Showing papers on "Fractional Fourier transform published in 2014"

Integral transforms and their applications

Abstract: INTEGRAL TRANSFORMS Brief Historical Introduction Basic Concepts and Definitions FOURIER TRANSFORMS AND THEIR APPLICATIONS Introduction The Fourier Integral Formulas Definition of the Fourier Transform and Examples Fourier Transforms of Generalized Functions Basic Properties of Fourier Transforms Poisson's Summation Formula The Shannon Sampling Theorem Gibbs' Phenomenon Heisenberg's Uncertainty Principle Applications of Fourier Transforms to Ordinary Differential Eqn Solutions of Integral Equations Solutions of Partial Differential Equations Fourier Cosine and Sine Transforms with Examples Properties of Fourier Cosine and Sine Transforms Applications of Fourier Cosine and Sine Transforms to Partial DE Evaluation of Definite Integrals Applications of Fourier Transforms in Mathematical Statistics Multiple Fourier Transforms and Their Applications Exercises LAPLACE TRANSFORMS AND THEIR BASIC PROPERTIES Introduction Definition of the Laplace Transform and Examples Existence Conditions for the Laplace Transform Basic Properties of Laplace Transforms The Convolution Theorem and Properties of Convolution Differentiation and Integration of Laplace Transforms The Inverse Laplace Transform and Examples Tauberian Theorems and Watson's Lemma Exercises APPLICATIONS OF LAPLACE TRANSFORMS Introduction Solutions of Ordinary Differential Equations Partial Differential Equations, Initial and Boundary Value Problems Solutions of Integral Equations Solutions of Boundary Value Problems Evaluation of Definite Integrals Solutions of Difference and Differential-Difference Equations Applications of the Joint Laplace and Fourier Transform Summation of Infinite Series Transfer Function and Impulse Response Function Exercises FRACTIONAL CALCULUS AND ITS APPLICATIONS Introduction Historical Comments Fractional Derivatives and Integrals Applications of Fractional Calculus Exercises APPLICATIONS OF INTEGRAL TRANSFORMS TO FRACTIONAL DIFFERENTIAL EQUATIONS Introduction Laplace Transforms of Fractional Integrals Fractional Ordinary Differential Equations Fractional Integral Equations Initial Value Problems for Fractional Differential Equations Green's Functions of Fractional Differential Equations Fractional Partial Differential Equations Exercises HANKEL TRANSFORMS AND THEIR APPLICATIONS Introduction The Hankel Transform and Examples Operational Properties of the Hankel Transform Applications of Hankel Transforms to Partial Differential Equations Exercises MELLIN TRANSFORMS AND THEIR APPLICATIONS Introduction Definition of the Mellin Transform and Examples Basic Operational Properties Applications of Mellin Transforms Mellin Transforms of the Weyl Fractional Integral and Derivative Application of Mellin Transforms to Summation of Series Generalized Mellin Transforms Exercises HILBERT AND STIELTJES TRANSFORMS Introduction Definition of the Hilbert Transform and Examples Basic Properties of Hilbert Transforms Hilbert Transforms in the Complex Plane Applications of Hilbert Transforms Asymptotic Expansions of One-Sided Hilbert Transforms Definition of the Stieltjes Transform