The fractional Fourier transform: theory, implementation and error analysis
TL;DR: It is hoped that this implementation and fixed-point error analysis will lead to a better understanding of the issues involved in finite register length implementation of the discrete fractional Fourier transform and will help the signal processing community make better use of the transform.
About: This article is published in Microprocessors and Microsystems.The article was published on 2003-11-03. It has received 128 citations till now. The article focuses on the topics: Fractional Fourier transform & Non-uniform discrete Fourier transform.
Citations
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TL;DR: In this paper, the concept of fractional sums, fractional powers of a square matrix, tolerant computing and FracSets, negative probabilities and fractional delay discrete-time linear systems are introduced.
Abstract: This paper studies several topics related with the concept of “fractional” that are not directly related with Fractional Calculus, but can help the reader in pursuit new research directions We introduce the concept of non-integer positional number systems, fractional sums, fractional powers of a square matrix, tolerant computing and FracSets, negative probabilities, fractional delay discrete-time linear systems, and fractional Fourier transform
34 citations
TL;DR: In this paper, a basis dependent rigged Hilbert spaces (RHS) of the line and half-line is constructed, and the complete sub-structure of both RHS and of the operators acting on them is obtained from their algebraic structures or from related fractional Fourier transforms.
Abstract: Simultaneous use of discrete and continuous bases in quantum systems is not possible in the context of Hilbert spaces, but only in the more general structure of rigged Hilbert spaces (RHS). In addition, the relevant operators in RHS (but not in Hilbert space) are a realization of elements of a Lie enveloping algebra and support representations of semigroups. We explicitly construct here basis dependent RHS of the line and half-line and relate them to the universal enveloping algebras of the Weyl-Heisenberg algebra and su(1, 1), respectively. The complete sub-structure of both RHS and of the operators acting on them is obtained from their algebraic structures or from the related fractional Fourier transforms. This allows us to describe both quantum and signal processing states and their dynamics. Two relevant improvements are introduced: (i) new kinds of filters related to restrictions to subspaces and/or the elimination of high frequency fluctuations and (ii) an operatorial structure that, starting from fix objects, describes their time evolution.
33 citations
TL;DR: This paper systematically analyze and compare the main DFRFT types: sampling-type DFR FTs and eigenvector decomposition-typeDFRFTs and discrete counterparts of the linear canonical transform (LCT), simplified FRFT (SFRFT) are summarized and classified.
33 citations
TL;DR: A novel algorithm to denoise medical images using the technique of blind source separation (BSS) and the fractional Fourier transform to provide superior and stable denoising of medical images is proposed.
32 citations
TL;DR: The proposed spectral–spatial hyperspectral anomaly detection method, based on fractional Fourier transform and saliency weighted collaborative representation, outperforms other nine well-known compared methods in terms of area under the receiver operating characteristic (ROC) curve values, visual detection characteristics, ROC curve, and separability.
Abstract: Anomaly target detection methods for hyperspectral images (HSI) often have the problems of potential anomalies and noise contamination when representing background. Therefore, a spectral–spatial hyperspectral anomaly detection method is proposed in this article, which is based on fractional Fourier transform (FrFT) and saliency weighted collaborative representation. First, hyperspectral pixels are projected to the fractional Fourier domain by the FrFT, which can enhance the capability of the detector to suppress the noise and make anomalies to be more distinctive. Then, a saliency weighted matrix is designed as the regularization matrix referring to context-aware saliency theory and combined with the FrFT-based collaborative representation detector. The saliency-weighted regularization matrix assigns different pixels with different weights by using both spectral and spatial information, which can reduce the influence of the potential anomalous pixels embedded in the background when applying collaborative representation theory. Finally, to further improve the performance of the proposed method, a spectral–spatial detection procedure is employed to calculate final anomaly scores by using both spectral information and spatial information. The proposed method is compared with nine state-of-the-art hyperspectral anomaly detection methods on six HSI datasets, including two synthetic HSI datasets and four real-world HSI datasets. Extensive experimental results illustrate that the proposed method's detection performance outperforms other nine well-known compared methods in terms of area under the receiver operating characteristic (ROC) curve values, visual detection characteristics, ROC curve, and separability.
28 citations
Cites methods from "The fractional Fourier transform: t..."
...The FrFT is an extension version of the traditional Fourier transform (FT) [37]....
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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
"The fractional Fourier transform: t..." refers background in this paper
...It is a well-documented fact that the FRFT of a signal corresponds to the rotation of the Wigner distribution of that signal by the required angle a [1]....
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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
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
TL;DR: This definition is based on a particular set of eigenvectors of the DFT matrix, which constitutes the discrete counterpart of the set of Hermite-Gaussian functions, and is exactly unitary, index additive, and reduces to the D FT for unit order.
Abstract: We propose and consolidate a definition of the discrete fractional Fourier transform that generalizes the discrete Fourier transform (DFT) in the same sense that the continuous fractional Fourier transform generalizes the continuous ordinary Fourier transform. This definition is based on a particular set of eigenvectors of the DFT matrix, which constitutes the discrete counterpart of the set of Hermite-Gaussian functions. The definition is exactly unitary, index additive, and reduces to the DFT for unit order. The fact that this definition satisfies all the desirable properties expected of the discrete fractional Fourier transform supports our confidence that it will be accepted as the definitive definition of this transform.
604 citations
01 Aug 1972
TL;DR: The groundwork is set through a discussion of the relationship between the binary representation of numbers and truncation or rounding, and a formulation of a statistical model for arithmetic roundoff, to illustrate techniques of working with particular models.
Abstract: When digital signal processing operations are implemented on a computer or with special-purpose hardware, errors and constraints due to finite word length are unavoidable. The main categories of finite register length effects are errors due to A/D conversion, errors due to roundoffs in the arithmetic, constraints on signal levels imposed by the need to prevent overflow, and quantization of system coefficients. The effects of finite register length on implementations of linear recursive difference equation digital filters, and the fast Fourier transform (FFT), are discussed in some detail. For these algorithms, the differing quantization effects of fixed point, floating point, and block floating point arithmetic are examined and compared. The paper is intended primarily as a tutorial review of a subject which has received considerable attention over the past few years. The groundwork is set through a discussion of the relationship between the binary representation of numbers and truncation or rounding, and a formulation of a statistical model for arithmetic roundoff. The analyses presented here are intended to illustrate techniques of working with particular models. Results of previous work are discussed and summarized when appropriate. Some examples are presented to indicate how the results developed for simple digital filters and the FFT can be applied to the analysis of more complicated systems which use these algorithms as building blocks.
333 citations
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