Showing papers on "Fractional Fourier transform published in 2011"

Wavelet Transform With Tunable Q-Factor

Abstract: This paper describes a discrete-time wavelet transform for which the Q-factor is easily specified. Hence, the transform can be tuned according to the oscillatory behavior of the signal to which it is applied. The transform is based on a real-valued scaling factor (dilation-factor) and is implemented using a perfect reconstruction over-sampled filter bank with real-valued sampling factors. Two forms of the transform are presented. The first form is defined for discrete-time signals defined on all of Z. The second form is defined for discrete-time signals of finite-length and can be implemented efficiently with FFTs. The transform is parameterized by its Q-factor and its oversampling rate (redundancy), with modest oversampling rates (e.g., three to four times overcomplete) being sufficient for the analysis/synthesis functions to be well localized.

Gabor deconvolution: Estimating reflectivity by nonstationary deconvolution of seismic data

Abstract: We have extended the method of stationary spiking deconvolution of seismic data to the context of nonstationary signals in which the nonstationarity is due to attenuation processes. As in the stationary case, we have assumed a statistically white reflectivity and a minimum-phase source and attenuation process. This extension is based on a nonstationary convolutional model, which we have developed and related to the stationary convolutional model. To facilitate our method, we have devised a simple numerical approach to calculate the discrete Gabor transform, or complex-valued time-frequency decomposition, of any signal. Although the Fourier transform renders stationary convolution into exact, multiplicative factors, the Gabor transform, or windowed Fourier transform, induces only an approximate factorization of the nonstationary convolutional model. This factorization serves as a guide to develop a smoothing process that, when applied to the Gabor transform of the nonstationary seismic trace, estimates the magnitude of the time-frequency attenuation function and the source wavelet. By assuming that both are minimum-phase processes, their phases can be determined. Gabor deconvolution is accomplished by spectral division in the time-frequency domain. The complex-valued Gabor transform of the seismic trace is divided by the complex-valued estimates of attenuation and source wavelet to estimate the Gabor transform of the reflectivity. An inverse Gabor transform recovers the time-domain reflectivity. The technique has applications to synthetic data and real data.

Lv's Distribution: Principle, Implementation, Properties, and Performance

Abstract: This paper proposes a novel representation, known as Lv's distribution (LVD), of linear frequency modulated (LFM) signals. It has been well known that a monocomponent LFM signal can be uniquely determined by two important physical quantities, centroid frequency and chirp rate (CFCR). The basic reason for expressing a LFM signal in the CFCR domain is that these two quantities may not be apparent in the time or time-frequency (TF) domain. The goal of the LVD is to naturally and accurately represent a mono- or multicomponent LFM in the CFCR domain. The proposed LVD is simple and only requires a two-dimensional (2-D) Fourier transform of a parametric scaled symmetric instantaneous autocorrelation function. It can be easily implemented by using the complex multiplications and fast Fourier transforms (FFT) based on the scaling principle. The computational complexity, properties, detection performance and representation errors are analyzed for this new distribution. Comparisons with three other popular methods, Radon-Wigner transform (RWT), Radon-Ambiguity transform (RAT), and fractional Fourier transform (FRFT) are performed. With several numerical examples, our distribution is demonstrated to be a CFCR representation that is computed without using any searching operation. The main significance of the LVD is to convert a 1-D LFM into a 2-D single-frequency signal. One of the most important applications of the LVD is to generate a new TF representation, called inverse LVD (ILVD), and a new ambiguity function, called Lv's ambiguity function (LVAF), both of which may break through the tradeoff between resolution and cross terms.

Fractional Processes and Fractional-Order Signal Processing: Techniques and Applications

Abstract: Fractional processes are widely found in science, technology and engineering systems. In Fractional Processes and Fractional-order Signal Processing, some complex random signals, characterized by the presence of a heavy-tailed distribution or non-negligible dependence between distant observations (local and long memory), are introduced and examined from the fractional perspective using simulation, fractional-order modeling and filtering and realization of fractional-order systems. These fractional-order signal processing (FOSP) techniques are based on fractional calculus, the fractional Fourier transform and fractional lower-order moments. Fractional Processes and Fractional-order Signal Processing: presents fractional processes of fixed, variable and distributed order studied as the output of fractional-order differential systems; introduces FOSP techniques and the fractional signals and fractional systems point of view; details real-world-application examples of FOSP techniques to demonstrate their utility; and provides important background material on MittagLeffler functions, the use of numerical inverse Laplace transform algorithms and supporting MATLAB codes together with a helpful survey of relevant webpages. Readers will be able to use the techniques presented to re-examine their signals and signal-processing methods. This text offers an extended toolbox for complex signals from diverse fields in science and engineering. It will give academic researchers and practitioners a novel insight into the complex random signals characterized by fractional properties, and some powerful tools to analyze those signals.

