Showing papers on "Fractional Fourier transform published in 2003"
TL;DR: A new technique based on a random shifting, or jigsaw, algorithm is proposed, which does not require the use of phase keys for decrypting data and shows comparable or superior robustness to blind decryption.
Abstract: A number of methods have recently been proposed in the literature for the encryption of two-dimensional information by use of optical systems based on the fractional Fourier transform. Typically, these methods require random phase screen keys for decrypting the data, which must be stored at the receiver and must be carefully aligned with the received encrypted data. A new technique based on a random shifting, or jigsaw, algorithm is proposed. This method does not require the use of phase keys. The image is encrypted by juxtaposition of sections of the image in fractional Fourier domains. The new method has been compared with existing methods and shows comparable or superior robustness to blind decryption. Optical implementation is discussed, and the sensitivity of the various encryption keys to blind decryption is examined.
434 citations
TL;DR: A simple implementation of plane wave method for modeling photonic crystals with arbitrary shaped 'atoms' shows that using analytical Fourier transform when available can improve accuracy and avoid the grid resolution iteration.
Abstract: A simple implementation of plane wave method is presented for modeling photonic crystals with arbitrary shaped ‘atoms’ The Fourier transform for a single ‘atom’ is first calculated either by analytical Fourier transform or numerical FFT, then the shift property is used to obtain the Fourier transform for any arbitrary supercell consisting of a finite number of ‘atoms’ To ensure accurate results, generally, two iterating processes including the plane wave iteration and grid resolution iteration must converge Analysis shows that using analytical Fourier transform when available can improve accuracy and avoid the grid resolution iteration It converges to the accurate results quickly using a small number of plane waves Coordinate conversion is used to treat non-orthogonal unit cell with non-regular ‘atom’ and then is treated by standard numerical FFT MATLAB source code for the implementation requires about less than 150 statements, and is freely available at http://wwwlionsoduedu/~sguox002
251 citations
Book•
01 Jan 2003
TL;DR: This chapter discusses Signals: Analog, Discrete, and Digital, which is concerned with systems defined by Difference Equations, and its applications, including LTI Systems, Impulse Response, and Convolution.
Abstract: Preface. Acknowledgments. 1 Signals: Analog, Discrete, and Digital. 1.1 Introduction to Signals. 1.1.1 Basic Concepts. 1.1.2 Time-Domain Description of Signals. 1.1.3 Analysis in the Time-Frequency Plane. 1.1.4 Other Domains: Frequency and Scale. 1.2 Analog Signals. 1.2.1 Definitions and Notation. 1.2.2 Examples. 1.2.3 Special Analog Signals. 1.3 Discrete Signals. 1.3.1 Definitions and Notation. 1.3.2 Examples. 1.3.3 Special Discrete Signals. 1.4 Sampling and Interpolation. 1.4.1 Introduction. 1.4.2 Sampling Sinusoidal Signals. 1.4.3 Interpolation. 1.4.4 Cubic Splines. 1.5 Periodic Signals. 1.5.1 Fundamental Period and Frequency. 1.5.2 Discrete Signal Frequency. 1.5.3 Frequency Domain. 1.5.4 Time and Frequency Combined. 1.6 Special Signal Classes. 1.6.1 Basic Classes. 1.6.2 Summable and Integrable Signals. 1.6.3 Finite Energy Signals. 1.6.4 Scale Description. 1.6.5 Scale and Structure. 1.7 Signals and Complex Numbers. 1.7.1 Introduction. 1.7.2 Analytic Functions. 1.7.3 Complex Integration. 1.8 Random Signals and Noise. 1.8.1 Probability Theory. 1.8.2 Random Variables. 1.8.3 Random Signals. 1.9 Summary. 1.9.1 Historical Notes. 1.9.2 Resources. 1.9.3 Looking Forward. 1.9.4 Guide to Problems. References. Problems. 2 Discrete Systems and Signal Spaces. 2.1 Operations on Signals. 2.1.1 Operations on Signals and Discrete Systems. 2.1.2 Operations on Systems. 