and Examples Basic Operational Properties of Stieltjes Transforms Inversion Theorems for Stieltjes Transforms Applications of Stieltjes Transforms The Generalized Stieltjes Transform Basic Properties of the Generalized Stieltjes Transform Exercises FINITE FOURIER SINE AND COSINE TRANSFORMS Introduction Definitions of the Finite Fourier Sine and Cosine Transforms and Examples Basic Properties of Finite Fourier Sine and Cosine Transforms Applications of Finite Fourier Sine and Cosine Transforms Multiple Finite Fourier Transforms and Their Applications Exercises FINITE LAPLACE TRANSFORMS Introduction Definition of the Finite Laplace Transform and Examples Basic Operational Properties of the Finite Laplace Transform Applications of Finite Laplace Transforms Tauberian Theorems Exercises Z TRANSFORMS Introduction Dynamic Linear Systems and Impulse Response Definition of the Z Transform and Examples Basic Operational Properties The Inverse Z Transform and Examples Applications of Z Transforms to Finite Difference Equations Summation of Infinite Series Exercises FINITE HANKEL TRANSFORMS Introduction Definition of the Finite Hankel Transform and Examples Basic Operational Properties Applications of Finite Hankel Transforms Exercises LEGENDRE TRANSFORMS Introduction Definition of the Legendre Transform and examples Basic Operational Properties of Legendre Transforms Applications of Legendre Transforms to Boundary Value Problems Exercises JACOBI AND GEGENBAUER TRANSFORMS Introduction Definition of the Jacobi Transform and Examples Basic Operational Properties Applications of Jacobi Transforms to the Generalized Heat Conduction Problem The Gegenbauer Transform and its Basic Operational Properties Application of the Gegenbauer Transform LAGUERRE TRANSFORMS Introduction Definition of the Laguerre Transform and Examples Basic Operational Properties Applications of Laguerre Transforms Exercises HERMITE TRANSFORMS Introduction Definition of the Hermite Transform and Examples Basic Operational Properties Exercises THE RADON TRANSFORM AND ITS APPLICATION Introduction Radon Transform Properties of Radon Transform Radon Transform of Derivatives Derivatives of Radon Transform Convolution Theorem for Radon Transform Inverse of Radon Transform Exercises WAVELETS AND WAVELET TRANSFORMS Brief Historical Remarks Continuous Wavelet Transforms The Discrete Wavelet Transform Examples of Orthonormal Wavelets Exercises Appendix A Some Special Functions and Their Properties A-1 Gamma, Beta, and Error Functions A-2 Bessel and Airy Functions A-3 Legendre and Associated Legendre Functions A-4 Jacobi and Gegenbauer Polynomials A-5 Laguerre and Associated Laguerre Functions A-6 Hermite and Weber-Hermite Functions A-7 Hurwitz and Riemann zeta Functions Appendix B Tables of Integral Transforms B-1 Fourier Transforms B-2 Fourier Cosine Transforms B-3 Fourier Sine Transforms B-4 Laplace Transforms B-5 Hankel Transforms B-6 Mellin Transforms B-7 Hilbert Transforms B-8 Stieltjes Transforms B-9 Finite Fourier Cosine Transforms B-10 Finite Fourier Sine Transforms B-11 Finite Laplace Transforms B-12 Z Transforms B-13 Finite Hankel Transforms Answers and Hints to Selected Exercises Bibliography Index

