribution function (WDF) provide an efficient method for finding the position, spatial extent, spatial frequency extent and SBP of the encrypted signal.
TL;DR: Time-frequency filtering and time-time filtering are presented, in which a filter is applied to a time-frequency distribution instead of the Fourier spectrum.
Patent•
15 Jul 2005TL;DR: In this paper, a transform coder is described that performs a time-split transform in addition to a discrete cosine type transform, selectively performed based on characteristics of media data, and a decoder performs an inverse of the described transforms.
Abstract: A transform coder is described that performs a time-split transform in addition to a discrete cosine type transform. A time-split transform is selectively performed based on characteristics of media data. Transient detection identifies a changing signal characteristic, such as a transient in media data. After encoding an input signal from a time domain to a transform domain, a time-splitting transformer selectively perform an orthogonal sum-difference transform on adjacent coefficients indicated by a changing signal characteristic location. The orthogonal sum-difference transform on adjacent coefficients results in transforming a vector of coefficients in the transform domain as if they were multiplied by an identity matrix including at least one 2×2 time-split block along a diagonal of the matrix. A decoder performs an inverse of the described transforms.
Book•
07 Jul 2005
TL;DR: In this paper, the Fourier Transform on Finite Non-Abelian Group (FFT-NAG) is used to represent the non-abelian groups and the Gibbs Derivative on NAG.
Abstract: Preface. Acknowledgments. Acronyms. 1 Signals and Their Mathematical Models. 1.1 Systems. 1.2 Signals. 1.3 Mathematical Models of Signals. References. 2 Fourier Analysis. 2.1 Representations of Groups. 2.1.1 Complete Reducibility. 2.2 Fourier Transform on Finite Groups. 2.3 Properties of the Fourier Transform. 2.4 Matrix Interpretation of the Fourier Transform on Finite Non-Abelian Groups. 2.5 Fast Fourier Transform on Finite Non-Abelian Groups. References. 3 Matrix Interpretation of the FFT. 3.1 Matrix Interpretation of FFT on Finite Non-Abelian Groups. 3.2 Illustrative Examples. 3.3 Complexity of the FFT. 3.3.1 Complexity of Calculations of the FFT. 3.3.2 Remarks on Programming Implememtation of FFT. 3.4 FFT Through Decision Diagrams. 3.4.1 Decision Diagrams. 3.4.2 FFT on Finite Non-Abelian Groups Through DDs. 3.4.3 MMTDs for the Fourier Spectrum. 3.4.4 Complexity of DDs Calculation Methods. References. 4 Optimization of Decision Diagrams. 4.1 Reduction Possibilities in Decision Diagrams. 4.2 Group-Theoretic Interpretation of DD. 4.3 Fourier Decision Diagrams. 4.3.1 Fourier Decision Trees. 4.3.2 Fourier Decision Diagrams. 4.4 Discussion of Different Decompositions. 4.4.1 Algorithm for Optimization of DDs. 4.5 Representation of Two-Variable Function Generator. 4.6 Representation of Adders by Fourier DD. 4.7 Representation of Multipliers by Fourier DD. 4.8 Complexity of NADD. 4.9 Fourier DDs with Preprocessing. 4.9.1 Matrix-valued Functions. 4.9.2 Fourier Transform for Matrix-Valued Functions. 4.10 Fourier Decision Trees with Preprocessing. 4.11 Fourier Decision Diagrams with Preprocessing. 4.12 Construction of FNAPDD. 4.13 Algorithm for Construction of FNAPDD. 4.13.1 Algorithm for Representation. 4.14 Optimization of FNAPDD. References. 5 Functional Expressions on Quaternion Groups. 5.1 Fourier Expressions on Finite Dyadic Groups. 5.1.1 Finite Dyadic Groups. 5.2 Fourier Expressions on Q2. 5.3 Arithmetic Expressions. 5.4 Arithmetic Expressions from Walsh Expansions. 5.5 Arithmetic Expressions on Q2. 5.5.1 Arithmetic Expressions and Arithmetic-Haar Expressions. 5.5.2 Arithmetic-Haar Expressions and Kronecker Expressions. 5.6 Different Polarity Polynomials Expressions. 5.6.1 Fixed-Polarity Fourier Expressions in C(Q2). 5.6.2 Fixed-Polarity Arithmetic-Haar Expressions. 5.7 Calculation of the Arithmetic-Haar Coefficients. 5.7.1 FFT-like Algorithm. 5.7.2 Calculation of Arithmetic-Haar Coefficients Through Decision Diagrams. References. 6 Gibbs Derivatives on Finite Groups. 6.1 Definition and Properties of Gibbs Derivatives on Finite Non-Abelian Groups. 6.2 Gibbs Anti-Derivative. 6.3 Partial Gibbs Derivatives. 6.4 Gibbs Differential Equations. 6.5 Matrix Interpretation of Gibbs Derivatives. 6.6 Fast Algorithms for Calculation of Gibbs Derivatives on Finite Groups. 6.6.1 Complexity of Calculation of Gibbs Derivatives. 6.7 Calculation of Gibbs Derivatives Through DDs. 6.7.1 Calculation of Partial Gibbs Derivatives. References. 7 Linear Systems on Finite Non-Abelian Groups. 7.1 Linear Shift-Invariant Systems on Groups. 7.2 Linear Shift-Invariant Systems on Finite Non-Abelian Groups. 7.3 Gibbs Derivatives and Linear Systems. 7.3.1 Discussion. References. 8 Hilbert Transform on Finite Groups. 8.1 Some Results of Fourier Analysis on Finite Non-Abelian Groups. 8.2 Hilbert Transform on Finite Non-Abelian Groups. 8.3 Hilbert Transform in Finite Fields. References. Index.