2.1.3 Types of Systems. 2.2 Linear Systems. 2.2.1 Properties. 2.2.2 Decomposition. 2.3 Translation Invariant Systems. 2.4 Convolutional Systems. 2.4.1 Linear, Translation-Invariant Systems. 2.4.2 Systems Defined by Difference Equations. 2.4.3 Convolution Properties. 2.4.4 Application: Echo Cancellation in Digital Telephony. 2.5 The l p Signal Spaces. 2.5.1 l p Signals. 2.5.2 Stable Systems. 2.5.3 Toward Abstract Signal Spaces. 2.5.4 Normed Spaces. 2.5.5 Banach Spaces. 2.6 Inner Product Spaces. 2.6.1 Definitions and Examples. 2.6.2 Norm and Metric. 2.6.3 Orthogonality. 2.7 Hilbert Spaces. 2.7.1 Definitions and Examples. 2.7.2 Decomposition and Direct Sums. 2.7.3 Orthonormal Bases. 2.8 Summary. References. Problems. 3 Analog Systems and Signal Spaces. 3.1 Analog Systems. 3.1.1 Operations on Analog Signals. 3.1.2 Extensions to the Analog World. 3.1.3 Cross-Correlation, Autocorrelation, and Convolution. 3.1.4 Miscellaneous Operations. 3.2 Convolution and Analog LTI Systems. 3.2.1 Linearity and Translation-Invariance. 3.2.2 LTI Systems, Impulse Response, and Convolution. 3.2.3 Convolution Properties. 3.2.4 Dirac Delta Properties. 3.2.5 Splines. 3.3 Analog Signal Spaces. 3.3.1 L p Spaces. 3.3.2 Inner Product and Hilbert Spaces. 3.3.3 Orthonormal Bases. 3.3.4 Frames. 3.4 Modern Integration Theory. 3.4.1 Measure Theory. 3.4.2 Lebesgue Integration. 3.5 Distributions. 3.5.1 From Function to Functional. 3.5.2 From Functional to Distribution. 3.5.3 The Dirac Delta. 3.5.4 Distributions and Convolution. 3.5.5 Distributions as a Limit of a Sequence. 3.6 Summary. 3.6.1 Historical Notes. 3.6.2 Looking Forward. 3.6.3 Guide to Problems. References. Problems. 4 Time-Domain Signal Analysis. 4.1 Segmentation. 4.1.1 Basic Concepts. 4.1.2 Examples. 4.1.3 Classification. 4.1.4 Region Merging and Splitting. 4.2 Thresholding. 4.2.1 Global Methods. 4.2.2 Histograms. 4.2.3 Optimal Thresholding. 4.2.4 Local Thresholding. 4.3 Texture. 4.3.1 Statistical Measures. 4.3.2 Spectral Methods. 4.3.3 Structural Approaches. 4.4 Filtering and Enhancement. 4.4.1 Convolutional Smoothing. 4.4.2 Optimal Filtering. 4.4.3 Nonlinear Filters. 4.5 Edge Detection. 4.5.1 Edge Detection on a Simple Step Edge. 4.5.2 Signal Derivatives and Edges. 4.5.3 Conditions for Optimality. 4.5.4 Retrospective. 4.6 Pattern Detection. 4.6.1 Signal Correlation. 4.6.2 Structural Pattern Recognition. 4.6.3 Statistical Pattern Recognition. 4.7 Scale Space. 4.7.1 Signal Shape, Concavity, and Scale. 4.7.2 Gaussian Smoothing. 4.8 Summary. References. Problems. 5 Fourier Transforms of Analog Signals. 5.1 Fourier Series. 5.1.1 Exponential Fourier Series. 5.1.2 Fourier Series Convergence. 5.1.3 Trigonometric Fourier Series. 5.2 Fourier Transform. 5.2.1 Motivation and Definition. 5.2.2 Inverse Fourier Transform. 5.2.3 Properties. 5.2.4 Symmetry Properties. 5.3 Extension to L 2 (R). 5.3.1 Fourier Transforms in L 1 (R) &cap L 2 (R). 5.3.2 Definition. 5.3.3 Isometry. 5.4 Summary. 5.4.1 Historical Notes. 5.4.2 Looking Forward. References. Problems. 6 Generalized Fourier Transforms of Analog Signals. 6.1 Distribution Theory and Fourier Transforms. 6.1.1 Examples. 6.1.2 The Generalized Inverse Fourier Transform. 6.1.3 Generalized Transform Properties. 6.2 Generalized Functions and Fourier Series Coefficients. 6.2.1 Dirac Comb: A Fourier Series Expansion. 6.2.2 Evaluating the Fourier Coefficients: Examples. 6.3 Linear Systems in the Frequency Domain. 6.3.1 Convolution Theorem. 6.3.2 Modulation Theorem. 6.4 Introduction to Filters. 6.4.1 Ideal Low-pass Filter. 6.4.2 Ideal High-pass Filter. 6.4.3 Ideal Bandpass Filter. 6.5 Modulation. 6.5.1 Frequency Translation and Amplitude Modulation. 6.5.2 Baseband Signal Recovery. 6.5.3 Angle Modulation. 6.6 Summary. References. Problems. 7 Discrete Fourier Transforms. 7.1 Discrete Fourier Transform. 7.1.1 Introduction. 7.1.2 The DFT's Analog Frequency-Domain Roots. 7.1.3 Properties. 7.1.4 Fast Fourier Transform. 7.2 Discrete-Time Fourier Transform. 7.2.1 Introduction. 7.2.2 Properties. 