Showing papers on "Fractional Fourier transform published in 2006"

Research progress of the fractional Fourier transform in signal processing

Abstract: The fractional Fourier transform is a generalization of the classical Fourier transform, which is introduced from the mathematic aspect by Namias at first and has many applications in optics quickly. Whereas its potential appears to have remained largely unknown to the signal processing community until 1990s. The fractional Fourier transform can be viewed as the chirp-basis expansion directly from its definition, but essentially it can be interpreted as a rotation in the time-frequency plane, i.e. the unified time-frequency transform. With the order from 0 increasing to 1, the fractional Fourier transform can show the characteristics of the signal changing from the time domain to the frequency domain. In this research paper, the fractional Fourier transform has been comprehensively and systematically treated from the signal processing point of view. Our aim is to provide a course from the definition to the applications of the fractional Fourier transform, especially as a reference and an introduction for researchers and interested readers.

Convolution theorems for the linear canonical transform and their applications

Abstract: As generalization of the fractional Fourier transform (FRFT), the linear canonical transform (LCT) has been used in several areas, including optics and signal processing. Many properties for this transform are already known, but the convolution theorems, similar to the version of the Fourier transform, are still to be determined. In this paper, the authors derive the convolution theorems for the LCT, and explore the sampling theorem and multiplicative filter for the band limited signal in the linear canonical domain. Finally, the sampling and reconstruction formulas are deduced, together with the construction methodology for the above mentioned multiplicative filter in the time domain based on fast Fourier transform (FFT), which has much lower computational load than the construction method in the linear canonical domain.