Analysing and compensating the effects of range and Doppler frequency migrations in linear frequency modulation pulse compression radar

Abstract: The pulse compression and Doppler processing (PCDP) method has been extensively used to detect low-speed and uniform-speed targets in linear frequency modulation (LFM) pulse compression radar. However, the PCDP method is affected by range migration (RM) and Doppler frequency migration (DFM) when detecting high-speed and accelerating targets. In this paper, the authors analyse and quantify these effects, and obtain the relationships of the optimal number of pulses and the threshold number of pulses with RM and DFM. It shows that when the number of pulses equals its optimal value, the maximum output signal-to-noise ratio can be obtained; when the number of pulses is greater than its threshold value, the migrations should be compensated. Then the authors propose a method based on the scaling processing and the fractional Fourier transform to remove the two migrations. In the end, the authors give a target detection experiment to show that the proposed method can effectively compensate these migrations.

Discrete Fourier transform-based watermarking method with an optimal implementation radius

Abstract: In this paper, we evaluate the degradation of an image due to the implementation of a watermark in the frequency domain of the image. As a result, a watermarking method, which minimizes the impact of the watermark implementation on the overall quality of an image, is developed. The watermark is embedded in magnitudes of the Fourier transform. A peak signal-to-noise ratio is used to evaluate quality degradation. The obtained results were used to develop a watermarking strategy that chooses the optimal radius of the implementation to minimize quality degradation. The robustness of the proposed method was evaluated on the dataset of 1000 images. Detection rates and receiver operating characteristic performance showed considerable robustness against the print-scan process, print-cam process, amplitude modulated, halftoning, and attacks from the StirMark benchmark software.

Application of New Transform "Elzaki Transform" to Partial Differential Equations

Abstract: The ELzaki transform of partial derivatives is derived, and its applicability demonstrated using four different partial differential equations. In this paper we find the particular solutions of these equations.