TL;DR: Simulation results have proved that the new algorithm is robust and also computationally efficient as compared with previously reported algorithms such as joint time-frequency (JTF) imaging.
Abstract: A novel adaptive inverse synthetic aperture radar (ISAR) imaging technique is proposed for targets with nonuniform motion. The proposed algorithm is referred to as the generalized range-Doppler (GRD) ISAR imaging technique and is based on the fractional Fourier transform (FRFT). By utilizing this technique, clear ISAR imaging can be achieved for nonuniformly moving targets without involvement of complex motion compensation. Simulation results have proved that the new algorithm is robust and also computationally efficient as compared with previously reported algorithms such as joint time-frequency (JTF) imaging.
TL;DR: In this article, new inversion techniques for the Lorentz Integral Transform are introduced and it is shown that they all contain a regularization scheme, which is necessary to overcome the ill-posed problem.
Abstract: The Lorentz Integral Transform approach allows microscopic calculations of electromagnetic reaction cross-sections without explicit knowledge of final-state wave functions. The necessary inversion of the transform has to be treated with great care, since it constitutes a so-called ill-posed problem. In this work new inversion techniques for the Lorentz Integral Transform are introduced. It is shown that they all contain a regularization scheme, which is necessary to overcome the ill-posed problem. In addition, it is illustrated that the new techniques have a much broader range of application than the present standard inversion method of the Lorentz Integral Transform.
TL;DR: A new class of FRFT is introduced that includes the conventional fractional Fourier transforms and the weighted-type fractional fourier transforms as special cases, and is structurally well organized.
Abstract: The paper studies the possibility of giving a general multiplicity of the fractional Fourier transform (FRFT) with the intention of combining existing finite versions of the FRFT. We introduce a new class of FRFT that includes the conventional fractional Fourier transforms (CFRFTs) and the weighted-type fractional Fourier transforms (WFRFTs) as special cases. The class is structurally well organized because these new FRFTs, which are called general multifractional Fourier transform (GMFRFTs), are related with one another by the Generalized Permutation Matrix Group (GPMG), and their kernels are related with that of CFRFTs as the finite combination by the recursion of matrix. In addition, we have computer simulations of some GMFRFTs on a rectangular function as a simple application of GMFRFTs to signal processing.
TL;DR: In this paper, the authors prove the boundedness of a general class of Fourier multipliers, in particular of the Hilbert transform, on modulation spaces, using Gabor frames and methods from time-frequency analysis.
TL;DR: A new discrete polynomial transform constructed from the rows of Pascal's triangle is introduced, and applications of the transform in digital image processing, such as bump and edge detection are discussed.
Abstract: We introduce a new discrete polynomial transform constructed from the rows of Pascal's triangle. The forward and inverse transforms are computed the same way in both the one- and two-dimensional cases, and the transform matrix can be factored into binary matrices for efficient hardware implementation. We conclude by discussing applications of the transform in digital image processing, such as bump and edge detection.
Book•
07 Nov 2005
TL;DR: In this article, the Laplace transform is used for transient analysis of basic circuits and the Fourier transform is applied to transform a circuit into an RLC circuit under dc-input-output (RLC) conditions.