Diffraction, Fourier Optics and Imaging

Abstract: Preface. 1. Diffraction, Fourier Optics and Imaging. 1.1 Introduction. 1.2 Examples of Emerging Applications with Growing Significance. 2. Linear Systems and Transforms. 2.1 Introduction. 2.2 Linear Systems and Shift Invariance. 2.3 Continuous-Space Fourier Transform. 2.4 Existence of Fourier Transform. 2.5 Properties of the Fourier Transform. 2.6 Real Fourier Transform. 2.7 Amplitude and Phase Spectra. 2.8 Hankel Transforms. 3. Fundamentals of Wave Propagation. 3.1 Introduction. 3.2 Waves. 3.3 Electromagnetic Waves. 3.4 Phasor Representation. 3.5 Wave Equations in a Charge-Free Medium. 3.6 Wave Equations in Phasor Representation in a Charge-Free Medium. 3.7 Plane EM Waves. 4. Scalar Diffraction Theory. 4.1 Introduction. 4.2 Helmholtz Equation. 4.3 Angular Spectrum of Plane Waves. 4.4 Fast Fourier Transform (FFT) Implementation of the Angular Spectrum of Plane Waves. 4.5 The Kirchoff Theory of Diffraction. 4.6 The Rayleigh-Sommerfeld Theory of Diffraction. 4.7 Another Derivation of the First Rayleigh-Sommerfeld Diffraction Integral. 4.8 The Rayleigh-Sommerfeld Diffraction Integral For Nonmonochromatic Waves. 5. Fresnel and Fraunhofer Approximations. 5.1 Introduction. 5.2 Diffraction in the Fresnel Region. 5.3 FFT Implementation of Fresnel Diffraction. 5.4 Paraxial Wave Equation. 5.5 Diffraction in the Fraunhofer Region. 5.6 Diffraction Gratings. 5.7 Fraunhofer Diffraction By a Sinusoidal Amplitude Grating. 5.8 Fresnel Diffraction By a Sinusoidal Amplitude Grating. 5.9 Fraunhofer Diffraction with a Sinusoidal Phase Grating. 5.10 Diffraction Gratings Made of Slits. 6. Inverse Diffraction. 6.1 Introduction. 6.2 Inversion of the Fresnel and Fraunhofer Representations. 6.3 Inversion of the Angular Spectrum Representation. 6.4 Analysis. 7. Wide-Angle Near and Far Field Approximations for Scalar Diffraction. 7.1 Introduction. 7.2 A Review of Fresnel and Fraunhofer Approximations. 7.3 The Radial Set of Approximations. 7.4 Higher Order Improvements and Analysis. 7.5 Inverse Diffraction and Iterative Optimization. 7.6 Numerical Examples. 7.7 More Accurate Approximations. 7.8 Conclusions. 8. Geometrical Optics. 8.1 Introduction. 8.2 Propagation of Rays. 8.3 The Ray Equations. 8.4 The Eikonal Equation. 8.5 Local Spatial Frequencies and Rays. 8.6 Matrix Representation of Meridional Rays. 8.7 Thick Lenses. 8.8 Entrance and Exit Pupils of an Optical System. 9. Fourier Transforms and Imaging with Coherent Optical Systems. 9.1 Introduction. 9.2 Phase Transformation With a Thin Lens. 9.3 Fourier Transforms With Lenses. 9.4 Image Formation As 2-D Linear Filtering. 9.5 Phase Contrast Microscopy. 9.6 Scanning Confocal Microscopy. 9.7 Operator Algebra for Complex Optical Systems. 10. Imaging with Quasi-Monochromatic Waves. 10.1 Introduction. 10.2 Hilbert Transform. 10.3 Analytic Signal. 10.4 Analytic Signal Representation of a Nonmonochromatic Wave Field. 10.5 Quasi-Monochromatic, Coherent, and Incoherent Waves. 10.6 Diffraction Effects in a General Imaging System. 10.7 Imaging With Quasi-Monochromatic Waves. 10.8 Frequency Response of a Diffraction-Limited Imaging System. 10.9 Computer Computation of the Optical Transfer Function. 10.10 Aberrations. 11. Optical Devices Based on Wave Modulation. 11.1 Introduction. 11.2 Photographic Films and Plates. 11.3 Transmittance of Light by Film. 11.4 Modulation Transfer Function. 11.5 Bleaching. 11.6 Diffractive Optics, Binary Optics, and Digital Optics. 11.7 E-Beam Lithography. 12. Wave Propagation in Inhomogeneous Media. 12.1 Introduction. 12.4 Beam Propagation Method. 12.5 Wave Propagation in a Directional Coupler. 13. Holography. 13.1 Introduction. 13.2 Coherent Wave Front Recording. 13.3 Types of Holograms. 13.4 Computer Simulation of Holographic Reconstruction. 13.5 Analysis of Holographic Imaging and Magnification. 13.6 Aberrations. 14. Apodization, Superresolution, and Recovery of Missing Information. 14.1 Introduction. 14.2 Apodization. 14.2.1 Discrete-Time Windows. 14.3 Two-Point Resolution and Recovery of Signals. 14.4 Contractions. 14.5 An Iterative Method of Contractions for Signal Recovery. 14.6 Iterative Constrained Deconvolution. 14.7 Method of Projections. 14.8 Method of Projections onto Convex Sets. 14.9 Gerchberg-Papoulis (GP) Algorithm. 14.10 Other POCS Algorithms. 14.11 Restoration From Phase. 14.12 Reconstruction From a Discretized Phase Function by Using the DFT. 14.13 Generalized Projections. 14.14 Restoration From Magnitude. 14.15 Image Recovery By Least Squares and the Generalized Inverse. 14.16 Computation of Hp By Singular Value Decomposition (SVD). 14.17 The Steepest Descent Algorithm. 14.18 The Conjugate Gradient Method. 15. Diffractive Optics I. 15.1 Introduction. 15.2 Lohmann Method. 15.3 Approximations in the Lohmann Method. 15.4 Constant Amplitude Lohmann Method. 15.5 Quantized Lohmann Method. 15.6 Computer Simulations with the Lohmann Method. 15.7 A Fourier Method Based on Hard-Clipping. 15.8 A Simple Algorithm for Construction of 3-D Point Images. 15.9 The Fast Weighted Zero-Crossing Algorithm. 15.10 One-Image-Only Holography. 15.11 Fresnel Zone Plates. 16. Diffractive Optics II. 16.1 Introduction. 16.2 Virtual Holography. 16.3 The Method of POCS for the Design of Binary DOE. 16.4 Iterative Interlacing Technique (IIT). 16.5 Optimal Decimation-in-Frequency Iterative Interlacing Technique (ODIFIIT). 16.5.1 Experiments with ODIFIIT. 16.6 Combined Lohmann-ODIFIIT Method. 17. Computerized Imaging Techniques I: Synthetic Aperture Radar. 17.1 Introduction. 17.2 Synthetic Aperture Radar. 17.3 Range Resolution. 17.4 Choice of Pulse Waveform. 17.5 The Matched Filter. 17.6 Pulse Compression by Matched Filtering. 17.7 Cross-Range Resolution. 17.8 A Simplified Theory of SAR Imaging. 17.9 Image Reconstruction with Fresnel Approximation. 17.10 Algorithms for Digital Image Reconstruction. 18. Computerized Imaging II: Image Reconstruction from Projections. 18.1 Introduction. 18.2 The Radon Transform. 18.3 The Projection Slice Theorem. 18.4 The Inverse Radon Transform. 18.5 Properties of the Radon Transform. 18.6 Reconstruction of a Signal From its Projections. 18.7 The Fourier Reconstruction Method. 18.8 The Filtered-Backprojection Algorithm. 19. Dense Wavelength Division Multiplexing. 19.1 Introduction. 19.2 Array Waveguide Grating. 19.3 Method of Irregularly Sampled Zero-Crossings (MISZC). 19.4 Analysis of MISZC. 19.4.1 Dispersion Analysis. 19.4.2 Finite-Sized Apertures. 19.5 Computer Experiments. 19.6 Implementational Issues. 20. Numerical Methods for Rigorous Diffraction Theory. 20.1 Introduction. 20.2 BPM Based on Finite Differences. 20.3 Wide Angle BPM. 20.4 Finite Differences. 20.5 Finite Difference Time Domain Method. 20.6 Computer Experiments. 20.7 Fourier Modal Methods. Appendix A: The Impulse Function. Appendix B: Linear Vector Spaces. Appendix C: The Discrete-Time Fourier Transform, The Discrete Fourier Transform and The Fast Fourier Transform. References. Index.

