Local Fractional Fourier Research Articles

Local discrete fractional fourier transform: An algorithm for calculating partial points of DFrFT

Fractional Fourier transform (FrFT) is widely utilized in nonstationary signal processing owing to its linearity and flexibility. Various discrete algorithms focus on the global discrete FrFT (DFrFT) points. However, only a part of the DFrFT points are considered because of a priori knowledge of applications and limited memory space of digital machines. Therefore, a local DFrFT (LDFrFT) was proposed in this study. LDFrFT for the first Q continuous DFrFT points is proposed, and its complexity was reduced from O(Mlog2(M)) to O(Mlog2(Q)) for an M-point sequence with Q being a factor of M. It is further extended to the LDFrFT for any Q continuous DFrFT points having complexity less than that of the DFrFT. These two algorithms were then generalized to local discrete linear canonical transform algorithms. The LDFrFT uses all sampling points, thus maintaining the resolution of the DFrFT. Moreover, LDFrFT is not limited by the starting point and is more flexible than the recently proposed local DFrFT algorithm. Finally, simulated frequency-modulated continuous-wave radar signals, gravitational waves, and real-world bat echolocation chirp data were utilized to demonstrate the effectiveness of the LDFrFT, which works well when the SNR is over -30 dB.

Hyperspectral anomaly detection with local correlation fractional Fourier transform and vector pulse coupled neural network

Hyperspectral anomaly detection has attracted widespread attention because of its significant practical application value. However, the existence of noise and the lack of target spatial shape and texture features in hyperspectral image (HSI) limit the performance of hyperspectral anomaly detection. Here, we present a novel hyperspectral anomaly detection method by local correlation fractional Fourier transform (FrFT) and vector pulse coupled neural network (VPCNN). First, combined with the correlation between local pixels, the HSI is transformed into the fractional Fourier domain through the proposed local correlation FrFT, which can suppress noise and background while preserving anomaly targets features in the HSI. Second, the VPCNN, which can effectively deal with the lack of spatial shape and texture features of the target, is proposed to segment the transformed HSI into different regions to obtain the segmentation image. Finally, anomaly detection result is extracted from the segmentation image. Experimental results on four real-world datasets show that the proposed method has better superiority and advancement compared with other state-of-the-art alternative methods. We hope that our research will inspire the application of segmentation methods in related fields.

A Novel High-Frequency Vibration Error Estimation and Compensation Algorithm for THz-SAR Imaging Based on Local FrFT.

Compared with microwave synthetic aperture radar (SAR), terahertz SAR (THz-SAR) is easier to achieve ultrahigh-resolution image due to its higher frequency and shorter wavelength. However, higher carrier frequency makes THz-SAR image quality very sensitive to high-frequency vibration error of motion platform. Therefore, this paper proposes a novel high-frequency vibration error estimation and compensation algorithm for THz-SAR imaging based on local fractional Fourier transform (LFrFT). Firstly, the high-frequency vibration error of the motion platform is modeled as a simple harmonic motion and THz-SAR echo signal received in each range pixel can be considered as a sinusoidal frequency modulation (SFM) signal. A novel algorithm for the parameter estimation of the SFM signal based on LFrFT is proposed. The instantaneous chirp rate of the SFM signal is estimated by determining the matched order of LFrFT in a sliding small-time window and the vibration acceleration is obtained. Hence, the vibration frequency can be estimated by the spectrum analysis of estimated vibration acceleration. With the estimated vibration acceleration and vibration frequency, the SFM signal is reconstructed. Then, the corresponding THz-SAR imaging algorithm is proposed to estimate and compensate the phase error caused by the high-frequency vibration error of the motion platform and realize high-frequency vibration error estimation and compensation for THz-SAR imaging. Finally, the effectiveness of the novel algorithm proposed in this paper is demonstrated by simulation results.

Application of local fractional fourier sine transform for 1-D local fractional heat transfer equation

This paper proposes a new method called the local fractional Fourier sine transform to solve fractional differential equations on a fractal space. The method takes full advantages of the Yang-Fourier transform, the local fractional Fourier cosine, and sine transforms. A 1-D local fractional heat transfer equation is used as an example to reveal the merits of the new technology, and the example can be used as a paradigm for other applications.