7.2.3 LTI Systems and the DTFT. 7.3 The Sampling Theorem. 7.3.1 Band-Limited Signals. 7.3.2 Recovering Analog Signals from Their Samples. 7.3.3 Reconstruction. 7.3.4 Uncertainty Principle. 7.4 Summary. References. Problems. 8 The z-Transform. 8.1 Conceptual Foundations. 8.1.1 Definition and Basic Examples. 8.1.2 Existence. 8.1.3 Properties. 8.2 Inversion Methods. 8.2.1 Contour Integration. 8.2.2 Direct Laurent Series Computation. 8.2.3 Properties and z-Transform Table Lookup. 8.2.4 Application: Systems Governed by Difference Equations. 8.3 Related Transforms. 8.3.1 Chirp z-Transform. 8.3.2 Zak Transform. 8.4 Summary. 8.4.1 Historical Notes. 8.4.2 Guide to Problems. References. Problems. 9 Frequency-Domain Signal Analysis. 9.1 Narrowband Signal Analysis. 9.1.1 Single Oscillatory Component: Sinusoidal Signals. 9.1.2 Application: Digital Telephony DTMF. 9.1.3 Filter Frequency Response. 9.1.4 Delay. 9.2 Frequency and Phase Estimation. 9.2.1 Windowing. 9.2.2 Windowing Methods. 9.2.3 Power Spectrum Estimation. 9.2.4 Application: Interferometry. 9.3 Discrete filter design and implementation. 9.3.1 Ideal Filters. 9.3.2 Design Using Window Functions. 9.3.3 Approximation. 9.3.4 Z-Transform Design Techniques. 9.3.5 Low-Pass Filter Design. 9.3.6 Frequency Transformations. 9.3.7 Linear Phase. 9.4 Wideband Signal Analysis. 9.4.1 Chirp Detection. 9.4.2 Speech Analysis. 9.4.3 Problematic Examples. 9.5 Analog Filters. 9.5.1 Introduction. 9.5.2 Basic Low-Pass Filters. 9.5.3 Butterworth. 9.5.4 Chebyshev. 9.5.5 Inverse Chebyshev. 9.5.6 Elliptic Filters. 9.5.7 Application: Optimal Filters. 9.6 Specialized Frequency-Domain Techniques. 9.6.1 Chirp-z Transform Application. 9.6.2 Hilbert Transform. 9.6.3 Perfect Reconstruction Filter Banks. 9.7 Summary. References. Problems. 10 Time-Frequency Signal Transforms. 10.1 Gabor Transforms. 10.1.1 Introduction. 10.1.2 Interpretations. 10.1.3 Gabor Elementary Functions. 10.1.4 Inversion. 10.1.5 Applications. 10.1.6 Properties. 10.2 Short-Time Fourier Transforms. 10.2.1 Window Functions. 10.2.2 Transforming with a General Window. 10.2.3 Properties. 10.2.4 Time-Frequency Localization. 10.3 Discretization. 10.3.1 Transforming Discrete Signals. 10.3.2 Sampling the Short-Time Fourier Transform. 10.3.3 Extracting Signal Structure. 10.3.4 A Fundamental Limitation. 10.3.5 Frames of Windowed Fourier Atoms. 10.3.6 Status of Gabor's Problem. 10.4 Quadratic Time-Frequency Transforms. 10.4.1 Spectrogram. 10.4.2 Wigner-Ville Distribution. 10.4.3 Ambiguity Function. 10.4.4 Cross-Term Problems. 10.4.5 Kernel Construction Method. 10.5 The Balian-Low Theorem. 10.5.1 Orthonormal Basis Decomposition. 10.5.2 Frame Decomposition. 10.5.3 Avoiding the Balian-Low Trap. 10.6 Summary. 10.6.1 Historical Notes. 10.6.2 Resources. 10.6.3 Looking Forward. References. Problems. 11 Time-Scale Signal Transforms. 11.1 Signal Scale. 11.2 Continuous Wavelet Transforms. 11.2.1 An Unlikely Discovery. 11.2.2 Basic Theory. 11.2.3 Examples. 11.3 Frames. 11.3.1 Discretization. 11.3.2 Conditions on Wavelet Frames. 11.3.3 Constructing Wavelet Frames. 11.3.4 Better Localization. 11.4 Multiresolution Analysis and Orthogonal Wavelets. 11.4.1 Multiresolution Analysis. 11.4.2 Scaling Function. 11.4.3 Discrete Low-Pass Filter. 11.4.4 Orthonormal Wavelet. 11.5 Summary. References. Problems. 12 Mixed-Domain Signal Analysis. 12.1 Wavelet Methods for Signal Structure. 12.1.1 Discrete Wavelet Transform. 12.1.2 Wavelet Pyramid Decomposition. 12.1.3 Application: Multiresolution Shape Recognition. 12.2 Mixed-Domain Signal Processing. 12.2.1 Filtering Methods. 12.2.2 Enhancement Techniques. 12.3 Biophysical Applications. 12.3.1 David Marr's Program. 12.3.2 Psychophysics. 12.4 Discovering Signal Structure. 12.4.1 Edge Detection. 12.4.2 Local Frequency Detection. 12.4.3 Texture Analysis. 12.5 Pattern Recognition Networks. 12.5.1 Coarse-to-Fine Methods. 12.5.2 Pattern Recognition Networks. 12.5.3 Neural Networks. 12.5.4 Application: Process Control. 12.6 Signal Modeling and Matching. 12.6.1 Hidden Markov Models. 12.6.2 Matching Pursuit. 12.6.3 Applications. 12.7 Afterword. References. Problems. Index.