Signal processing for neuroscientists : introduction to the analysis of physiological signals

Abstract: Introduction Data Acquisition Noise Signal Averaging Real and Complex Fourier Series Continuous, Discrete, and Fast Fourier Transform Fourier Transform Applications LTI systems, Convolution, Correlation, and Coherence Laplace and z-Transform Introduction to Filters: the RC-Circuit Filters: Analysis Filters: Specification, Bode plot, Nyquist plot Filters: Digital Filters Spike Train Analysis Wavelet Analysis: Time Domain Properties Wavelet Analysis: Frequency Domain Properties Nonlinear Techniques

The multiple-parameter discrete fractional Fourier transform

Abstract: The discrete fractional Fourier transform (DFRFT) is a generalization of the discrete Fourier transform (DFT) with one additional order parameter. In this letter, we extend the DFRFT to have N order parameters, where N is the number of the input data points. The proposed multiple-parameter discrete fractional Fourier transform (MPDFRFT) is shown to have all of the desired properties for fractional transforms. In fact, the MPDFRFT reduces to the DFRFT when all of its order parameters are the same. To show an application example of the MPDFRFT, we exploit its multiple-parameter feature and propose the double random phase encoding in the MPDFRFT domain for encrypting digital data. The proposed encoding scheme in the MPDFRFT domain significantly enhances data security.

A discrete fractional random transform

Abstract: We propose a discrete fractional random transform based on a generalization of the discrete fractional Fourier transform with an intrinsic randomness. Such discrete fractional random transform inheres excellent mathematical properties of the fractional Fourier transform along with some fantastic features of its own. As a primary application, the discrete fractional random transform has been used for image encryption and decryption.