Fourier Methods in Imaging

Abstract: Series Editor s Preface. Preface. 1 Introduction. 1.1 Signals, Operators, and Imaging Systems. 1.2 The Three Imaging Tasks. 1.3 Examples of Optical Imaging. 1.4 ImagingTasks inMedical Imaging. 2 Operators and Functions. 2.1 Classes of Imaging Operators. 2.2 Continuous and Discrete Functions. Problems. 3 Vectors with Real-Valued Components. 3.1 Scalar Products. 3.2 Matrices. 3.3 Vector Spaces. Problems. 4 Complex Numbers and Functions. 4.1 Arithmetic of Complex Numbers. 4.2 Graphical Representation of Complex Numbers. 4.3 Complex Functions. 4.4 Generalized Spatial Frequency Negative Frequencies. 4.5 Argand Diagrams of Complex-Valued Functions. Problems. 5 Complex-Valued Matrices and Systems. 5.1 Vectors with Complex-Valued Components. 5.2 Matrix Analogues of Shift-Invariant Systems. 5.3 Matrix Formulation of ImagingTasks. 5.4 Continuous Analogues of Vector Operations. Problems. 6 1-D Special Functions. 6.1 Definitions of 1-D Special Functions. 6.2 1-D Dirac Delta Function. 6.3 1-D Complex-Valued Special Functions. 6.4 1-D Stochastic Functions Noise. 6.5 Appendix A: Area of SINC[x] and SINC2[x]. 6.6 Appendix B: Series Solutions for Bessel Functions J0[x] and J1[x]. Problems. 7 2-D Special Functions. 7.1 2-D Separable Functions. 7.2 Definitions of 2-D Special Functions. 7.3 2-D Dirac Delta Function and its Relatives. 7.4 2-D Functions with Circular Symmetry. 7.5 Complex-Valued 2-D Functions. 7.6 Special Functions of Three (orMore) Variables. Problems. 8 Linear Operators. 8.1 Linear Operators. 8.2 Shift-Invariant.Operators. 8.3 Linear Shift-Invariant (LSI) Operators. 8.4 Calculating Convolutions. 8.5 Properties of Convolutions. 8.6 Autocorrelation. 8.7 Crosscorrelation. 8.8 2-DLSIOperations. 8.9 Crosscorrelations of 2-D Functions. 8.10 Autocorrelations of 2-D.Functions. Problems. 9 Fourier Transforms of 1-D Functions. 9.1 Transforms of Continuous-Domain Functions. 9.2 Linear Combinations of Reference Functions. 9.3 Complex-Valued Reference Functions. 9.4 Transforms of Complex-Valued Functions. 9.5 Fourier Analysis of Dirac Delta Functions. 9.6 Inverse Fourier Transform. 9.7 Fourier Transforms of 1-D Special Functions. 9.8 Theorems of the Fourier Transform. 9.9 Appendix: Spectrum of Gaussian via Path Integral. Problems. 10 Multidimensional Fourier Transforms. 10.1 2-D Fourier Transforms. 10.2 Spectra of Separable 2-D Functions. 10.3 Theorems of 2-D Fourier Transforms. Problems. 11 Spectra of Circular Functions. 11.1 The Hankel Transform. 11.2 Inverse Hankel Transform. 11.3 Theorems of Hankel Transforms. 11.4 Hankel Transforms of Special Functions. 11.5 Appendix: Derivations of Equations (11.12) and (11.14). Problems. 12 The Radon Transform. 12.1 Line-Integral Projections onto Radial Axes. 12.2 Radon Transforms of Special Functions. 12.3 Theorems of the Radon Transform. 12.4 Inverse Radon Transform. 12.5 Central-Slice Transform. 12.6 Three Transforms of Four Functions. 12.7 Fourier and Radon Transforms of Images. Problems. 13 Approximations to Fourier Transforms. 13.1 Moment Theorem. 13.2 1-D Spectra via Method of Stationary Phase. 13.3 Central-Limit Theorem. 13.4 Width Metrics and Uncertainty Relations. Problems. 14 Discrete Systems, Sampling, and Quantization. 14.1 Ideal Sampling. 14.2 Ideal Sampling of Special Functions. 14.3 Interpolation of Sampled Functions. 14.4 Whittaker Shannon Sampling Theorem. 14.5 Aliasingand Interpolation. 14.6 Prefiltering to Prevent Aliasing. 14.7 Realistic Sampling. 14.8 Realistic Interpolation. 14.9 Quantization. 14.10 Discrete Convolution. Problems. 15 Discrete Fourier Transforms. 15.1 Inverse of the Infinite-Support DFT. 15.2 DFT over Finite Interval. 15.3 Fourier Series Derived from Fourier Transform. 15.4 Efficient Evaluation of the Finite DFT. 15.5 Practical Considerations for DFT and FFT. 15.6 FFTs of 2-D Arrays. 15.7 Discrete Cosine Transform. Problems. 16 Magnitude Filtering. 16.1 Classes of Filters. 16.2 Eigenfunctions of Convolution. 16.3 Power Transmission of Filters. 16.4 Lowpass Filters. 16.5 Highpass Filters. 16.6 Bandpass Filters. 16.7 Fourier Transform as a Bandpass Filter. 16.8 Bandboost and Bandstop Filters. 16.9 Wavelet Transform. Problems. 17 Allpass (Phase) Filters. 17.1 Power-Series Expansion for Allpass Filters. 17.2 Constant-Phase Allpass Filter. 17.3 Linear-Phase Allpass Filter. 17.4 Quadratic-Phase Filter. 17.5 Allpass Filters with Higher-Order Phase. 17.6 Allpass Random-Phase Filter. 17.7 Relative Importance of Magnitude and Phase. 17.8 Imaging of Phase Objects. 17.9 Chirp Fourier Transform. Problems. 18 Magnitude Phase Filters. 18.1 Transfer Functions of Three Operations. 18.2 Fourier Transform of Ramp Function. 18.3 Causal Filters. 18.4 Damped Harmonic Oscillator. 18.5 Mixed Filters with Linear or Random Phase. 18.6 Mixed Filter with Quadratic Phase. Problems. 19 Applications of Linear Filters. 19.1 Linear Filters for the Imaging Tasks. 19.2 Deconvolution Inverse Filtering . 19.3 Optimum Estimators for Signals in Noise. 19.4 Detection of Known Signals Matched Filter. 19.5 Analogies of Inverse and Matched Filters. 19.6 Approximations to Reciprocal Filters. 19.7 Inverse Filtering of Shift-Variant Blur. Problems. 20 Filtering in Discrete Systems. 20.1 Translation, Leakage, and Interpolation. 20.2 Averaging Operators Lowpass Filters. 20.3 Differencing Operators Highpass Filters. 20.4 Discrete Sharpening Operators. 20.5 2-DGradient. 20.6 Pattern Matching. 20.7 Approximate Discrete Reciprocal Filters. Problems. 21 Optical Imaging in Monochromatic Light. 21.1 Imaging Systems Based on Ray Optics Model. 21.2 Mathematical Model of Light Propagation. 21.3 Fraunhofer Diffraction. 21.4 Imaging System based on Fraunhofer Diffraction. 21.5 Transmissive Optical Elements. 21.6 Monochromatic Optical Systems. 21.7 Shift-Variant Imaging Systems. Problems. 22 Incoherent Optical Imaging Systems. 22.1 Coherence. 22.2 Polychromatic Source Temporal Coherence. 22.3 Imaging in Incoherent Light. 22.4 System Function in Incoherent Light. Problems. 23 Holography. 23.1 Fraunhofer Holography. 23.2 Holography in Fresnel Diffraction Region. 23.3 Computer-Generated Holography. 23.4 Matched Filtering with Cell-Type CGH. 23.5 Synthetic-Aperture Radar (SAR). Problems. References. Index.