GESPAR: Efficient Phase Retrieval of Sparse Signals

Abstract: We consider the problem of phase retrieval, namely, recovery of a signal from the magnitude of its Fourier transform, or of any other linear transform. Due to the loss of Fourier phase information, this problem is ill-posed. Therefore, prior information on the signal is needed in order to enable its recovery. In this work we consider the case in which the signal is known to be sparse, i.e., it consists of a small number of nonzero elements in an appropriate basis. We propose a fast local search method for recovering a sparse signal from measurements of its Fourier transform (or other linear transform) magnitude which we refer to as GESPAR: GrEedy Sparse PhAse Retrieval. Our algorithm does not require matrix lifting, unlike previous approaches, and therefore is potentially suitable for large scale problems such as images. Simulation results indicate that GESPAR is fast and more accurate than existing techniques in a variety of settings.

Maneuvering Target Detection via Radon-Fractional Fourier Transform-Based Long-Time Coherent Integration

Abstract: Long-time coherent integration technique is one of the most important methods for the improvement of radar detection ability of a weak maneuvering target, whereas the integration performance may be greatly influenced by the across range unit (ARU) and Doppler frequency migration (DFM) effects. In this paper, a novel representation known as Radon-fractional Fourier transform (RFRFT) is proposed and investigated to solve the above problems simultaneously. It can not only eliminate the effect of DFM by selecting a proper rotation angle but also achieve long-time coherent integration without ARU effect. The RFRFT can be regarded as a special Doppler filter bank composed of filters with different rotation angles, which indicates a generalization of the traditional moving target detection (MTD) and FRFT methods. Some useful properties and the likelihood ratio test detector of RFRFT are derived for maneuvering target detection. Finally, numerical experiments of aerial target and marine target detection are carried out using simulated and real radar datasets. The results demonstrate that for integration gain and detection ability, the proposed method is superior to MTD, FRFT, and Radon-Fourier transform under low signal-to-clutter/noise ratio (SCR/SNR) environments. Moreover, the trajectory of target can be easily obtained via RFRFT as well.

Information Transmission Using the Nonlinear Fourier Transform, Part I: Mathematical Tools

Abstract: The nonlinear Fourier transform (NFT), a powerful tool in soliton theory and exactly solvable models, is a method for solving integrable partial differential equations governing wave propagation in certain nonlinear media. The NFT decorrelates signal degrees-of-freedom in such models, in much the same way that the Fourier transform does for linear systems. In this three-part series of papers, this observation is exploited for data transmission over integrable channels, such as optical fibers, where pulse propagation is governed by the nonlinear Schrodinger equation. In this transmission scheme, which can be viewed as a nonlinear analogue of orthogonal frequency-division multiplexing commonly used in linear channels, information is encoded in the nonlinear frequencies and their spectral amplitudes. Unlike most other fiber-optic transmission schemes, this technique deals with both dispersion and nonlinearity directly and unconditionally without the need for dispersion or nonlinearity compensation methods. This paper explains the mathematical tools that underlie the method.

Recent Developments in the Sparse Fourier Transform: A compressed Fourier transform for big data

Abstract: The discrete Fourier transform (DFT) is a fundamental component of numerous computational techniques in signal processing and scientific computing. The most popular means of computing the DFT is the fast Fourier transform (FFT). However, with the emergence of big data problems, in which the size of the processed data sets can easily exceed terabytes, the "fast" in FFT is often no longer fast enough. In addition, in many big data applications it is hard to acquire a sufficient amount of data to compute the desired Fourier transform in the first place. The sparse Fourier transform (SFT) addresses the big data setting by computing a compressed Fourier transform using only a subset of the input data, in time smaller than the data set size. The goal of this article is to survey these recent developments, explain the basic techniques with examples and applications in big data, demonstrate tradeoffs in empirical performance of the algorithms, and discuss the connection between the SFT and other techniques for massive data analysis such as streaming algorithms and compressive sensing.

Applications of the synchrosqueezing transform in seismic time-frequency analysis

Abstract: Time-frequency representation of seismic signals provides a source of information that is usually hidden in the Fourier spectrum. The short-time Fourier transform and the wavelet transform are the principal approaches to simultaneously decompose a signal into time and frequency components. Known limitations, such as trade-offs between time and frequency resolution, may be overcome by alternative techniques that extract instantaneous modal components. Empirical mode decomposition aims to decompose a signal into components that are well separated in the time-frequency plane allowing the reconstruction of these components. On the other hand, a recently proposed method called the “synchrosqueezing transform” (SST) is an extension of the wavelet transform incorporating elements of empirical mode decomposition and frequency reassignment techniques. This new tool produces a well-defined time-frequency representation allowing the identification of instantaneous frequencies in seismic signals to highlight ...

Detection and Extraction of Target With Micromotion in Spiky Sea Clutter Via Short-Time Fractional Fourier Transform