The Two-Dimensional Clifford-Fourier Transform

Abstract: Recently several generalizations to higher dimension of the Fourier transform using Clifford algebra have been introduced, including the Clifford-Fourier transform by the authors, defined as an operator exponential with a Clifford algebra-valued kernel. In this paper an overview is given of all these generalizations and an in depth study of the two-dimensional Clifford-Fourier transform of the authors is presented. In this special two-dimensional case a closed form for the integral kernel may be obtained, leading to further properties, both in the L 1 and in the L 2 context. Furthermore, based on this Clifford-Fourier transform Clifford-Gabor filters are introduced.

Advanced continuous wavelet transform algorithm for digital interferogram analysis and processing

Abstract: An advanced continuous wavelet transform algorithm for digital interferogram analysis and processing is proposed. The algorithm is an extension of the traditional wavelet transform; the mother wavelet and normalization parameter are selected based on the characteristics of optical interferograms. To reduce the processing time, a fast Fourier transform scheme is employed to implement the wavelet transform calculation. The algorithm is simple and is a robust tool for interferogram filtering and for whole-field fringe and phase information detection. The concept is verified by computer simulation and actual experimental interferogram analysis.

Discrete Fractional Fourier Transform Based on New Nearly Tridiagonal Commuting Matrices

Abstract: Based on discrete Hermite-Gaussian-like functions, a discrete fractional Fourier transform (DFRFT), which provides sample approximations of the continuous fractional Fourier transform, was defined and investigated recently. In this paper, we propose a new nearly tridiagonal matrix, which commutes with the discrete Fourier transform (DFT) matrix. The eigenvectors of the new nearly tridiagonal matrix are shown to be DFT eigenvectors, which are more similar to the continuous Hermite-Gaussian functions than those developed before. Rigorous discussions on the relations between the eigendecomposition of the newly proposed nearly tridiagonal matrix and the DFT matrix are described. Furthermore, by appropriately combining two linearly independent matrices that both commute with the DFT matrix, we develop a method to obtain DFT eigenvectors even more similar to the continuous Hermite-Gaussian functions (HGFs). Then, new versions of DFRFT produce their transform outputs closer to the samples of the continuous fractional Fourier transform, and their applications are described. Related computer experiments are performed to illustrate the validity of the works in this paper

Operational Calculus and Related Topics

Abstract: INTEGRAL TRANSFORMS Introduction to Operational Calculus Integral Transforms - Introductory Remarks The Fourier Transform The Laplace Transform The Mellin Transform The Stieltjes Transform The Hilbert Transform Bessel Transforms The Mehler-Fock Transform Finite Integral Transforms OPERATIONAL CALCULUS Introduction The Theorem of Titchmarsh Operators Bases of the Operator Analysis Operators Reducible to Functions Application of Operational Calculus GENERALIZED FUNCTIONS Introduction Generalized Functions - Functional Approach Generalized Functions - Sequential Approach Delta Sequences Convergent Sequences Local Properties Irregular Operations Hilbert Transform and Multiplication Forms

Polarization analysis and polarization filtering of three-component signals with the time–frequency S transform

Abstract: SUMMARY From basic Fourier theory, a one-component signal can be expressed as a superposition of sinusoidal oscillations in time, with the Fourier amplitude and phase spectra describing the contribution of each sinusoid to the total signal. By extension, three-component signals can be thought of as superpositions of sinusoids oscillating in the x-, y-, and z-directions, which, when considered one frequency at a time, trace out elliptical motion in three-space. Thus the total three-component signal can be thought of as a superposition of ellipses. The information contained in the Fourier spectra of the x-, y-, and z-components of the signal can then be re-expressed as Fourier spectra of the elements of these ellipses, namely: the lengths of their semi-major and semi-minor axes, the strike and dip of each ellipse plane, the pitch of the major axis, and the phase of the particle motion at each frequency. The same type of reasoning can be used with windowed Fourier transforms (such as the S transform), to give time-varying spectra of the elliptical elements. These can be used to design signal-adaptive polarization filters that reject signal components with specific polarization properties. Filters of this type are not restricted to reducing the whole amplitude of any particular ellipse; for example, the ‘linear’ part of the ellipse can be retained while the ‘circular’ part is rejected. This paper describes the mathematics behind this technique, and presents three examples: an earthquake seismogram that is first separated into linear and circular parts, and is later filtered specifically to remove the Rayleigh wave; and two shot gathers, to which similar Rayleigh-wave filters have been applied on a trace-by-trace basis.

