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A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "General Mathematics, Analysis".

Deadline for manuscript submissions: closed (31 October 2023) | Viewed by 17657

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Special Issue Editors

Prof. Dr. Zunwei Fu

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Department of Computer Science, The University of Suwon, Hwaseong-si, Gyeonggi-do 18323, Republic of Korea
Interests: fractional Fourier transform; fractional integral operator and applications
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Prof. Dr. Bingzhao Li

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Guest Editor

School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China
Interests: signal and image processing; fractional Fourier transform and linear canonical transform theory and method; statistical data analysis and processing
Special Issues, Collections and Topics in MDPI journals

Dr. Xiangyang Lu

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Guest Editor

School of Electronics and Information , Zhongyuan University of Technology, Zhengzhou 450007, China
Interests: signal and image processing; fractional Fourier transform
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

With the rapid development of modern signal processing theory, the processed signal has gradually developed from the early stationary signal to the non-stationary, non-Gaussian, non-single sampling complex signal. As one of the important branches of non-stationary signal processing theory, fractional Fourier transform (FRFT) is favored by many researchers due to its unique characteristics. In recent decades, new research results have emerged in an endless stream. At present, FRFT has been widely used in many fields of scientific research and engineering technology, such as such as swept filter, artificial neural network, wavelet transform, time-frequency analysis, time-varying filtering, complex transmission, partial differential equations, quantum mechanics, etc. In addition, FRFT can also be used to define fractional convolution, correlation, Hilbert transform, Riesz transform, and other operations, and can also be further generalized into the linear canonical transformation.

This Special Issue aims to continue to advance research on topics relating to the theory, algorithm development and application of fractional Fourier transform. Topics that are invited for submission include (but are not limited to):

Mathematical theory of FRFT;
Fractional integral transformation based on FRFT, such as Hilbert transform, Riesz transform;
Applications of FRFT in signal processing, PDE, information security and other fields;
Numerical algorithm of FRFT;
The generalization of FRFT (e.g., the linear canonical transform (LCT), fractional wavelet transforms, and chirp Fourier transform) in theory and applications.

Prof. Dr. Zunwei Fu
Prof. Dr. Bingzhao Li
Dr. Xiangyang Lu
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

fractional Fourier transform
linear canonical transform
digital signal processing
fractional integral operator
partial differential equations

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Related Special Issue

Recent Advances in Fractional Fourier Transforms and Applications, 2nd Edition in Fractal and Fractional (2 articles)

Published Papers (11 papers)

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15 pages, 7046 KiB

Open AccessArticle

Fast Encryption Algorithm Based on Chaotic System and Cyclic Shift in Integer Wavelet Domain

by Yuan-Min Li, Yang Deng, Mingjie Jiang and Deyun Wei

Fractal Fract. 2024, 8(2), 75; https://doi.org/10.3390/fractalfract8020075 - 24 Jan 2024

Cited by 4 | Viewed by 1400

Abstract

This paper introduces a new fast image encryption scheme based on a chaotic system and cyclic shift in the integer wavelet domain. In order to increase the effectiveness and security of encryption, we propose a new diffusion scheme by using bidirectional diffusion and cyclic shift and apply it to our encryption scheme. First, a two-level integer wavelet transform is used to split the plaintext picture into four low-frequency components. Second, we use random sequences generated by Chen’s hyper-chaotic system to scramble four low-frequency components. The initial value is determined by Secure Hash Algorithm 256-bit (SHA256) and user-defined parameters, which increases the plaintext sensitivity. Then, the new diffusion scheme is applied to the matrix containing most of the information and matrices are transformed by a one-level inverse integer wavelet. Finally, to create the ciphertext image, the diffused matrices are subjected to the one-level inverse integer wavelet transform. In the simulation part, we examine the suggested algorithm’s encryption impact. The findings demonstrate that the suggested technique has a sufficient key space and can successfully fend off common attacks. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Fourier Transforms and Applications)