Abstract: In order to effectively detect moving targets in heavy sea clutter, the micro-Doppler (m-D) effect is studied and an effective algorithm based on short-time fractional Fourier transform (STFRFT) is proposed for target detection and m-D signal extraction. Firstly, the mathematical model of target with micromotion at sea, including translation and rotation movement, is established, which can be approximated as the sum of linear-frequency-modulated signals within a short time. Then, due to the high-power, time-varying, and target-like properties of sea spikes, which may result in poor detection performance, sea spikes are identified and eliminated before target detection to improve signal-to-clutter ratio (SCR). By taking the absolute amplitude of signals in the best STFRFT domain (STFRFD) as the test statistic, and comparing it with the threshold determined by a constant false alarm rate detector, micromotion target can be declared or not. STFRFT with Gaussian window is employed to provide time-frequency distribution of m-D signals, and the instantaneous frequency of each component can be extracted and estimated precisely by STFRFD filtering. In the end, datasets from the intelligent pixel processing radar with HH and VV polarizations are used to verify the validity of this proposed algorithm. Two shore-based experiments are also conducted using an X-band sea search radar and an S-band sea surveillance radar, respectively. The results demonstrate that the proposed method not only achieves high detection probability in a low-SCR environment but also outperforms the short-time Fourier transform-based method.

General Parameterized Time-Frequency Transform

Abstract: Interest in parameterized time-frequency analysis for non-stationary signal processing is increasing steadily. An important advantage of such analysis is to provide highly concentrated time-frequency representation with signal-dependent resolution. In this paper, a general scheme, named as general parameterized time-frequency transform (GPTF transform), is proposed for carrying out parameterized time-frequency analysis. The GPTF transform is defined by applying generalized kernel based rotation operator and shift operator. It provides the availability of a single generalized time-frequency transform for applications on signals of different natures. Furthermore, by replacing kernel function, it facilitates the implementation of various parameterized time – frequency transforms from the same standpoint. The desirable properties and the dual definition in the frequency domain of GPTF transform are also described in this paper. One of the advantages of the GPTF transform is that the generalized kernel can be customized to characterize the time – frequency signature of non-stationary signal. As different kernel formulation has bias toward the signal to be analyzed, a proper kernel is vital to the GPTF. Thus, several potential kernels are provided and discussed in this paper to develop the desired parameterized time – frequency transforms. In real applications, it is desired to identify proper kernel with respect to the considered signal. This motivates us to propose an effective method to identify the kernel for the GPTF.

Sparse Discrete Fractional Fourier Transform and Its Applications

Abstract: The discrete fractional Fourier transform is a powerful signal processing tool with broad applications for nonstationary signals. In this paper, we propose a sparse discrete fractional Fourier transform (SDFrFT) algorithm to reduce the computational complexity when dealing with large data sets that are sparsely represented in the fractional Fourier domain. The proposed technique achieves multicomponent resolution in addition to its low computational complexity and robustness against noise. In addition, we apply the SDFrFT to the synchronization of high dynamic direct-sequence spread-spectrum signals. Furthermore, a sparse fractional cross ambiguity function (SFrCAF) is developed, and the application of SFrCAF to a passive coherent location system is presented. The experiment results confirm that the proposed approach can substantially reduce the computation complexity without degrading the precision.

A Novel Method for Parameter Estimation of Space Moving Targets

Abstract: This letter proposes a novel parameter estimation method based on keystone transform (KT) and Radon-Fourier transform (RFT) for space moving targets with high-speed maneuvering performance. In this method, second-order KT is used to correct the range curvature and part of the range walk for all targets simultaneously. Then, fractional Fourier transform is employed to estimate the targets' radial acceleration, followed by the quadric phase term compensation. Finally, RFT and Clean technique are carried out to correct the residual range walk, and the initial range and radial velocity of moving targets are further obtained. The advantage of the proposed method is that it can overcome the limitation of Doppler frequency ambiguity and correct range curvature for all targets in one processing step, which simplifies the operation procedure. Simulation results are presented to demonstrate the validity of the proposed method.