Discrete Radon transform has an exact, fast inverse and generalizes to operations other than sums along lines

Abstract: Gotz, Druckmuller, and, independently, Brady have defined a discrete Radon transform (DRT) that sums an image9s pixel values along a set of aptly chosen discrete lines, complete in slope and intercept. The transform is fast, O ( N 2 log N ) for an N × N image; it uses only addition, not multiplication or interpolation, and it admits a fast, exact algorithm for the adjoint operation, namely backprojection. This paper shows that the transform additionally has a fast, exact (although iterative) inverse. The inverse reproduces to machine accuracy the pixel-by-pixel values of the original image from its DRT, without artifacts or a finite point-spread function. Fourier or fast Fourier transform methods are not used. The inverse can also be calculated from sampled sinograms and is well conditioned in the presence of noise. Also introduced are generalizations of the DRT that combine pixel values along lines by operations other than addition. For example, there is a fast transform that calculates median values along all discrete lines and is able to detect linear features at low signal-to-noise ratios in the presence of pointlike clutter features of arbitrarily large amplitude.

Fast 2-Dimensional 4 $times$ 4 Forward Integer Transform Implementation for H.264/AVC

Abstract: In this paper, the novel two-dimensional (2-D) fast algorithm for realization of 4 $times$ 4 forward integer transform in H.264 is proposed. Based on matrix operations with Kronecker product and direct sum, the efficient fast 2-D 4 $times$ 4 forward integer transform can be derived from the proposed one-dimensional fast 4 $times$ 4 forward integer transform through matrix decompositions. The proposed fast 2-D 4 $times$ 4 forward integer transform design doesn't need transpose memory for direct parallel pipelined architecture. The fast 2-D 4 $times$ 4 forward integer transform requires fewer latency delays than the state-of-the-art methods. With regular modularity, the proposed fast algorithm is suitable for VLSI implementation to achieve real-time H.264/advanced video coding (AVC) signal processing.

Constrained surrogate time series with preservation of the mean and variance structure.

Abstract: A method is presented for generating surrogates that are constrained realizations of a time series but which preserve the local mean and variance of the original signal. The method is based on the popular iterated amplitude adjusted Fourier transform method but makes use of a wavelet transform to constrain behavior in the time domain. Using this method it is possible to test for local changes in the nonlinear properties of the signal. We present an example for a change in Hurst exponent in a time series produced by fractional Brownian motion.

Signal reconstruction from two close fractional Fourier power spectra

Abstract: Based on the definition of the instantaneous fre quency (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 per mits 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 applica tions of the angular derivative of the fractional power spectra for signal analysis are discussed briefly. 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.

Pseudo-log-polar Fourier transform for image registration

Abstract: A new registration algorithm based on pseudo-log-polar Fourier transform (PLPFT) for estimating large translations, rotations, and scalings in images is developed. The PLPFT, which is calculated at points distributed at nonlinear increased concentric squares, approximates log-polar Fourier representations of images accurately. In addition, it can be calculated quickly by utilizing the Fourier separability property and the fractional fast Fourier transform. Using the log-polar Fourier representations and cross-power spectrum method, we can estimate the rotations and scalings in images and obtain translations later. Experimental results have verified the robustness and high accuracy of this algorithm.

Efficient computation of quadratic-phase integrals in optics

Abstract: We present a fast NlogN time algorithm for computing quadratic-phase integrals. This three-parameter class of integrals models propagation in free space in the Fresnel approximation, passage through thin lenses, and propagation in quadratic graded-index media as well as any combination of any number of these and is therefore of importance in optics. By carefully managing the sampling rate, one need not choose N much larger than the space-bandwidth product of the signals, despite the highly oscillatory integral kernel. The only deviation from exactness arises from the approximation of a continuous Fourier transform with the discrete Fourier transform. Thus the algorithm computes quadratic-phase integrals with a performance similar to that of the fast-Fourier-transform algorithm in computing the Fourier transform, in terms of both speed and accuracy.

