Fractional Fourier transform as a signal processing tool: An overview of recent developments
TL;DR: This paper is geared toward signal processing practitioners by emphasizing the practical digital realizations and applications of the FRFT, which is closely related to other mathematical transforms, such as time-frequency and linear canonical transforms.
About: This article is published in Signal Processing.The article was published on 2011-06-01 and is currently open access. It has received 335 citations till now. The article focuses on the topics: Fractional Fourier transform.
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Book•
28 Feb 2013
TL;DR: Introduction to Fourier Analysis Linear Time-Frequency Representations Quadratic Time- frequency Distributions Higher Order Time-f frequency Representations Analysis of Non-Stationary Noisy Signals Some Applications of Time- Frequency Analysis.
Abstract: The culmination of more than twenty years of research, this authoritative resource provides you with a practical understanding of time-frequency signal analysis. The book offers in-depth coverage of critical concepts and principles, along with discussions on key applications in a wide range of signal processing areas, from communications and optics, to radar and biomedicine. Supported with over 140 illustrations and more than 1,700 equations, this detailed reference explores the topics you need to understand for your work in the field, such as Fourier analysis, linear time frequency representations, quadratic time-frequency distributions, higher order time-frequency representations, and analysis of non-stationary noisy signals. This unique book also serves as an excellent text for courses in this area, featuring numerous examples and problems at the end of each chapter.
187 citations
TL;DR: The deep neural network is adopted to recognize faults in bogies and provides a new paradigm for fault diagnosis of the high-speed train with big data and plays an important role in this field.
Abstract: Bogies are an important component of high-speed trains. The level of mechanical performance of bogies has a major influence on the safety and reliability of high-speed train. Therefore, conducting fault diagnoses on bogies with big data is very important. Fault mechanisms of bogies are very complex, and feature signals are nonobvious. For these reasons, fault information of bogies cannot be effectively extracted using the traditional signal processing method. Therefore, this paper adopted the deep neural network to recognize faults in bogies. The deep neural network offers numerous benefits in this context. Using deep neural networks, fault information in a signal spectrum can be extracted in a selfadaptive method. This technique is free of dependence on extensive signal processing knowledge and diagnostic experience. Compared with the traditional intelligent diagnosis method, the deep neural network can obtain a higher diagnostic accuracy. Additionally, the deep neural network does not depend on the sample size, and it can obtain high diagnostic accuracy even when the sample size is relatively small. It also achieves very high diagnostic accuracy applied to high-speed trains with different speeds and different faults, which shows that the method is extensively applicable. Furthermore, the recognition accuracy rate of the deep neural network under normal conditions can reach 100%. This method provides a new paradigm for fault diagnosis of the high-speed train with big data and plays an important role in this field.
169 citations
Cites methods from "Fractional Fourier transform as a s..."
...FRFT [28], [29] is only a signal processing method which is used to convert a time-domain signal into the frequency-...
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TL;DR: Some of the most important developments in the last two decades related to the concept of the IF, performance analysis ofIF estimators, and development of IF estimators for low SNR environments are reviewed.
137 citations
TL;DR: In this paper, the authors proposed an orthogonal chirp-division multiplexing (OCDM) for high-speed communication, which can efficiently exploit multipath diversity and thus outperform the OFDM, and that it is more resilient against the interference due to insufficient guard interval than single-carrier frequency domain equalization.
Abstract: Chirp waveform plays a significant role in radar and communication systems for its ability of pulse compression and spread spectrum. This paper presents a principle of multiplexing a bank of orthogonal chirps, termed orthogonal chirp-division multiplexing (OCDM) for high-speed communication. As Fourier transform is the kernel of orthogonal frequency-division multiplexing (OFDM), which achieves the maximum spectral efficiency (SE) of frequency-division multiplexing, Fresnel transform underlies the proposed OCDM system, which achieves the maximum SE of chirp spread spectrum. By using discrete Fresnel transform, digital implementation of OCDM is introduced. According to the properties of Fresnel transform, the transmission of OCDM signal in linear time-invariant channel is studied. Efficient digital signal processing is proposed for channel dispersion compensation. The implementation of the OCDM system is discussed with the emphasis on its compatibility to the OFDM system; it is shown that it can be easily integrated into the existing OFDM systems. Finally, the simulations are provided to validate the feasibility of the proposed OCDM. It is shown that the OCDM system can efficiently exploit multipath diversity and thus outperforms the OFDM, and that it is more resilient against the interference due to insufficient guard interval than single-carrier frequency-domain equalization.
131 citations
TL;DR: A class of methods in TFA, parameterised TFA is focused on, summarizing its latest research progress and related engineering applications, so as to provide reference and guidance for researchers applying parametric TFA in different fields.
130 citations
References
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TL;DR: The authors briefly introduce the functional Fourier transform and a number of its properties and present some new results: the interpretation as a rotation in the time-frequency plane, and the FRFT's relationships with time- frequencies such as the Wigner distribution, the ambiguity function, the short-time Fouriertransform and the spectrogram.