Fast Numerical Nonlinear Fourier Transforms

Abstract: The nonlinear Fourier transform, which is also known as the forward scattering transform, decomposes a periodic signal into nonlinearly interacting waves. In contrast to the common Fourier transform, these waves no longer have to be sinusoidal. Physically relevant waveforms are often available for the analysis instead. The details of the transform depend on the waveforms underlying the analysis, which in turn are specified through the implicit assumption that the signal is governed by a certain evolution equation. For example, water waves generated by the Korteweg-de Vries equation can be expressed in terms of cnoidal waves. Light waves in optical fiber governed by the nonlinear Schrodinger equation (NSE) are another example. Nonlinear analogs of classic problems such as spectral analysis and filtering arise in many applications, with information transmission in optical fiber, as proposed by Yousefi and Kschischang, being a very recent one. The nonlinear Fourier transform is eminently suited to address them -- at least from a theoretical point of view. Although numerical algorithms are available for computing the transform, a "fast" nonlinear Fourier transform that is similarly effective as the fast Fourier transform is for computing the common Fourier transform has not been available so far. The goal of this paper is to address this problem. Two fast numerical methods for computing the nonlinear Fourier transform with respect to the NSE are presented. The first method achieves a runtime of $O(D^2)$ floating point operations, where $D$ is the number of sample points. The second method applies only to the case where the NSE is defocusing, but it achieves an $O(D\log^2D)$ runtime. Extensions of the results to other evolution equations are discussed as well.

Detection and Extraction of Target With Micromotion in Spiky Sea Clutter via Short-Time Fractional

Abstract: In order to effectively detect moving targets in heavy sea clutter, the micro-Doppler (m-D) effect is studied and an effective algorithm based on short-time fractional Fourier transform (STFRFT) is proposed for target detection and m-D signal extraction. Firstly, the mathematical model of target with micromotion at sea, including translation and rotation movement, is established, which can be approximated as the sum of linear- frequency-modulated signals within a short time. Then, due to the high-power, time-varying, and target-like properties of sea spikes, which may result in poor detection performance, sea spikes are identified and eliminated before target detection to improve signal-to-clutter ratio (SCR). By taking the absolute amplitude of signals in the best STFRFT domain (STFRFD) as the test statistic, and comparing it with the threshold determined by a constant false alarm rate detector, micromotion target can be declared or not. STFRFT with Gaussian window is employed to provide time-frequency distribution of m-D signals, and the instantaneous frequency of each component can be extracted and estimated precisely by STFRFD filtering. In the end, datasets from the intelligent pixel processing radar with HH and VV polarizations are used to verify the validity of this proposed algorithm. Two shore-based experiments are also conducted using an X-band sea search radar and an S-band sea surveillance radar, respectively. The results demonstrate that the proposed method not only achieves high detection probability in a low- SCR environment but also outperforms the short-time Fourier transform-based method.

Extracting Micro-Doppler Radar Signatures From Rotating Targets Using Fourier–Bessel Transform and Time–Frequency Analysis

Abstract: In this paper, we report the efficiency of the Fourier-Bessel transform (FBT) and time-frequency (TF)-based method in conjunction with the fractional Fourier transform (FrFT), for extracting micro-Doppler (m-D) radar signatures from the rotating targets. This approach comprises mainly of two processes, with the first being the decomposition of the radar return, in order to extract m-D features, and the second being the TF analysis to estimate motion parameters of the target. In order to extract m-D features from the radar signal returns, the time domain radar signal is decomposed into stationary and nonstationary components using the FBT in conjunction with the FrFT. The components are then reconstructed by applying the inverse Fourier-Bessel transform (IFBT). After the extraction of the m-D features from the target's original radar return, TF analysis is used to estimate the target's motion parameters. This proposed method is also an effective tool for detecting maneuvering air targets in strong sea clutter and is also applied to both simulated data and real-world experimental data.

Detection of a Low Observable Sea-Surface Target With Micromotion via the Radon-Linear Canonical Transform

Abstract: In this letter, a novel long-time coherent integration method, known as the Radon-linear canonical transform (RLCT), is proposed for detection of a low observable moving target in sea clutter. The micro-Doppler (m-D) of a sea-surface target is studied and modeled as multiple linear-frequency-modulated signals, which result from the accelerated and 3-D rotated movements. The RLCT-based algorithm employs m-D as a useful signature for target detection and can simultaneously compensate the range and Doppler migrations during long observation time, which simplifies the operational procedure. By searching along the moving trajectory and using extra three degrees of freedom, the observation values of m-D signals can be well matched and accumulated as peaks in the RLCT domain. Then, the target can be declared by comparing the peak value with an adaptive threshold. The definition of the RLCT demonstrates that it is the generalization of the popular moving target detection, Radon-Fourier transform, fractional Fourier transform, and linear canonical transform methods. Finally, experiments using a real sea clutter data set show that the proposed method can achieve high integration gain and detection probability of a micromotion target in heavy sea clutter.