Sparse signal representations using the tunable Q-factor wavelet transform

Abstract: The tunable Q-factor wavelet transform (TQWT) is a fully-discrete wavelet transform for which the Q-factor, Q, of the underlying wavelet and the asymptotic redundancy (over-sampling rate), r, of the transform are easily and independently specified. In particular, the specified parameters Q and r can be real-valued. Therefore, by tuning Q, the oscillatory behavior of the wavelet can be chosen to match the oscillatory behavior of the signal of interest, so as to enhance the sparsity of a sparse signal representation. The TQWT is well suited to fast algorithms for sparsity-based inverse problems because it is a Parseval frame, easily invertible, and can be efficiently implemented using radix-2 FFTs. The TQWT can also be used as an easily-invertible discrete approximation of the continuous wavelet transform.

Security enhancement of a phase-truncation based image encryption algorithm

Abstract: The asymmetric cryptosystem, which is based on phase-truncated Fourier transforms (PTFTs), can break the linearity of conventional systems. However, it has been proven to be vulnerable to a specific attack based on iterative Fourier transforms when the two random phase masks are used as public keys to encrypt different plaintexts. An improvement from the asymmetric cryptosystem may be taken by relocating the amplitude values in the output plane. In this paper, two different methods are adopted to realize the amplitude modulation of the output image. The first one is to extend the PTFT-based asymmetrical cryptosystem into the anamorphic fractional Fourier transform domain directly, and the second is to add an amplitude mask in the Fourier plane of the encryption scheme. Some numerical simulations are presented to prove the good performance of the proposed cryptosystems.

On the stability of the hyperbolic cross discrete Fourier transform

Abstract: A straightforward discretisation of problems in high dimensions often leads to an exponential growth in the number of degrees of freedom. Sparse grid approximations allow for a severe decrease in the number of used Fourier coefficients to represent functions with bounded mixed derivatives and the fast Fourier transform (FFT) has been adapted to this thin discretisation. We show that this so called hyperbolic cross FFT suffers from an increase of its condition number for both increasing refinement and increasing spatial dimension.

Analysis of financial data series using fractional Fourier transform and multidimensional scaling

Abstract: The goal of this study is the analysis of the dynamical properties of financial data series from worldwide stock market indexes during the period 2000–2009. We analyze, under a regional criterium, ten main indexes at a daily time horizon. The methods and algorithms that have been explored for the description of dynamical phenomena become an effective background in the analysis of economical data. We start by applying the classical concepts of signal analysis, fractional Fourier transform, and methods of fractional calculus. In a second phase we adopt the multidimensional scaling approach. Stock market indexes are examples of complex interacting systems for which a huge amount of data exists. Therefore, these indexes, viewed from a different perspectives, lead to new classification patterns.