237 citations
TL;DR: By extending the time-bandwidth product concept to fractional Fourier domains, a generalized time- bandwidth product (GTBP) is defined and it is shown that GTBP provides a rotation independent measure of compactness.
Abstract: Shift and rotation invariance properties of linear time-frequency representations are investigated. It is shown that among all linear time-frequency representations, only the short-time Fourier transform (STFT) family with the Hermite-Gaussian kernels satisfies both the shift invariance and rotation invariance properties that are satisfied by the Wigner distribution (WD). By extending the time-bandwidth product (TBP) concept to fractional Fourier domains, a generalized time-bandwidth product (GTBP) is defined. For mono-component signals, it is shown that GTBP provides a rotation independent measure of compactness. Similar to the TBP optimal STFT, the GTBP optimal STFT that causes the least amount of increase in the GTBP of the signal is obtained. Finally, a linear canonical decomposition of the obtained GTBP optimal STFT analysis is presented to identify its relation to the rotationally invariant STFT.
219 citations
TL;DR: Two short-time FrFT variants which are suited to the analysis of multicomponent and nonlinear chirp signals are developed and comparative variance measures based on the Gaussian function are given and are shown to be consistent with the uncertainty principle in fractional domains.
Abstract: The fractional Fourier transform (FrFT) provides a valuable tool for the analysis of linear chirp signals. This paper develops two short-time FrFT variants which are suited to the analysis of multicomponent and nonlinear chirp signals. Outputs have similar properties to the short-time Fourier transform (STFT) but show improved time-frequency resolution. The FrFT is a parameterized transform with parameter, a, related to chirp rate. The two short-time implementations differ in how the value of a is chosen. In the first, a global optimization procedure selects one value of a with reference to the entire signal. In the second, a values are selected independently for each windowed section. Comparative variance measures based on the Gaussian function are given and are shown to be consistent with the uncertainty principle in fractional domains. For appropriately chosen FrFT orders, the derived fractional domain uncertainty relationship is minimized for Gaussian windowed linear chirp signals. The two short-time FrFT algorithms have complementary strengths demonstrated by time-frequency representations for a multicomponent bat chirp, a highly nonlinear quadratic chirp, and an output pulse from a finite-difference sonar model with dispersive change. These representations illustrate the improvements obtained in using FrFT based algorithms compared to the STFT.
188 citations
TL;DR: In this article, the authors extend the uncertainty principle due to Beurling into a characterization of Hermite functions, and obtain similar results for the windowed Fourier transform (up to elementary changes of functions, as the radar ambiguity function or the Wigner transform).
Abstract: We extend an uncertainty principle due to Beurling into a characterization of Hermite functions. More precisely, all functions $f$ on $\mathbb{R}^d$ which may be written as $P(x)\exp (-\langle Ax, x\rangle)$, with $A$ a real symmetric definite positive matrix, are characterized by integrability conditions on the product $f(x) \widehat{f}(y)$. We then obtain similar results for the windowed Fourier transform (also known, up to elementary changes of functions, as the radar ambiguity function or the Wigner transform). We complete the paper with a sharp version of Heisenberg's inequality for this transform.
183 citations
TL;DR: A measure of the strength/robustness of the level of encryption of the various techniques is proposed and a comparison is carried out between the methods.