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14 pages, 3239 KiB

Open AccessArticle

The Early Diagnosis of Rolling Bearings’ Faults Using Fractional Fourier Transform Information Fusion and a Lightweight Neural Network

by Fengyun Xie, Gang Li, Chengjie Song and Minghua Song

Fractal Fract. 2023, 7(12), 875; https://doi.org/10.3390/fractalfract7120875 - 10 Dec 2023

Cited by 5 | Viewed by 1549

Abstract

In response to challenges associated with feature extraction and diagnostic models’ complexity in the early diagnosis of bearings’ faults, this paper presents an innovative approach for the early fault diagnosis of rolling bearings. This method combined concepts from frequency domain signal analysis with lightweight neural networks. To begin, vibration signals from rolling bearings were collected using vibration sensors, and the mean square value was utilized as an indicator for accurate early fault signal extraction. Subsequently, employing the fractional Fourier transform, the time domain signal was converted into a frequency domain signal, which provided more detailed frequency feature information. The fusion process combined amplitude frequency and phase frequency information, and was visualized as a Gram angle field map. The lightweight neural network Xception was selected as the primary fault diagnosis tool. Xception, a convolutional neural network (CNN) variant, was chosen for its lightweight design, which maintains excellent performance while significantly reducing model parameters. The experimental results demonstrated that the Xception model excelled in rolling bearing fault diagnosis, particularly when utilizing fused information datasets. This outcome underscores the advantages of combining information fusion and the Xception model to enhance the accuracy of early rolling bearing fault diagnosis, and offers a viable solution for health monitoring and fault diagnosis in industrial settings. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Fourier Transforms and Applications)

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21 pages, 10940 KiB

Open AccessArticle

Parameter Estimation of LFM Signals Based on FOTD-CFRFT under Impulsive Noise

by Houyou Wang, Yong Guo and Lidong Yang

Fractal Fract. 2023, 7(11), 822; https://doi.org/10.3390/fractalfract7110822 - 15 Nov 2023

Cited by 1 | Viewed by 1135

Abstract

Due to the short duration and high amplitude characteristics of impulsive noise, these parameter estimation methods based on Gaussian assumptions are ineffective in the presence of impulsive noise. To address this issue, a LFM signal parameter estimation method is proposed based on FOTD and CFRFT. Firstly, the mathematical expression of FOTD is presented and its tracking performance is verified. Secondly, the tracked signal is subjected to discrete time CFRFT, and a mathematical optimization model for LFM signal parameter estimation is established on the fractional spectrum characteristic. Finally, a correction method for non-standard SαS distributed noise is proposed, and the performance of parameter estimation under both standard and non-standard SαS distributions are analyzed. The simulation results show that this method not only effectively suppresses the impact of impulsive noise on the fractional spectrum of LFM signal, but also has better parameter estimation accuracy and stability in the low GSNR. The proposed method is particularly effective under the measured noise environment, as it successfully suppresses the impact of impulsive noise and achieves high-precision parameter estimation. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Fourier Transforms and Applications)

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15 pages, 1325 KiB

Open AccessArticle

Circuit of Quantum Fractional Fourier Transform

by Tieyu Zhao and Yingying Chi

Fractal Fract. 2023, 7(10), 743; https://doi.org/10.3390/fractalfract7100743 - 8 Oct 2023

Cited by 1 | Viewed by 1602

Abstract

In this paper, we first use the quantum Fourier transform (QFT) and quantum phase estimation (QPE) to realize the quantum fractional Fourier transform (QFrFT). As diverse definitions of the discrete fractional Fourier transform (DFrFT) exist, the relationship between the QFrFT and a classical algorithm is then established; that is, we determine the classical algorithm corresponding to the QFrFT. Second, we observe that many definitions of the multi-fractional Fourier transform (mFrFT) are flawed: when we attempt to propose a design scheme for the quantum mFrFT, we find that there are many invalid weighting terms in the definition of the mFrFT. This flaw may have very significant impacts on relevant algorithms for signal processing and image encryption. Finally, we analyze the circuit of the QFrFT and the reasons for the observed defects. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Fourier Transforms and Applications)