Convolution, correlation, and sampling theorems for the offset linear canonical transform

Abstract: The offset linear canonical transform (OLCT), which is a time-shifted and frequency-modulated version of the linear canonical transform, has been shown to be a powerful tool for signal processing and optics. However, some basic results for this transform, such as convolution and correlation theorems, remain unknown. In this paper, based on a new convolution operation, we formulate convolution and correlation theorems for the OLCT. Moreover, we use the convolution theorem to investigate the sampling theorem for the band-limited signal in the OLCT domain. The formulas of uniform sampling and low-pass reconstruction related to the OLCT are obtained. We also discuss the design method of the multiplicative filter in the OLCT domain. Based on the model of the multiplicative filter in the OLCT domain, a practical method to achieve multiplicative filtering through convolution in the time domain is proposed.

Optical image encryption based on compressive sensing and chaos in the fractional Fourier domain

Abstract: We propose a novel image encryption algorithm based on compressive sensing (CS) and chaos in the fractional Fourier domain. The original image is dimensionality reduction measured using CS. The measured values are then encrypted using chaotic-based double-random-phase encoding technique in the fractional Fourier transform domain. The measurement matrix and the random-phase masks used in the encryption process are formed from pseudo-random sequences generated by the chaotic map. In this proposed algorithm, the final result is compressed and encrypted. The proposed cryptosystem decreases the volume of data to be transmitted and simplifies the keys for distribution simultaneously. Numerical experiments verify the validity and security of the proposed algorithm.

Multiple-image encryption based on optical wavelet transform and multichannel fractional Fourier transform

Abstract: A multiple-image encryption scheme based on the optical wavelet transform (OWT) and the multichannel fractional Fourier transform (MFrFT) is proposed. The scheme can make full use of multi-resolution decomposition of wavelet transform (WT) and multichannel processing of MFrFT. The mentioned properties can achieve the encryption of multi-image and the encryption of single image. When encryption finished, each image gets its own fractional order and independent keys. Analysis of encrypted effects has been completed. Furthermore, the influence of WT type and order are analyzed, and the application and analysis of MFrFT are accomplished as well. Numerical simulation verifies the feasibility of the scheme and shows that the problem of insufficient capacity is better solved, and the flexibility of scheme increases. A simple opto-electronic mixed device to realize the scheme is proposed.

Detection of high-speed and accelerated target based on the linear frequency modulation radar

Abstract: The high-speed movement of a target may cause range migration and Doppler frequency migration of the radar echo, which has a serious impact on the detection performance of the radar. To resolve the problem of detecting a high-speed target in linear frequency modulation radar, this study analyses the effect on the integration gain caused by range migration and Doppler frequency migration, and proposes a corresponding compensation method according to the different input signal-to-noise ratios (SNRs) of the echo signal. To compensate for range migration in high SNRs, two-dimensional median filtering and constant false alarm rate technology are combined to estimate the speed. For low SNRs, based on coarse valuations, the authors use the discrete Fourier transformation (DFT) to realise the fractional delay cell to improve speed accuracy. Furthermore, to compensate for Doppler frequency migration, an instantaneous cross-correlation method is proposed for high SNRs, which is combined with the fractional Fourier transform method to estimate the acceleration for low SNRs. The input SNR threshold for the different algorithms is then analysed using simulation data, and the theoretical reference value is shown. Finally, the study verifies the effectiveness of the proposed methods through simulation and measured data.

