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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17 Sep 2018
TL;DR: A sampling approach based on Compressive Sensing (CS) in the context of reflectometry is proposed and allows the detection of multiple reflection peaks caused by the defects at a sampling frequency 10 times lower than the actual sampling rate with a relative fault location error of 2%.
Abstract: Reflectometry is a structural health monitoring technique that allows to efficiently detect and localize electrical defects in wire networks. The main challenge in reflectometry is to improve the precision of defect localization and characterization, especially in the case of complex networks. The solution is to increase the frequency of the injected signal since the spatial resolution is inversely proportional to the injected signal frequency. However, such solution applicability is limited by the sampling capabilities of existing Analog-to-Digital Converters (ADC). In this paper, we propose a sampling approach based on Compressive Sensing (CS) in the context of reflectometry. The resulting methodology offers the possibility to inject high frequency signals and later to reconstruct the reflected waveform from a lower set of samples than that required in the classical sampling scheme. In that respect, a complex linear chirp signal is considered as a testing signal and injected in a complex Y-branches network with a hard defect at the edges. In order to have a sparse representation, the reflected chirp signal is decomposed in the Fractional Fourier Transform (FrFT) domain. The main result is that the new acquisition scheme allows the detection of multiple reflection peaks caused by the defects at a sampling frequency 10 times lower than the actual sampling rate with a relative fault location error of 2%.
5 citations
Cites methods from "The fractional Fourier transform: t..."
...Here, the fractional powers F of the ordinary Fourier transform operation F correspond to the rotation by the angle π 2 in the time-frequency plane [15]....
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23 Sep 2011
TL;DR: A novel blind separation approach using fractional Fourier transform (FRFT) is presented, which is particularly suitable for nonlinear signal and simulation results have shown that the method is feasible.
Abstract: In this paper, a novel blind separation approach using fractional Fourier transform(FRFT) is presented. Fractional Fourier transform is particularly suitable for nonlinear signal. Blind Source Separation using FRFT is suitable for dealing with non-stationary signal. And simulation results have shown that the method is feasible.
4 citations
TL;DR: In this article, a blind third-order dispersion estimation method based on fractional Fourier transformation (FrFT) was proposed for optical fiber communication system, which is robust against nonlinear and amplified spontaneous emission (ASE) noise.
Abstract: In this paper, we propose a blind third-order dispersion estimation method based on fractional Fourier transformation (FrFT) in optical fiber communication system. By measuring the chromatic dispersion (CD) at different wavelengths, this method can estimation dispersion slope and further calculate the third-order dispersion. The simulation results demonstrate that the estimation error is less than 2 % in 28GBaud dual polarization quadrature phase-shift keying (DP-QPSK) and 28GBaud dual polarization 16 quadrature amplitude modulation (DP-16QAM) system. Through simulations, the proposed third-order dispersion estimation method is shown to be robust against nonlinear and amplified spontaneous emission (ASE) noise. In addition, to reduce the computational complexity, searching step with coarse and fine granularity is chosen to search optimal order of FrFT. The third-order dispersion estimation method based on FrFT can be used to monitor the third-order dispersion in optical fiber system.
4 citations
TL;DR: A new JTF-based method is proposed for image formation in inverse synthetic aperture radars (ISAR), which uses minimum entropy criterion for optimum parameter adjustment of JTF algorithms and shows that α-order FrFT with local adjustment has much better performance than the other methods in this category even in low SNR.
Abstract: Conventional radar imaging systems use Fourier transform for image formation, but due to the target's complicated motion the Doppler spectrum is time-varying and thus the reconstructed image becomes blurred even after applying standard motion compensation algorithms. Therefore, sophisticated algorithms such as polar reformatting are usually employed to produce clear images. Alternatively, Joint Time-Frequency (JTF) analysis can be used for image formation which produces clear image without using polar reformatting algorithm. In this paper, a new JTF-based method is proposed for image formation in inverse synthetic aperture radars (ISAR). This method uses minimum entropy criterion for optimum parameter adjustment of JTF algorithms. Short Time Fourier Transform (STFT) and Fractional Fourier Transform (FrFT) are applied as JTF for time-varying Doppler spectrum analysis. Both the width of Gaussian window of STFT and the order of FrFT, α, are adjusted using minimum entropy as local and total measures. Furthermore, a new statistical parameter, called normalized correlation, is defined for comparison of images reconstructed by different methods. Simulation results show that α-order FrFT with local adjustment has much better performance than the other methods in this category even in low SNR.
4 citations
TL;DR: In this paper, the fractional Fourier transform is applied to the spectral bands of two component mixture containing oxfendazole and oxyclozanide to provide the multicomponent quantitative prediction of the related substances.
Abstract: In this paper, the fractional Fourier transform (FrFT) is applied to the spectral bands of two component mixture containing oxfendazole and oxyclozanide to provide the multicomponent quantitative prediction of the related substances. With this aim in mind, the modulus of FrFT spectral bands are processed by the continuous Mexican Hat family of wavelets, being denoted by MEXH-CWT-MOFrFT. Four modulus sets are obtained for the parameter $$a$$
of the FrFT going from 0.6 up to 0.9 in order to compare their effects upon the spectral and quantitative resolutions. Four linear regression plots for each substance were obtained by measuring the MEXH-CWT-MOFrFT amplitudes in the application of the MEXH family to the modulus of the FrFT. This new combined powerful tool is validated by analyzing the artificial samples of the related drugs, and it is applied to the quality control of the commercial veterinary samples.
4 citations
Cites background from "The fractional Fourier transform: t..."
...There is not a ‘best’ definition of the FrFT, and one should rather try to take the most suitable one while modeling a process or considering a mathematical problem [11–13]....
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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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