141 citations
TL;DR: It is hoped that this implementation and fixed-point error analysis will lead to a better understanding of the issues involved in finite register length implementation of the discrete fractional Fourier transform and will help the signal processing community make better use of the transform.
128 citations
TL;DR: A fully phase encryption system, using fractional Fourier transform to encrypt and decrypt a 2-D phase image obtained from an amplitude image, and experimental results in support of the proposed idea are presented.
Abstract: We implement a fully phase encryption system, using fractional Fourier transform to encrypt and decrypt a 2-D phase image obtained from an amplitude image. The encrypted image is holographically recorded in a barium titanate crystal and is then decrypted by generating through phase conjugation, a conjugate of the encrypted image. The decrypted phase image is converted into an amplitude image by the phase contrast technique using an electrically addressed spatial light modulator. Experimental results in support of the proposed idea are presented.
117 citations
TL;DR: In this article, the authors proposed radial harmonic Fourier moments, which are shifting, scaling, rotation, and intensity invariant compared with Chebyshev-Fourier moments.
Abstract: We propose radial harmonic Fourier moments, which are shifting, scaling, rotation, and intensity invariant Compared with Chebyshev–Fourier moments, the new moments have superior performance near the origin and better ability to describe small images in terms of image-reconstruction errors and noise sensitivity A multidistortion-invariant pattern-recognition experiment was performed with radial harmonic Fourier moments
108 citations
01 Jan 2003
TL;DR: In this paper, the authors discuss the controversial claim that any continuous periodic signal could be represented by the sum of properly chosen sinusoidal waves, and discuss the use of the Fourier transform for digital signal processing.
Abstract: Publisher Summary This chapter discusses the controversial claim that any continuous periodic signal could be represented by the sum of properly chosen sinusoidal waves. Fourier analysis forms the basis for much of digital signal processing. Simply stated, the Fourier transform (there are actually several members of this family) allows a time domain signal to be converted into its equivalent representation in the frequency domain. Conversely, if the frequency response of a signal is known, the inverse Fourier transform allows the corresponding time domain signal to be determined. In addition to frequency analysis, these transforms are useful in filter design, because the frequency response of a filter can be obtained by taking the Fourier transform of its impulse response. Conversely, if the frequency response is specified, the required impulse response can be obtained by taking the inverse Fourier transform of the frequency response. Digital filters can be constructed based on their impulse response, because the coefficients of an FIR filter and its impulse response are identical.
Book•
04 Dec 2003
TL;DR: In this paper, the nonparametric Radon transform and Fourier transform on algebraic varieties have been studied in the context of geometry and analysis, where the Fourier transforms have been applied to algebraic manifolds.
Abstract: PREFACE CHAPTERS I-X I. Introduction I.1 Functions, Geometry and Spaces I.2 Parametric Radon transform I.3 Geometry of the nonparametric Radon transform I.4 Parametrization problems I.5 Differential equations I.6 Lie groups I.7 Fourier transform on varieties: The projection slice theorem and the Poisson summation Formula I.8 Tensor products and direct integrals II. The nonparametric Radon transform II.1 Radon transform and Fourier transform II.2 Tensor products and their topology II.3 Support conditions III. Harmonic functions in Rn III.1 Algebraic theory III.2 Analytic theory III.3 Fourier series expansions on spheres III.4 Fourier expansions on hyperbolas III.5 Deformation theory IV. Harmonic functions and Radon transform on algebraic varieties IV.1 Algebraic theory and finite Cauchy problem IV.2 The compact Watergate problem IV.3 The noncompact Watergate problem V. The nonlinear Radon and Fourier transforms V.1 Nonlinear Radon transform V.2 Nonconvex support and regularity V.3 Wave front set V.4 Microglobal analysis VI. The parametric Radon transform VI.1 The John and invariance equations VI.2 Characterization by John equations VI.3 Non-Fourier analysis approach VI.4 Some other parametric linear Radon transforms VII. Radon transform on groups VII.1 Affine and projection methods VII.2 The nilpotent (horocyclic) Radon transform on G/K VIII. Radon transform as the interrelation of geometry and analysis VIII.1 Integral geometry and differential equations VIII.2 The Poisson summation formula and exotic intertwining VIII.3 The Euler-MacLaurin summation formula IX. Extension of solutions of differential equations IX.1 Formulation of the problem IX.2 Hartogs-Lewy extension IX.3 Wave front sets and the Caucy problem X. Periods of Eisenstein and Poincare series X.1 The Lorentz group, Minowski geometry and a nonlinear projection-slice theorem X.2 Spreads and cylindrical coordinates in Minowski geometry X.3 Eisenstein series and their periods X.4 Poincareseries and their periods X.5 Hyperbolic Eisenstein and Poincare series X.6 The four dimensional representation X.7 Higher dimensional groups BIBILIOGRAPHY OF CHAPTERS I-X XI. Some problems of integral geometry arising in tomography XI.1 Introduction XI.2 X-ray tomography XI.3 Attenuated and exponential Radon transforms XI.4 Hyperbolic integral geometry and electrical impedance tomography INDEX
TL;DR: This paper presents an image encryption technique, which is based on a recently proposed method of phase retrieval using the FRT, and outlines the implementation of the algorithm and examines the sensitivities of the various encryption keys.