Abstract: The functional Fourier transform (FRFT), which is a generalization of the classical Fourier transform, was introduced a number of years ago in the mathematics literature but appears to have remained largely unknown to the signal processing community, to which it may, however, be potentially useful. The FRFT depends on a parameter /spl alpha/ and can be interpreted as a rotation by an angle /spl alpha/ in the time-frequency plane. An FRFT with /spl alpha/=/spl pi//2 corresponds to the classical Fourier transform, and an FRFT with /spl alpha/=0 corresponds to the identity operator. On the other hand, the angles of successively performed FRFTs simply add up, as do the angles of successive rotations. The FRFT of a signal can also be interpreted as a decomposition of the signal in terms of chirps. The authors briefly introduce the FRFT and a number of its properties and then present some new results: the interpretation as a rotation in the time-frequency plane, and the FRFT's relationships with time-frequency representations such as the Wigner distribution, the ambiguity function, the short-time Fourier transform and the spectrogram. These relationships have a very simple and natural form and support the FRFT's interpretation as a rotation operator. Examples of FRFTs of some simple signals are given. An example of the application of the FRFT is also given. >
1,698 citations
"Fractional Fourier transform as a s..." refers background in this paper
...In very simple terms, the fractional Fourier transform (FRFT) is a generalization of the ordinary Fourier transform [1]....
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Book•
01 Jan 2000
TL;DR: The Handbook of Image and Video Processing contains a comprehensive and highly accessible presentation of all essential mathematics, techniques, and algorithms for every type of image and video processing used by scientists and engineers.
Abstract: 1.0 INTRODUCTION 1.1 Introduction to Image and Video Processing (Bovik) 2.0 BASIC IMAGE PROCESSING TECHNIQUES 2.1 Basic Gray-Level Image Processing (Bovik) 2.2 Basic Binary Image Processing (Desai/Bovik) 2.3 Basic Image Fourier Analysis and Convolution (Bovik) 3.0 IMAGE AND VIDEO PROCESSING Image and Video Enhancement and Restoration 3.1 Basic Linear Filtering for Image Enhancement (Acton/Bovik) 3.2 Nonlinear Filtering for Image Enhancement (Arce) 3.3 Morphological Filtering for Image Enhancement and Detection (Maragos) 3.4 Wavelet Denoising for Image Enhancement (Wei) 3.5 Basic Methods for Image Restoration and Identification (Biemond) 3.6 Regularization for Image Restoration and Reconstruction (Karl) 3.7 Multi-Channel Image Recovery (Galatsanos) 3.8 Multi-Frame Image Restoration (Schulz) 3.9 Iterative Image Restoration (Katsaggelos) 3.10 Motion Detection and Estimation (Konrad) 3.11 Video Enhancement and Restoration (Lagendijk) Reconstruction from Multiple Images 3.12 3-D Shape Reconstruction from Multiple Views (Aggarwal) 3.13 Image Stabilization and Mosaicking (Chellappa) 4.0 IMAGE AND VIDEO ANALYSIS Image Representations and Image Models 4.1 Computational Models of Early Human Vision (Cormack) 4.2 Multiscale Image Decomposition and Wavelets (Moulin) 4.3 Random Field Models (Zhang) 4.4 Modulation Models (Havlicek) 4.5 Image Noise Models (Boncelet) 4.6 Color and Multispectral Representations (Trussell) Image and Video Classification and Segmentation 4.7 Statistical Methods (Lakshmanan) 4.8 Multi-Band Techniques for Texture Classification and Segmentation (Manjunath) 4.9 Video Segmentation (Tekalp) 4.10 Adaptive and Neural Methods for Image Segmentation (Ghosh) Edge and Boundary Detection in Images 4.11 Gradient and Laplacian-Type Edge Detectors (Rodriguez) 4.12 Diffusion-Based Edge Detectors (Acton) Algorithms for Image Processing 4.13 Software for Image and Video Processing (Evans) 5.0 IMAGE COMPRESSION 5.1 Lossless Coding (Karam) 5.2 Block Truncation Coding (Delp) 5.3 Vector Quantization (Smith) 5.4 Wavelet Image Compression (Ramchandran) 5.5 The JPEG Lossy Standard (Ansari) 5.6 The JPEG Lossless Standard (Memon) 5.7 Multispectral Image Coding (Bouman) 6.0 VIDEO COMPRESSION 6.1 Basic Concepts and Techniques of Video Coding (Barnett/Bovik) 6.2 Spatiotemporal Subband/Wavelet Video Compression (Woods) 6.3 Object-Based Video Coding (Kunt) 6.4 MPEG-I and MPEG-II Video Standards (Ming-Ting Sun) 6.5 Emerging MPEG Standards: MPEG-IV and MPEG-VII (Kossentini) 7.0 IMAGE AND VIDEO ACQUISITION 7.1 Image Scanning, Sampling, and Interpolation (Allebach) 7.2 Video Sampling and Interpolation (Dubois) 8.0 IMAGE AND VIDEO RENDERING AND ASSESSMENT 8.1 Image Quantization, Halftoning, and Printing (Wong) 8.2 Perceptual Criteria for Image Quality Evaluation (Pappas) 9.0 IMAGE AND VIDEO STORAGE, RETRIEVAL AND COMMUNICATION 9.1 Image and Video Indexing and Retrieval (Tsuhan Chen) 9.2 A Unified Framework for Video Browsing and Retrieval (Huang) 9.3 Image and Video Communication Networks (Schonfeld) 9.4 Image Watermarking (Pitas) 10.0 APPLICATIONS OF IMAGE PROCESSING 10.1 Synthetic Aperture Radar Imaging (Goodman/Carrera) 10.2 Computed Tomography (Leahy) 10.3 Cardiac Imaging (Higgins) 10.4 Computer-Aided Detection for Screening Mammography (Bowyer) 10.5 Fingerprint Classification and Matching (Jain) 10.6 Probabilistic Models for Face Recognition (Pentland/Moghaddam) 10.7 Confocal Microscopy (Merchant/Bartels) 10.8 Automatic Target Recognition (Miller) Index
1,678 citations
TL;DR: In this article, a generalized operational calculus is developed, paralleling the familiar one for the ordinary transform, which provides a convenient technique for solving certain classes of ordinary and partial differential equations which arise in quantum mechanics from classical quadratic hamiltonians.
