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
More filters
TL;DR: In this article, a class of fractional Fourier integral operators on classes of function spaces known as ultra-Boehmians is investigated and a convolution product is established as a product of different functions.
Abstract: In this paper, an attempt is being made to investigate a class of fractional Fourier integral operators on classes of function spaces known as ultraBoehmians. We introduce a convolution product and establish a convolution theorem as a product of different functions. By employing the convolution theorem and making use of an appropriate class of approximating identities, we provide necessary axioms and define function spaces where the fractional Fourier integral operator is an isomorphism connecting the different spaces. Further, we provide an inversion formula and obtain various properties of the cited integral in the generalized sense.
11 citations
Cites background from "The fractional Fourier transform: t..."
...Indeed, it has applications in the solution of ordinary differential equations, quantum optics (Garcia et al. [11]), quantum mechanics (Andez [12]), optical systems (Narayanana and Prabhu [13]), time filtering (Narayanana and Prabhu [13]), and some pattern recognitions (see Zayed and Garcia [14]) and Zayed [15, 16]) as well)....
[...]
...Narayanana, V.A., Prabhu, K.: The fractional Fourier transform: theory, implementation and error analysis....
[...]
...[11]), quantum mechanics (Andez [12]), optical systems (Narayanana and Prabhu [13]), time filtering (Narayanana and Prabhu [13]), and some pattern recognitions (see Zayed and Garcia [14]) and Zayed [15, 16]) as well)....
[...]
Journal Article•
TL;DR: In this article, a pseudo Wigner distribution in the Fractional Fourier transform (FT) domain is proposed to analyze single or multi-component chirp signals.
Abstract: A new pseudo Wigner distribution (PWD) in the Fractional Fourier transform (FT) domain is proposed to analyze single or multi-component chirp signals. Rotated short-time Fourier transform (STFT) is used to realize the proposed distribution, and the appropriate fractional domain is found from the knowledge of the second-order fractional FT moments. By analyzing the signal in the most appropriate fractional domain, the proposed time-frequency distribution preserves the WD auto-terms and cancels the cross-terms when there are several signal components. The proposed method is easy to perform without significant additional computational cost. Simulation results prove its qualitative advantage in the time-frequency representation when the calculation is done in the optimal fractional domain.
11 citations
TL;DR: This paper introduces a novel technique based on fractional Fourier transform to discriminate between the true target echo and those false targets in the case of frequency-shifting jammers.
Abstract: Self-protection deceptive jammers create at the radar receiver output multiple-false targets that are impossible to isolate in both time and frequency domains. In this paper, we introduce a novel technique based on fractional Fourier transform (FrFT) to discriminate between the true target echo and those false targets in the case of frequency-shifting jammers. In fact, we exploit the capability of the FrFT to resolve, in a matched manner, spectra that are overlapping in time and frequency. This is a property that cannot be achieved using a standard matched filter. The theoretical analysis of this technique is presented and its effectiveness is verified by simulation.
10 citations
01 Jun 2016
TL;DR: In this paper, the authors proposed a new method combining the Short Time Fourier Transform (STFT) and Zoom-FRFT, to estimate the MLFM signal parameters, which can improve both the parameters estimation precision and the computation cost significantly.
Abstract: Traditional parameters estimation methods for the Multi-component Linear Frequency Modulation (MLFM) signal based on Fractional Fourier Transform (FRFT) could not achieve the satisfactory precision and the lesser computation cost. In this paper, we propose a new method combining the Short Time Fourier Transform (STFT) and Zoom-FRFT, to estimate the MLFM signal parameters. Firstly, the coarse estimation can be achieved from the straight line detection of short time Fourier spectrum. Then, we could calculate the analysis range of transform order and Fractional domain spectrum by the FRFT domain spectrum distribution characteristics of interference signal. Finally, we can estimate the optimal order and precise peak position by the optimum seeking method in the Zoom-FRFT domain. The simulation results show that, this method can improve both the parameters estimation precision and the computation cost significantly, and can flexibly choose the window width and the zoom times.
10 citations
Cites methods from "The fractional Fourier transform: t..."
...Compared with the Fourier transform, the time information can be integrated into the spectrum change using STFT. 112978-1-5090-1781-2/16/ $31.00 ©2016 IEEE The STFT is defined as: 2, j fuSTFT t f x t g u t e du (1) where x t is the signal, and g t is the window function....
[...]
TL;DR: In this article, the authors employ scanning near-field optical microscopy, full-vector finite difference time domain numerical simulations and fractional Fourier transformation to investigate the near field and propagation behavior of the electromagnetic energy scattered at 1.56 µm by dielectric arrays of silicon nitride nanopillars with chiral 1-Vogel spiral geometry.
Abstract: In this work, we employ scanning near-field optical microscopy, full-vector finite difference time domain numerical simulations and fractional Fourier transformation to investigate the near-field and propagation behavior of the electromagnetic energy scattered at 1.56µm by dielectric arrays of silicon nitride nanopillars with chiral 1-Vogel spiral geometry. In particular, we experimentally study the spatial evolution of scattered radiation and demonstrate near-field coupling between adjacent nanopillars along the parastichies arms. Moreover, by measuring the spatial distribution of the scattered radiation at different heights from the array plane, we demonstrate a characteristic rotation of the scattered field pattern consistent with net transfer of orbital angular momentum in the Fresnel zone, within a few micrometers from the plane of the array. Our experimental results agree with the simulations we performed and may be of interest to nanophotonics applications.
9 citations
Cites background from "The fractional Fourier transform: t..."
...Given a function f(u), under the same conditions in which the standard Fourier transform exists, we can define the ath order FRFT fa(u) with a being a real number in several equivalent ways [20, 24]....
[...]
...More information on alternative definitions, generalizations and the many properties of FRFTs can be found in [20, 24]....
[...]
References
More filters
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]....
[...]
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
Additional excerpts
...[9–11]....
[...]