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16 pages, 3925 KiB

Open AccessArticle

Sliding-Window TD-FrFT Algorithm for High-Precision Ranging of LFM Signals in the Presence of Impulse Noise

by Bo Xiao, Xuelian Liu, Chunyang Wang, Yuchao Wang and Tingsheng Huang

Fractal Fract. 2023, 7(9), 679; https://doi.org/10.3390/fractalfract7090679 - 11 Sep 2023

Cited by 2 | Viewed by 1170

Abstract

To address the performance degradation of the conventional linear frequency modulation signal ranging method in the presence of impulse noise, this paper proposes a novel technique that integrates a sliding-window tracking differentiator (TD) with the fractional Fourier transform (FrFT) ranging method. First, the sliding-window TD filtering algorithm is used to suppress the noise in the echo. Subsequently, the filtered signal is subjected to FrFT to calculate the time delay based on the difference in the peak point positions in the fractional domain for realizing target ranging. The simulation results show that the proposed method can effectively suppress impulse noise of different intensities and achieve an accurate and robust ranging of the target. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Fourier Transforms and Applications)

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25 pages, 6647 KiB

Open AccessArticle

Numerical Method for Multi-Dimensional Coupled Forward-Backward Stochastic Differential Equations Based on Fractional Fourier Fast Transform

by Xiaoxiao Zeng, Kexin Fu, Xiaofei Li, Junjie Du and Weiran Fan

Fractal Fract. 2023, 7(6), 441; https://doi.org/10.3390/fractalfract7060441 - 30 May 2023

Cited by 2 | Viewed by 1381

Abstract

Forward-backward stochastic differential equations (FBSDEs) have received more and more attention in the past two decades. FBSDEs can be applied to many fields, such as economics and finance, engineering control, population dynamics analysis, and so on. In most cases, FBSDEs are nonlinear and high-dimensional and cannot be obtained as analytic solutions. Therefore, it is necessary and important to design their numerical approximation methods. In this paper, a novel numerical method of multi-dimensional coupled FBSDEs is proposed based on a fractional Fourier fast transform (FrFFT) algorithm, which is used to compute the Fourier and inverse Fourier transforms. For the forward component of FBSDEs, time discretization is used as well as the backward equation to yield a recursive system with terminal conditions. For the numerical experiments to be successful, three types of numerical methods were used to solve the problem, which ensured the efficiency and speed of computation. Finally, the numerical methods for different examples are verified. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Fourier Transforms and Applications)

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18 pages, 798 KiB

Open AccessArticle

Fast Linear Canonical Transform for Nonequispaced Data

by Yannan Sun and Wenchao Qian

Fractal Fract. 2023, 7(5), 353; https://doi.org/10.3390/fractalfract7050353 - 26 Apr 2023

Cited by 1 | Viewed by 1107

Abstract

The investigations of the discrete and fast linear canonical transform (LCT) are becoming one of the hottest research topics in modern signal processing and optics. Among them, the fast calculation of LCT for non-uniform data is one of key problems. Focus on this problem, a new fast algorithm of the LCT has been proposed in this paper firstly by interpolation and approximation theory. The proposed algorithms can calculate quickly the LCT of the data, whether the input or output data is uniform. Secondly, the complexity and precision of derived algorithms have been analyzed for different situations. Finally, the experimental results are presented to verify the correctness of the obtained results. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Fourier Transforms and Applications)

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22 pages, 3367 KiB

Open AccessArticle

2D Linear Canonical Transforms on L^p and Applications

by Yinuo Yang, Qingyan Wu and Seong-Tae Jhang

Fractal Fract. 2023, 7(2), 100; https://doi.org/10.3390/fractalfract7020100 - 17 Jan 2023

Cited by 5 | Viewed by 1516

Abstract

As Fourier transformations of

L^{p}

functions are the mathematical basis of various applications, it is necessary to develop

L^{p}

theory for 2D-LCT before any further rigorous mathematical investigation of such transformations. In this paper, we study this

L^{p}

theory for [...] Read more.