Abstract: We introduce the concept of Fourier transforms of fractional order, the ordinary Fourier transform being a transform of order 1. The integral representation of this transform can be used to construct a table of fractional order Fourier transforms. A generalized operational calculus is developed, paralleling the familiar one for the ordinary transform. Its application provides a convenient technique for solving certain classes of ordinary and partial differential equations which arise in quantum mechanics from classical quadratic hamiltonians. The method of solution is first illustrated by its application to the free and to the forced quantum mechanical harmonic oscillator. The corresponding Green's functions are obtained in closed form. The new technique is then extended to three-dimensional problems and applied to the quantum mechanical description of the motion of electrons in a constant magnetic field. The stationary states, energy levels and the evolution of an initial wave packet are obtained by a systematic application of the rules of the generalized operational calculus. Finally, the method is applied to the second order partial differential equation with time-dependent coefficients describing the quantum mechanical dynamics of electrons in a time-varying magnetic field.
1,523 citations
"Fractional Fourier transform as a s..." refers background or methods in this paper
...The FRFT can be useful in terms of differential equations [7,126], especially for solving these equations....
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...More specifically, the eigenvalues of the FRFT are the a-th root of the eigenvalues of the Fourier transform [7]:...
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...In particular, Namias solved several Shrodinger equations using this assumption [7]....
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...Namias provided an elegant generalization of the Fourier transform to the FRFT [7], by deriving the FRFT from the eigenfunctions of the Fourier transform....
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...the Mehler formula, Namias in fact shows that we obtain the FRFT from (8) [7]....
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Book•
31 Jan 2001
TL;DR: The fractional Fourier transform (FFT) as discussed by the authors has been used in a variety of applications, such as matching filtering, detection, and pattern recognition, as well as signal recovery.
Abstract: Preface. Acknowledgments. Introduction. Signals, Systems, and Transformations. Wigner Distributions and Linear Canonical Transforms. The Fractional Fourier Transform. Time-Order and Space-Order Representations. The Discrete Fractional Fourier Transform. Optical Signals and Systems. Phase-Space Optics. The Fractional Fourier Transform in Optics. Applications of the Fractional Fourier Transform to Filtering, Estimation, and Signal Recovery. Applications of the Fractional Fourier Transform to Matched Filtering, Detection, and Pattern Recognition. Bibliography on the Fractional Fourier Transform. Other Cited Works. Credits. Index.
1,287 citations
"Fractional Fourier transform as a s..." refers background or methods in this paper
...It should be mentioned that the LCT is also called the affine Fourier transform, the generalized Fresnel transform, the Collins formula, the ABCD transform, or the almost Fourier and almost Fresnel transformation [1,45]....
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...The LCT is defined by [1,45,46]...
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...Further examples can be found in [1,7,126]....
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...The above equation can be considered as the spectral decomposition of the kernel function of FRFT [1]....
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...Introductory and review contributions [1,4–15]...
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TL;DR: An algorithm for efficient and accurate computation of the fractional Fourier transform for signals with time-bandwidth product N, which computes the fractionsal transform in O(NlogN) time.
Abstract: An algorithm for efficient and accurate computation of the fractional Fourier transform is given. For signals with time-bandwidth product N, the presented algorithm computes the fractional transform in O(NlogN) time. A definition for the discrete fractional Fourier transform that emerges from our analysis is also discussed.
1,034 citations
"Fractional Fourier transform as a s..." refers background or methods in this paper
..., [57,58]) and the fact that most of these provide a satisfactory approximation to the continuous transform....
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...proposed two innovative approaches for obtaining the DFRFT through sampling of the FRFT [58]....
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...However, both presented cases assumed that the WD of x(t) is zero outside an origin-centered circle of diameter equal to the sampling period [58]....
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...This is a desirable property for a definition of the DFRFT matrix [58]....
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...In order to alleviate some of the problems associated with the DFRFT proposed in [58], a new type of DFRFT, which is unitary, reversible, and flexible, was proposed in [56]....
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