Sampling Theorem for Fractional Bandlimited Signals: A Self-Contained Proof. Application to Digital Holography

Abstract: This letter provides a self-contained proof of the sampling theorem for fractional bandlimited signals. An interpolation formula for the fractional spectrum of a given signal is explicitely written. The chosen proof also leads to an explicit process for recovering the original signal from its periodized fractional spectrum. Both interpolation and signal recovering process are shown to be useful in computing digital holograms in the Fresnel domain

Why is the Linear Canonical Transform so little known

Abstract: The linear canonical transform (LCT), is the name of a parameterized continuum of transforms which include, as particular cases, the most widely used linear transforms and operators in engineering and physics such as the Fourier transform, fractional Fourier transform (FRFT), Fresnel transform (FRST), time scaling, chirping, and others. Therefore the LCT provides a unified framework for studying the behavior of many practical transforms and system responses in optics and engineering in general. From the system‐engineering point of view the LCT provides a powerful tool for design and analysis of the characteristics of optical systems. Despite this fact only few authors take advantage of the powerful and general LCT theory for analysis and design of optical systems. In this paper we review some important properties about the continuous LCT and we present some new results regarding the discretization and computation of the LCT.

Analytical and numerical analysis of linear optical systems

Abstract: The numerical calculation of the Fresnel transform (FST) presents significant challenges due to the high sampling rate associated with the chirp function in the kernel. The development of an efficient algorithm is further complicated by the fact that the output extent of the FST is dependent on the propagation distance. In this paper, we implement a recently proposed technique for efficiently calculating the FST in which we apply the Wigner distribution function and the space bandwidth product to identify suitable sampling rates. This method is shown to be suitable for all propagation distances. Our method can also be applied to describe the effect of a thin lens modeled as a chirp modulation transform (CMT). Combining our results for the FST and the CMT, we numerically calculate the light distribution at the output of both Cai-Wang and Lohmann Type-I optical fractional Fourier transform (OFRT) systems. Analytic solutions for the OFRT of rectangular window and circular apertures are presented. The analytical solutions are compared to experimental data and to numerical results for equivalent cases. Finally the numerical method is applied to examine the effect that apertured lenses, in the OFRT system, have on the output distribution.

Linear Canonical Transform and Its Implication

Abstract: As an emerging tool for signal processing,the linear canonical transform(LCT) proves itself to be more general and flexible than the Fourier transform as well as the fractional Fourier transform.So it can slove problems that can't be dealt with well by the latter.In this paper,the definition of LCT and some special cases are given at first,followed by its properties as listed.Besides,the discrete linear canonical transform is introduced.The implication of LCT is illustrated finally,displaying(LCT's) potentials and capabilities in the field of signal processing.

Processing of ND NMR spectra sampled in polar coordinates: a simple Fourier transform instead of a reconstruction.

Abstract: In order to reduce the acquisition time of multidimensional NMR spectra of biological macromolecules, projected spectra (or in other words, spectra sampled in polar coordinates) can be used. Their standard processing involves a regular FFT of the projections followed by a reconstruction, i.e. a non-linear process. In this communication, we show that a 2D discrete Fourier transform can be implemented in polar coordinates to obtain directly a frequency domain spectrum. Aliasing due to local violations of the Nyquist sampling theorem gives rise to base line ridges but the peak line-shapes are not distorted as in most reconstruction methods. The sampling scheme is not linear and the data points in the time domain should thus be weighted accordingly in the polar FT; however, artifacts can be reduced by additional data weighting of the undersampled regions. This processing does not require any parameter tuning and is straightforward to use. The algorithm written for polar sampling can be adapted to any sampling scheme and will permit to investigate better compromises in terms of experimental time and lack of artifacts.