TL;DR: The eigenfunctions and the eigenvalues of the offset FT, FRFT, and LCT are derived and can be used to analyze the self-imaging phenomena of the optical system with free spaces and the media with the transfer function exp[j(h2x2 + h1x + h0)].
Abstract: The offset Fourier transform (offset FT), offset fractional Fourier transform (offset FRFT), and offset linear canonical transform (offset LCT) are the space-shifted and frequency-modulated versions of the original transforms. They are more general and flexible than the original ones. We derive the eigenfunctions and the eigenvalues of the offset FT, FRFT, and LCT. We can use their eigenfunctions to analyze the self-imaging phenomena of the optical system with free spaces and the media with the transfer function exp[j(h2x2+h1x+h0)] (such as lenses and shifted lenses). Their eigenfunctions are also useful for resonance phenomena analysis, fractal theory development, and phase retrieval.
TL;DR: In this article, the matching-pursuit algorithm is implemented to develop an extension of the split-operator Fourier transform method to a nonorthogonal, nonuniform and dynamically adaptive coherent-state representation.
Abstract: The matching-pursuit algorithm is implemented to develop an extension of the split-operator Fourier transform method to a nonorthogonal, nonuniform and dynamically adaptive coherent-state representation. The accuracy and efficiency of the computational approach are demonstrated in simulations of deep tunneling and long time dynamics by comparing our simulation results with the corresponding benchmark calculations.
TL;DR: In this paper, the generalized S transform (GST) is generalized with two steps, and two kinds of new transforms are obtained, which are called generalized s transform (gST) and gST2.
Abstract: S transform (ST) proposed by Stockwell et al. is the unique transform that provides frequency-dependent resolution while maintaining a direct relationship with the Fourier spectrum. This feature is very important for applications. However, the ST can't work well for seismic data analysis since its basic wavelet is not appropriate. In this paper, the ST is generalized with two steps, and two kinds of new transforms are obtained, which are called generalized S transform (GST). First, the basic wavelet in ST is replaced by a modulated harmonic wave with four undetermined coefficients, and then a new transform and its inverse are given, called GST1. Second, taking a linear combination of the basic wavelets in step 1 as a new basic wavelet, called GST2, and its inverse is constructed. To compare ST with GST, the ST and GST method are used to analyze several typical models of thin beds, respectively. The results show that the resolution of GST is better than that of ST. The GST method can determine accurately the location of interfaces of acoustic impedance in thin interbeds of thickness being only an eighth wavelength, while ST method can't. In this study, the effectiveness of GST method is also verified by processing results of real data.
TL;DR: A new signal-adaptive joint time-frequency distribution for the analysis of nonstationary signals is proposed, based on a fractional-Fourier-domain realization of the weighted Wigner distribution producing auto-terms close to the ones in the WignER distribution itself, but with reduced cross-terms.
TL;DR: The 3D discrete definition of the Radon transform is shown to be geometrically faithful as the planes used for summation exhibit no wraparound effects and there exists a special set of planes in the 3D case for which the transform is rapidly computable and invertible.
TL;DR: It is shown that interference between convergent phase factor and the kernel of the different numerical Fresnel transformation may impose serious restrictions on the range of distances, where diffraction patterns can be accurately calculated.
TL;DR: A notion of ridgelet transform for arrays of digital data in which the analysis operator uses true ridge functions, as does the synthesis operator, and what is view as the key issue: the summability of the kernel underlying the constructed frame is investigated.