Secure OFDM-PON System Based on Chaos and Fractional Fourier Transform Techniques

Abstract: We propose a novel method to simultaneously improve the physical layer security and the transmission performance of the orthogonal frequency division multiplexing (OFDM) passive optical network system by using chaos and fractional Fourier transform (FrFT) techniques. The designed 3-D chaotic sequences are used to form the training sequence for time synchronization, to perform the OFDM subcarriers masking, and to control the fractional order of the FrFT operation. The analyses show that the whole key space size of the proposed scheme could be beyond 1050, and the peak-to-average-power-ratio of the transmitted OFDM signal can be decreased by about 0.5 dB. Furthermore, we successfully demonstrate an 8.18 Gbps 16-quadrature-amplitude- modulation (QAM)-OFDM data transmission experiment with chaotic and FrFT operations over 25 km single mode fiber. The results show that the proposed scheme could effectively enhance the system security and the transmission performance without additional bandwidth requirement.

Nonlinear optical security system based on a joint transform correlator in the Fresnel domain

Abstract: A new optical security system for image encryption based on a nonlinear joint transform correlator (JTC) in the Fresnel domain (FrD) is proposed. The proposal of the encryption process is a lensless optical system that produces a real encrypted image and is a simplified version of some previous JTC-based encryption systems. We use a random complex mask as the key in the nonlinear system for the purpose of increasing the security of the encrypted image. In order to retrieve the primary image in the decryption process, a nonlinear operation has to be introduced in the encrypted function. The optical decryption process is implemented through the Fresnel transform and the fractional Fourier transform. The security system proposed in this paper preserves the shift-invariance property of the JTC-based encryption system in the Fourier domain, with respect to the lateral displacement of the key random mask in the decryption process. This system shows an improved resistance to chosen-plaintext and known-plaintext attacks, as they have been proposed in the cryptanalysis of the JTC encrypting system. Numerical simulations show the validity of this new optical security system.

High-precision Fourier forward modeling of potential fields

Abstract: We analyzed the numerical forward methods in the Fourier domain for potential fields. Existing Fourier-domain forward methods applied the standard fast Fourier transform (FFT) algorithm to inverse transform a conjugate symmetrical spectrum into a real field. It had significant speed advantages over space-domain forward methods but suffered from problems including aliasing, imposed periodicity, and edge effect. Usually, grid expansion was needed to reduce these errors, which was equivalent to the numerical evaluation of the oscillatory Fourier integral using the trapezoidal rule with smaller steps. We tested a high-precision Fourier-domain forward method based on a combined use of shift-sampling technique and Gaussian quadrature theory. The trapezoidal rule applied by the standard FFT algorithm to evaluate the continuous Fourier transform was modified by introducing a shift parameter ξ. By choosing optimum values of ξ as Gaussian quadrature nodes, we developed a Gauss-FFT method for Fourier forward modeling of potential fields. No grid expansion was needed, the sources can be set near the boundary of the fields or even go beyond the boundary. The Gauss-FFT method converged to the space-domain solution much faster than the standard FFT method with grid expansion. Forward modeling results almost identical to space-domain ones can be obtained in less time. Numerical examples, of both simple and complex 2D and 3D source forward modeling, revealed the reliability and adaptability of the method.

Polynomial Implementation of the Taylor–Fourier Transform for Harmonic Analysis

Abstract: Recently, the Taylor-Fourier transform (TFT) was proposed to analyze the spectrum of signals with oscillating harmonics. The coefficients of this linear transformation were obtained through the calculation of the pseudoinverse matrix, which provides the classical solution to the normal equations of the least-squares (LS) approximation. This paper presents a filtering design technique that obtains the coefficients of the filters at each harmonic by imposing the maximally flat conditions to the polynomials defining their frequency responses. This condition can be used to solve the LS problem at each particular harmonic frequency, without the need of obtaining the whole set, as in the classical pseudoinverse solution. In addition, the filter passband central frequency can follow the fluctuations of the fundamental frequency. Besides, the method offers a reduction of the computational burden of the pseudoinverse solution. An implementation of the proposed estimator as an adaptive algorithm using its own instantaneous frequency estimate to relocate its bands is shown, and several tests are used to compare its performance with that of the ordinary TFT.