Fractional Fourier Series Research Articles

Fractional Fourier Series on the Torus and Applications

Fractional Fourier Series on the Torus and Applications

Solution of fractional Laplace type equation in conformable sense using fractional fourier series with separation of variables technique

Solution of fractional Laplace type equation in conformable sense using fractional fourier series with separation of variables technique

The minimality of mean square error in chirp approximation using fractional fourier series and fractional fourier transform

Chirps are familiar in nature, have a built-in resistance to noise and interference, and are connected to a wide range of highly oscillatory processes. Detecting chirp oscillating patterns by traditional Fourier series is challenging because the chirp frequencies constantly change over time. Estimating such types of functions considering the partial sums of a Fourier series in Fourier analysis does not permit an approximate solution, which entails more Fourier coefficients required for signal reconstruction. The standard Fourier series, therefore, has a poor convergence rate and is an inadequate approximation. In this study, we use a parameterized orthonormal basis with an adjustable parameter to match the oscillating behavior of the chirp to approximate linear chirps using the partial sums of a generalized Fourier series known as fractional Fourier series, which gives the best approximation with only a small number of fractional Fourier coefficients. We used the fractional Fourier transform to compute the fractional Fourier coefficients at sample points. Additionally, we discover that the fractional parameter has the best value at which fractional Fourier coefficients of zero degrees have the most considerable magnitude, leading to the rapid decline of fractional Fourier coefficients of high degrees. Furthermore, fractional Fourier series approximation with optimal fractional parameters provides the minimum mean square error over the fractional Fourier parameter domain.

Synthesis of unequally-spaced arrays using the fractional Fourier series

This paper introduces an algebraic method to synthesize the radiation pattern of unequally-spaced arrays. In this method, first, the characteristic matrix, containing the sampled data of the desired pattern, is established. Then, the fractional Fourier series is calculated using the distinct eigenvalues and eigenvectors of the characteristic matrix. In the following, the location of the array elements is estimated using the eigenvalues by considering the mutual coupling effect. The magnitude and phase of the excitation currents are computed using the least square method. Also, it is proved that the number of elements can be reduced using the eigenvalue decomposition and differential spectrum method. Several important patterns are investigated to verify the performance of the proposed method, and then a comprehensive discussion is expressed for all cases. It is shown that using the proposed method, a greater average spacing is achieved which allows bettering mitigate the phenomenon of mutual coupling. Furthermore, the various results show that the proposed method offers a better approximation of the desired array factor especially for beam-shaped patterns. This method also has a noble ability to reduce the number of array elements compared to a generic reference array, while still retaining a good ability to approximate the desired pattern fairly faithfully.

Application of Fractional Fourier series in Evaluating Fractional Integrals

Application of Fractional Fourier series in Evaluating Fractional Integrals

Fractional Fourier series Expansion of Two Types of Fractional Trigonometric Functions

Fractional Fourier series Expansion of Two Types of Fractional Trigonometric Functions

Fractional Fourier Series with Separation of Variables Technique and Its Application on Fractional differential Equations

When using some classical methods, such us separation of variables; it is impossible to find a general solution for some differential equations. Therefore, we suggest adding conformable fractional Fourier series to get a new technique to solve fractional Benjamin Bana Mahony and Heat Equations. Furtheremore, we give new numerical approximation for functions using mathematica coding called conformable fractional Fourier series approximation

Tensor Product and Certain Solutions of Fractional Wave Type Equation

In this paper we Önd certain solutions of some fractional partial di§erential equations. Tensor product of Banach spaces is used to Önd some solutions where separation of variables does not work. We solve the fractional wave type equation using fractional Fourier Series

Fractional fourier series and its applications

In this article, we use a new multiplication to propose a definition of fractional Fourier series, regarding the Jumarie type of modified Riemann-Liouville fractional derivatives. Furthermore, several examples are provided to illustrate the applications of fractional Fourier series.

Conformable Derivatives in Laplace Equation and Fractional Fourier Series Solution

In this paper the solution of conformable Laplace equation, \frac{\partial^{\alpha}u(x,y)}{\partial x^{\alpha}}+ \frac{\partial^{\alpha}u(x,y)}{\partial y^{\alpha}}=0, where 1 < α ≤ 2 has been deduced by using fractional fourier series and separation of variables method. For special cases α =2 (Laplace's equation), α=1.9, and α=1.8 conformable fractional fourier coefficients have been calculated. To calculate coefficients, integrals are of type "conformable fractional integral".

Fractional Fourier Analysis Using the Möbius Inversion Formula

Compared with the Fourier analysis, the fractional Fourier analysis is more suitable to process linear frequency modulation type non-stationary functions. To the best of our knowledge, the theoretical framework of the fractional Fourier analysis has not well established yet, especially for the fractional Fourier series (FrFS) and the discrete fractional Fourier transform (DFrFT) algorithms. To tackle with these problems, the efficient FrFS and DFrFT algorithms based on the Mobius function are proposed. First, the existence and applicability of the FrFS are analyzed basing on the Mobius inversion formula. Second, two kinds of fast algorithms for the infinite/finite FrFS are proposed. Then, based on the amplitude scaling relationship between the FrFS and samples of the FrFT, two efficient DFrFT algorithms are obtained, which are noted as the arithmetic discrete fractional Fourier transform (ADFrFT)-I and the ADFrFT-II. Importantly, the multiplication complexity of the two proposed ADFrFT algorithms is reduced to O(M), which is less than that of the state-of-the-art DFrFT algorithms. The parallel butterfly structure of the ADFrFT-II algorithm is suitable for the very large scale integration implementation. Finally, the simulations justify the efficiency of the ADFrFT algorithms in filtering and parameter evaluation of radar and optical signals.

A multichannel compressed sampling method for fractional bandlimited signals

This paper presents a multichannel compressed sampling scheme based on the modulated wideband converter and the fractional Fourier transform. The system consists of modulators, analog filters, and analog-to-digital converters. The analog signal is multiplied by a chirp signal and a bank of periodic waveforms. After filtering, the product is uniformly sampled at a rate that is considerably lower than the Nyquist rate. The proposed method which is valid for fractional multiband signals is proven based on the fractional Fourier series. The robustness, recovery accuracy and influence of the fractional order are analyzed by the empirical probability of the successful recovery and the mean squared error.

SOLUTION OF SOME CONFORMABLE FRACTIONAL DIFFERENTIAL EQUATIONS

In this paper, we solve some conformable fractional partial dif- ferential equations including the wave equation. While the solution of such equations is often either numerical or too long, in our case we use fractional Fourier series and separation of variables, where solutions turned out to be easy and interesting.

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.

Resolution of the Gibbs Phenomenon for Fractional Fourier Series

Resolution of the Gibbs Phenomenon for Fractional Fourier Series

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.