The $\mathfrak {su}(2)_\alpha $ Hahn oscillator and a discrete Fourier–Hahn transform

Abstract: We define the quadratic algebra which is a one-parameter deformation of the Lie algebra extended by a parity operator. The odd-dimensional representations of (with representation label j, a positive integer) can be extended to representations of . We investigate a model of the finite one-dimensional harmonic oscillator based upon this algebra . It turns out that in this model the spectrum of the position and momentum operator can be computed explicitly, and that the corresponding (discrete) wavefunctions can be determined in terms of Hahn polynomials. The operation mapping position wavefunctions into momentum wavefunctions is studied, and this so-called discrete Fourier–Hahn transform is computed explicitly. The matrix of this discrete Fourier–Hahn transform has many interesting properties, similar to those of the traditional discrete Fourier transform.

Using EMD-FrFT Filtering to Mitigate Very High Power Interference in Chirp Tracking Radars

Abstract: This letter presents a new signal processing subsystem for conventional monopulse tracking radars that offers an improved solution to the problem of dealing with manmade high power interference (jamming). It is based on the hybrid use of empirical mode decomposition (EMD) and fractional Fourier transform (FrFT). EMD-FrFT filtering is carried out for complex noisy radar chirp signals to decrease the signal's noisy components. An improvement in the signal-to-noise ratio (SNR) of up to 18 dB for different target SNRs is achieved using the proposed EMD-FrFT algorithm.

The Class of Clifford-Fourier Transforms

Abstract: Recently, there has been an increasing interest in the study of hypercomplex signals and their Fourier transforms. This paper aims to study such integral transforms from general principles, using 4 different yet equivalent definitions of the classical Fourier transform. This is applied to the so-called Clifford-Fourier transform (see Brackx et al., J. Fourier Anal. Appl. 11:669–681, 2005). The integral kernel of this transform is a particular solution of a system of PDEs in a Clifford algebra, but is, contrary to the classical Fourier transform, not the unique solution. Here we determine an entire class of solutions of this system of PDEs, under certain constraints. For each solution, series expressions in terms of Gegenbauer polynomials and Bessel functions are obtained. This allows to compute explicitly the eigenvalues of the associated integral transforms. In the even-dimensional case, this also yields the inverse transform for each of the solutions. Finally, several properties of the entire class of solutions are proven.

Fast Walsh–Hadamard–Fourier Transform Algorithm

Abstract: An efficient fast Walsh-Hadamard-Fourier transform algorithm which combines the calculation of the Walsh-Hadamard transform (WHT) and the discrete Fourier transform (DFT) is introduced. This can be used in Walsh-Hadamard precoded orthogonal frequency division multiplexing systems (WHT-OFDM) to increase speed and reduce the implementation cost. The algorithm is developed through the sparse matrices factorization method using the Kronecker product technique, and implemented in an integrated butterfly structure. The proposed algorithm has significantly lower arithmetic complexity, shorter delays and simpler indexing schemes than existing algorithms based on the concatenation of the WHT and FFT, and saves about 70%-36% in computer run-time for transform lengths of 16-4096.

Product of two generalized pseudo-differential operators involving fractional Fourier transform

Abstract: A generalized pseudo-differential operator involving fractional Fourier transform associated with symbol a(x, y) is defined. The product of two generalized pseudo-differential operators is shown to be a generalized pseudo-differential operator.

Diagnostic ultrasound tooth imaging using fractional fourier transform

Abstract: An ultrasound contact imaging method is proposed to measure the enamel thickness in the human tooth. A delay-line transducer with a working frequency of 15 MHz is chosen to achieve a minimum resolvable distance of 400 μm in human enamel. To confirm the contact between the tooth and the transducer, a verification technique based on the phase shift upon reflection is used. Because of the high attenuation in human teeth, linear frequency-modulated chirp excitation and pulse compression are exploited to increase the penetration depth and improve the SNR. Preliminary measurements show that the enamel-dentin boundary creates numerous internal reflections, which cause the applied chirp signals to interfere arbitrarily. In this work, the fractional Fourier transform (FrFT) is employed for the first time in dental imaging to separate chirp signals overlapping in both time and frequency domains. The overlapped chirps are compressed using the FrFT and matched filter techniques. Micro-computed tomography is used for validation of the ultrasound measurements for both techniques. For a human molar, the thickness of the enamel layer is measured with an average error of 5.5% after compressing with the FrFT and 13.4% after compressing with the matched filter based on the average speed of sound in human teeth.