Abstract: Chapter 1 Classical approach to transient analysis. Introduction. Appearance of transients in electrical circuits. Differential equations describing electrical circuits. Exponential solution of a simple differential equation. Natural and forced responses. Characteristic equation and methods of its determinations. Roots of the characteristic equation and different kinds of transient responses. First order characteristic equation. Second order characteristic equation. Independent and dependent initial conditions. Two switching laws (rules). Methods of finding independent initial conditions. Methods of finding dependent initial conditions. Generalized initial conditions. Methods of finding integration constants. Chapter 2 Transient response of basic circuits. Introduction. Five steps of solving problems in transients analysis. First order RL circuits. RL circuits under dc supply. RL circuits under ac supply. Applying a continuous flux linkage law to inductive circuits. First order RC circuits. Discharging and charging a capacitor. RC circuits under dc supply. RC circuits under ac supply. Applying a continuous charge law to capacitance circuits. The application of a unit-step forcing function. Superposition principle in transient analysis. Second order RLC circuits. RLC circuits under dc supply. RLC circuits under ac supply. Transients in RLC resonant circuits. Switching-off in RLC circuits. Chapter 3 Transients in complicated circuits and the Laplace transform. Introduction. The Laplace transform. Properties of the Laplace transform. Laplace transform of basic time functions. Initial-value and final-value theorems. Examples of finding circuit responses. Inverse transform and partial fraction expansion. Ohm and Kirchhoff's laws with the Laplace transform. Equivalent circuits with Laplace transform techniques. More examples of finding circuit responses. Using nodal analysis. Using mesh analysis. Mutually coupled circuits. Some techniques for simplifying the solution. Chapter 4 Transient analysis using the Fourier transform. Introduction. The inter-relation between the transient behavior of electrical circuits and their spectral properties. The Fourier transform. The definition of the Fourier transform. Relationship between a discreet and continuous spectra. Symmetry properties of the Fourier transform. Energy characteristics of continuous spectra. The comparison between Fourier and Laplace transforms. Some properties of the Fourier transform. Some important transform pairs. Input-impulse (delta) function. Unit-step function. Decreasing sinusoid. Saw-tooth pulse. A periodic time function. Convolution integral in the time domain and its Fourier transform. Circuit analysis with Fourier transform. Ohm and Kirchhoff's laws with the Fourier transform. Inversion of the Fourier transform using the residues of complex functions. Approximate transient analysis with the Fourier transform. Chapter 5 State variable analysis. Introduction. The concept of state variables. Order of complexity of a network. State equations and trajectory. Basic considerations in writing state equations. Fundamental cut-set and loop matrixes. 'Proper tree' method for writing state equations. A systematic method for writing the state equation based on circuit matrix representation. Complete solution of the matrix equation. The natural solution. Matrix exponential. The particular solution. Basic considerations in determining functions of a matrix. Evaluating the matrix exponential by the Laplace transform. Chapter 6 Transients in three-phase circuits. Introduction. Short circuit transients in power systems. Base quantities and per-unit conversion in three-phase circuits. Equivalent circuits and their simplification. Using the superposition principle. Short-circuiting in a simple circuit. Short-circuiting of a power transformer. Short-circuiting
TL;DR: The first two parts of this article will discuss concepts associated with the fast Fourier transform (FFT), an implementation of the DFT, and two applications are analyzed: a bat chirp and atmospheric sea-level pressure differences in the Pacific Ocean.
Abstract: The discrete Fourier transform (DFT) provides a means for transforming data sampled in the time domain to an expression of this data in the frequency domain. The inverse transform reverses the process, converting frequency data into time-domain data. Such transformations can be applied in a wide variety of fields, from geophysics to astronomy, from the analysis of sound signals to CO/sub 2/ concentrations in the atmosphere. Over the course of three articles, our goal is to provide a convenient summary that the experimental practitioner will find useful. In the first two parts of this article, we'll discuss concepts associated with the fast Fourier transform (FFT), an implementation of the DFT. In the third part, we'll analyze two applications: a bat chirp and atmospheric sea-level pressure differences in the Pacific Ocean.
Journal Article•
TL;DR: Two engineering-oriented methods of the dimensional normalization of the fractional Fourier transform makes the FRFT more practical in digital signal processing.
Abstract: The fast algorithm of the digital computation of the fractional Fourier transform(FRFT) requires the dimensional normalization, but how to do it for practical discrete signal is not settled yet For this reason, the paper presents two engineering-oriented methods of the dimensional normalization One is called as the discrete scaling transform method, another is called as the data zero-padding/interception method Furthermore, their effects on the parameter estimation of the chirp signal are studied and for discrete scaling method, a relationship before and after the normalization is developed Finally, these methods are verified by simulation examples These engineering-oriented methods of the dimensional normalization makes the FRFT more practical in digital signal processing