Abstract: We study a notion of ridgelet transform for arrays of digital data in which the analysis operator uses true ridge functions, as does the synthesis operator. There are fast algorithms for analysis, for synthesis, and for partial reconstruction. Associated with this is a transform which is a digital analog of the orthonormal ridgelet transform (but not orthonormal for finite n). In either approach, we get an overcomplete frame; the result of ridgelet transforming an n × n array is a 2n × 2n array. The analysis operator is invertible on its range; the appropriately preconditioned operator has a tightly controlled spread of singular values. There is a near-parseval relationship. Our construction exploits the recent development by Averbuch et al. (2001) of the Fast Slant Stack, a Radon transform for digital image data; it may be viewed as following a Fast Slant Stack with fast 2-d wavelet transform. A consequence of this construction is that it offers discrete objects (discrete ridgelets, discrete Radon transform, discrete Pseudopolar Fourier domain) which obey inter-relationships paralleling those in the continuum ridgelet theory (between ridgelets, Radon transform, and polar Fourier domain). We make comparisons with other notions of ridgelet transform, and we investigate what we view as the key issue: the summability of the kernel underlying the constructed frame. The sparsity observed in our current implementation is not nearly as good as the sparsity of the underlying continuum theory, so there is room for substantial progress in future implementations.
TL;DR: This work presents much briefer and more direct and transparent derivations of some sampling and series expansion relations for fractional Fourier and other transforms.
TL;DR: This method encompasses the conventional minimum mean-squared error beamforming in the frequency domain or spatial domain as special cases and is especially useful for applications involving chirp signals such as signal enhancement problems with accelerating sinusoidal sources.
Abstract: We present a new method of beamforming using the fractional Fourier transform (FrFT). This method encompasses the conventional minimum mean-squared error (MMSE) beamforming in the frequency domain or spatial domain as special cases. It is especially useful for applications involving chirp signals such as signal enhancement problems with accelerating sinusoidal sources where the Doppler effect generates chirp signals and a frequency shift and active radar problems where chirp signals are transmitted. Numerical examples demonstrate the potential advantage of the proposed method over the ordinary frequency or spatial domain beamforming for a moving source scenario.
TL;DR: The Stockwell transform (ST), recently developed for geophysics, combines features of the Fourier, Gabor and wavelet transforms; it reveals frequency variation over time or space and is a potentially effective tool to visualize, analyze, and process medical imaging data.
Abstract: The Stockwell transform (ST), recently developed for geophysics, combines features of the Fourier, Gabor and wavelet transforms; it reveals frequency variation over time or space. This valuable information is obtained by Fourier analysis of a small segment of a signal at a time. Localization of the Fourier spectrum is achieved by filtering the signal with frequency-dependent Gaussian scaling windows. This multi-scale time–frequency analysis provides information about which frequencies occur and more importantly when they occur. Furthermore, the Stockwell domain can be directly inferred from the Fourier domain and vice versa. These features make the ST a potentially effective tool to visualize,analyze, and process medical imaging data. The ST has proven useful in noise reduction and tissue texture analysis. Herein, we focus on the theory and effectiveness of the ST for medical imaging. Its effectiveness and comparison with other linear time–frequency transforms, such as the Gabor and wavelet transforms, are discussed and demonstrated using functional magnetic resonance imaging data.
TL;DR: In this article, a fractional Fourier transform (FRT) is applied to study the transformation properties of flattened Gaussian beams (FGBs) and an analytical formula is derived for the FRT of FGBs based on the definition of FRT.
Abstract: The fractional Fourier transform (FRT) is applied to study the transformation properties of flattened Gaussian beams (FGBs). An analytical formula is derived for the FRT of FGBs based on the definition of FRT. By using the derived formula, we study the intensity distribution properties of FGB in the FRT plane. The influences of the fractional order on the intensity distribution properties of FGB with different order in the fractional Fourier plane are studied in detail. The evolution of the kurtosis parameter of FGB in the FRT plane is also studied.
12 Jan 2003
TL;DR: In this article, the authors present three examples of unknown shift problems that can be solved efficiently on a quantum computer using the quantum Fourier transform and define the hidden coset problem, which generalizes the hidden shift problem and the hidden subgroup problem.
Abstract: Almost all of the most successful quantum algorithms discovered to date exploit the ability of the Fourier transform to recover subgroup structure of functions, especially periodicity. The fact that Fourier transforms can also be used to capture shift structure has received far less attention in the context of quantum computation.In this paper, we present three examples of "unknown shift" problems that can be solved efficiently on a quantum computer using the quantum Fourier transform. We also define the hidden coset problem, which generalizes the hidden shift problem and the hidden subgroup problem. This framework provides a unified way of viewing the ability of the Fourier transform to capture subgroup and shift structure.
TL;DR: This work considers the problem of fast computation of the Fourier transform over a finite field by decomposing an arbitrary polynomial into a sum of linearized polynomials.
Abstract: We consider the problem of fast computation of the Fourier transform over a finite field by decomposing an arbitrary polynomial into a sum of linearized polynomials. Examples of algorithms for the Fourier transform with complexity less than that of the best known analogs are given.