Local fractional Fourier method for solving modified diffusion equations with local fractional derivative

Local fractional Fourier method for solving modified diffusion equations with local fractional derivative

Local Fractional Fourier Series Method for Solving Nonlinear Equations with Local Fractional Operators

We apply the local fractional Fourier series method for solving nonlinear equation with local fractional operators. This method is the coupling of the local fractional Fourier series expansion method with other methods, such as the Yang-Laplace transformation method and the local fractional power series method, which effectively separates the variables of partial differential equation. Some testing nonlinear equations and equation systems are given to demonstrate the accuracy and applicability of the proposed approach.

Solving Initial-Boundary Value Problems for Local Fractional Differential Equation by Local Fractional Fourier Series Method

The initial-boundary value problems for the local fractional differential equation are investigated in this paper. The local fractional Fourier series solutions with the nondifferential terms are obtained. Two illustrative examples are given to show efficiency and accuracy of the presented method to process the local fractional differential equations.

Local Fractional Fourier Series Solutions for Nonhomogeneous Heat Equations Arising in Fractal Heat Flow with Local Fractional Derivative

The fractal heat flow within local fractional derivative is investigated. The nonhomogeneous heat equations arising in fractal heat flow are discussed. The local fractional Fourier series solutions for one-dimensional nonhomogeneous heat equations are obtained. The nondifferentiable series solutions are given to show the efficiency and implementation of the present method.

Local Fractional Function Decomposition Method for Solving Inhomogeneous Wave Equations with Local Fractional Derivative

We propose the local fractional function decomposition method, which is derived from the coupling method of local fractional Fourier series and Yang-Laplace transform. The forms of solutions for local fractional differential equations are established. Some examples for inhomogeneous wave equations are given to show the accuracy and efficiency of the presented technique.

Signal Processing for Nondifferentiable Data Defined on Cantor Sets: A Local Fractional Fourier Series Approach

From the signal processing point of view, the nondifferentiable data defined on the Cantor sets are investigated in this paper. The local fractional Fourier series is used to process the signals, which are the local fractional continuous functions. Our results can be observed as significant extensions of the previously known results for the Fourier series in the framework of the local fractional calculus. Some examples are given to illustrate the efficiency and implementation of the present method.

Local Fractional Functional Method for Solving Diffusion Equations on Cantor Sets

The analytical solutions for the diffusion equations on Cantor sets with the nondifferentiable terms are discussed by using the local fractional functional method, which is a coupling method for local fractional Fourier series and Laplace transform.

Mathematical aspects of the Heisenberg uncertainty principle within local fractional Fourier analysis

In this paper, we discuss the mathematical aspects of the Heisenberg uncertainty principle within local fractional Fourier analysis. The Schrodinger equation and Heisenberg uncertainty principles are structured within local fractional operators.

Analysis of Fractal Wave Equations by Local Fractional Fourier Series Method

The fractal wave equations with local fractional derivatives are investigated in this paper. The analytical solutions are obtained by using local fractional Fourier series method. The present method is very efficient and accurate to process a class of local fractional differential equations.

Mappings for Special Functions on Cantor Sets and Special Integral Transforms via Local Fractional Operators

The mappings for some special functions on Cantor sets are investigated. Meanwhile, we apply the local fractional Fourier series, Fourier transforms, and Laplace transforms to solve three local fractional differential equations, and the corresponding nondifferentiable solutions were presented.

1-D heat conduction in a fractal medium: A solution by the local fractional Fourier series method

In this communication 1-D heat conduction in a fractal medium is solved by the local fractional Fourier series method. The solution developed allows relating the basic properties of the fractal medium to the local heat transfer mechanism.

Local Fractional Fourier Series with Application to Wave Equation in Fractal Vibrating String

We introduce the wave equation in fractal vibrating string in the framework of the local fractional calculus. Our particular attention is devoted to the technique of the local fractional Fourier series for processing these local fractional differential operators in a way accessible to applied scientists. By applying this technique we derive the local fractional Fourier series solution of the local fractional wave equation in fractal vibrating string and show the fundamental role of the Mittag‐Leffler function.