As Fourier transformations of

L^{p}

functions are the mathematical basis of various applications, it is necessary to develop

L^{p}

theory for 2D-LCT before any further rigorous mathematical investigation of such transformations. In this paper, we study this

L^{p}

theory for

1 \leq p < \infty

. By defining an appropriate convolution, we obtain a result about the inverse of 2D-LCT on

L^{1} (R^{2})

. Together with the Plancherel identity and Hausdorff–Young inequality, we establish

L^{p} (R^{2})

multiplier theory and Littlewood–Paley theorems associated with the 2D-LCT. As applications, we demonstrate the recovery of the

L^{1} (R^{2})

signal function by simulation. Moreover, we present a real-life application of such a theory of 2D-LCT by encrypting and decrypting real images. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Fourier Transforms and Applications)

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14 pages, 10034 KiB

Open AccessArticle

LFM Signal Parameter Estimation via FTD-FRFT in Impulse Noise

by Xuelian Liu, Xuemei Li, Bo Xiao, Chunyang Wang and Bo Ma

Fractal Fract. 2023, 7(1), 69; https://doi.org/10.3390/fractalfract7010069 - 7 Jan 2023

Cited by 3 | Viewed by 1560

Abstract

LFM signals are widely applied in radar, communication, sonar and many other fields. LFM signals received by these systems contain a lot of noise and outliers. In order to suppress the interference of strong impulse noise on target signals and realize the accurate estimation of LFM signal parameters, the impulse noise of echo signals need to be filtered. In this paper, to solve the problem of poor performance of LFM signal parameter estimation based on fractional Fourier transform in impulse noise, alpha stable distribution is used to establish the mathematical model of impulse noise. The proposed fastest tracking differentiator with an adaptive tracking factor is used to suppress the strong impulse noise, and fractional Fourier transform is used to estimate the parameter of the LFM signals. The experimental results show that the proposed fastest tracking differentiator with an adaptive tracking factor has a good filtering performance. It can effectively filter the impulse noise in the echo signal and allows the FrFT method to accurate estimate the parameters of the LFM signals in strong impulse noise. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Fourier Transforms and Applications)

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13 pages, 556 KiB

Open AccessArticle

Approximation Theorems Associated with Multidimensional Fractional Fourier Transform and Applications in Laplace and Heat Equations

by Yinuo Yang, Qingyan Wu, Seong Tae Jhang and Qianqian Kang

Fractal Fract. 2022, 6(11), 625; https://doi.org/10.3390/fractalfract6110625 - 26 Oct 2022

Cited by 11 | Viewed by 1525

Abstract

In this paper, we establish two approximation theorems for the multidimensional fractional Fourier transform via appropriate convolutions. As applications, we study the boundary and initial problems of the Laplace and heat equations with chirp functions. Furthermore, we obtain the general Heisenberg inequality with [...] Read more.

(This article belongs to the Special Issue Recent Advances in Fractional Fourier Transforms and Applications)

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Other

Jump to: Research

13 pages, 354 KiB

Open AccessBrief Report

Unlimited Sampling Theorem Based on Fractional Fourier Transform

by Hui Zhao and Bing-Zhao Li

Fractal Fract. 2023, 7(4), 338; https://doi.org/10.3390/fractalfract7040338 - 18 Apr 2023

Cited by 2 | Viewed by 1562

Abstract

The recovery of bandlimited signals with high dynamic range is a hot issue in sampling research. The unlimited sampling theory expands the recordable range of traditional analog-to-digital converters (ADCs) arbitrarily, and the signal is folded back into a low dynamic range measurement, avoiding the saturation problem. Since the non-bandlimited signal in the Fourier domain cannot be directly applied to its existing theory, the non-bandlimited signal in the Fourier domain may be bandlimited in the fractional Fourier domain. Therefore, this brief report studies the unlimited sampling problem of high dynamic non-bandlimited signals in the Fourier domain based on the fractional Fourier transform. Firstly, a mathematical signal model for unlimited sampling is proposed. Secondly, based on this mathematical model, the annihilation filtering method is used to estimate the arbitrary folding time. Finally, a novel fractional Fourier domain unlimited sampling theorem is obtained. The theory proves that, based on the folding characteristics of the self-reset ADC, the number of samples is not affected by the modulo threshold, and any folding time can be handled. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Fourier Transforms and Applications)

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