TL;DR: This paper presents an innovative way of using the two-dimensional (2-D) Fourier transform for speech enhancement and proposes a hybrid filter which effectively combines the one-dimensional Wiener filter with the 2-DWiener filter.
Abstract: This paper presents an innovative way of using the two-dimensional (2-D) Fourier transform for speech enhancement. The blocking and windowing of the speech data for the 2-D Fourier transform are explained in detail. Several techniques of filtering in the 2-D Fourier transform domain are also proposed. They include magnitude spectral subtraction, 2-D Wiener filtering as well as a hybrid filter which effectively combines the one-dimensional (1-D) Wiener filter with the 2-D Wiener filter. The proposed hybrid filter compares favorably against other techniques using an objective test.
21 Apr 2003
TL;DR: In this paper, it is shown that the fractional domains exhibiting minimum and maximum support for a given signal define the limit on joint time-frequency resolution available under the FrFT.
Abstract: The fractional Fourier transform (FrFT) provides an important extension to conventional Fourier theory for the analysis and synthesis of linear chirp signals. It is a parameterised transform which can be used to provide extremely compact representations. The representation is maximally compressed when the transform parameter, /spl alpha/, is matched to the chirp rate of the input signal. Existing proofs are extended to demonstrate that the fractional Fourier transform of the Gaussian function also has Gaussian support. Furthermore, expressions are developed which allow calculation of the spread of the signal representation for a Gaussian windowed linear chirp signal in any fractional domain. Both continuous and discrete cases are considered. The fractional domains exhibiting minimum and maximum support for a given signal define the limit on joint time-frequency resolution available under the FrFT. This is equated with a restatement of the uncertainty principle for linear chirp signals and the fractional Fourier domains. The calculated values for the fractional domain support are tested empirically through comparison with the discrete transform output for a synthetic signal with known parameters. It is shown that the same expressions are appropriate for predicting the support of the ordinary Fourier transform of a Gaussian windowed linear chirp signal.
Journal Article•
TL;DR: In this article, a generalized S transform (GST) was proposed for seismic data analysis, and two kinds of new transforms were obtained, which are called generalized S Transform (ST) and Generalized S Transform(GST), which is the unique transform that provides frequency-dependent resolution while maintaining a direct relationship with the Fourier spectrum.
Abstract: S transform (ST) proposed by Stockwell et al. is the unique transform that provides frequency-dependent resolution while maintaining a direct relationship with the Fourier spectrum. This feature is very important for applications. However, the ST can't work well for seismic data analysis since its basic wavelet is not appropriate. In this paper, the ST is generalized with two steps, and two kinds of new transforms are obtained, which are called generalized S transform (GST). First, the basic wavelet in ST is replaced by a modulated harmonic wave with four undetermined coefficients, and then a new transform and its inverse are given, called GST1. Second, taking a linear combination of the basic wavelets in step 1 as a new basic wavelet, called GST2, and its inverse is constructed. To compare ST and GST, the ST and GST methods are used to analyze several typical models of thin beds, respectively. The results show that GST has a better resolution than ST. The GST methods can detemine the location of interfaces of acoustic impedance in thin beds of thickness being only an eighth wavelength, while ST method can't. In this study, the effectiveness of GST method is also verified by real data.
TL;DR: It is shown that the proposed method works well for signal-to-noise ratios (SNRs) higher than about 3 dB and can be applied for phase retrieval in optics, where only the fractional power spectra associated with intensity distributions can be easily measured.
Abstract: Based on the definition of the instantaneous frequency (signal phase derivative) as a local moment of the Wigner distribution, we derive the relationship between the instantaneous frequency and the derivative of the squared modulus of the fractional Fourier transform (fractional Fourier transform power spectrum) with respect to the angle parameter. We show that the angular derivative of the fractional power spectrum can be found from the knowledge of two close fractional power spectra. It permits us to find the instantaneous frequency and to solve the phase retrieval problem up to a constant phase term, if only two close fractional power spectra are known. The proposed technique is noniterative and noninterferometric. The efficiency of the method is demonstrated on several examples including monocomponent, multicomponent, and noisy signals. It is shown that the proposed method works well for signal-to-noise ratios (SNRs) higher than about 3 dB. The appropriate angular difference of the fractional power spectra used for phase retrieval depends on the complexity of the signal and can usually reach several degrees. Other applications of the angular derivative of the fractional power spectra for signal analysis are discussed. The proposed technique can be applied for phase retrieval in optics, where only the fractional power spectra associated with intensity distributions can be easily measured.