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: It has been observed, that the FrFT based MFCC, with timbral features and SVM, efficiently classifies the two western genres of rock and classical music, from the GTZAN dataset, with fewer features and a higher classification accuracy.
Abstract: This paper presents the automatic genre classification of Indian Tamil music and western music using timbral features and fractional Fourier transform (FrFT) based Mel frequency cepstral coefficient (MFCC) features. The classifier model for the proposed system has been built using K-nearest neighbours and support vector machine (SVM) classifiers. In this work, the performance of various features extracted from music excerpts have been analyzed, to identify the appropriate feature descriptors for the two major genres of Indian Tamil music, namely classical music (Carnatic based devotional hymn compositions) and folk music. The results have shown that the feature combination of spectral roll off, spectral flux, spectral skewness and spectral kurtosis, combined with fractional MFCC features, outperforms all other feature combinations, to yield a higher classification accuracy of 96.05 %, as compared to the accuracy of 84.21 % with conventional MFCC. It has also been observed, that the FrFT based MFCC, with timbral features and SVM, efficiently classifies the two western genres of rock and classical music, from the GTZAN dataset, with fewer features and a higher classification accuracy of 96.25 %, as compared to the classification accuracy of 80 % with conventional MFCC.
13 citations
TL;DR: A novel direction of arrival (DOA) estimation algorithms in the Dechirping domain is proposed, which is extended from the conventional multiple signal classification (MUSIC) algorithm in the time domain and performs a better estimation performance, especially for large angular spread and low signal-to-noise ratio.
Abstract: In this paper, we consider the problem of localizing the multiple distributed wideband chirp sources using the fractional Fourier transform. The model in the time domain and that in the fractional Fourier domain derived by the Taylor series expansion are presented respectively. The representation of location vector in the Dechirping domain is illustrated which is only related to the central angle. A novel direction of arrival (DOA) estimation algorithm in the Dechirping domain is proposed, which is extended from the conventional multiple signal classification (MUSIC) algorithm in the time domain. Using this algorithm, except for estimating the DOA, the incidence source number can be determined as well which is allowed to exceed the sensor number in the array. To demonstrate the performance of proposed algorithm, numerical results are conducted. Compared to the previous FrFT-MUSIC algorithm based on the assumption of point source model, the proposed algorithm performs a better estimation performance, especially for large angular spread and low signal-to-noise ratio.
13 citations
Cites background from "The fractional Fourier transform: t..."
...1 Notation and definition The fractional Fourier transform as a linear integral transform with kernel Kα(u, t) [25, 26]:...
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17 Apr 2019
TL;DR: In this article, the authors present a proof corrected by the author available online since mercredi 17 avril 2019 and available since 2017 since the 17th of February 2019, 2017.
Abstract: IRBM - In Press.Proof corrected by the author Available online since mercredi 17 avril 2019
12 citations
TL;DR: In this article, the authors present a study on linear canonical transformation in the framework of a phase space representation of quantum mechanics that they have introduced in their previous work [1].
Abstract: We present a study on linear canonical transformation in the framework of a phase space representation of quantum mechanics that we have introduced in our previous work [1]. We begin with a brief recall about the so called phase space representation. We give the definition of linear canonical transformation with the transformation law of coordinate and momentum operators. We establish successively the transformation laws of mean values, dispersions, basis state and wave functions.Then we introduce the concept of isodispersion linear canonical transformation.
12 citations
Additional excerpts
...This relation looks like a fractional Fourier transformation [13] Y = m1 − s 2O L :<[ 04hi4 4 i0 g/ ]...
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TL;DR: Experimental results show that the gear fault analysis method proposed in this paper is able to effectively extract the weak fault signal.
Abstract: An intelligent analysis method for gear faults based on fractional wavelet transform (FRWT) and support vector machine (SVM) is proposed. Based on this method, FRWT is used to eliminate noise from the gear vibration signal, and the vibration signal after noise elimination is carried thought wavelet packet decomposition and reconstruction. A sequence corresponding to the signal is constructed consisting of the module with the highest level wavelet coefficients after decomposition and feature vectors corresponding to the energy sequence which were obtained by calculation. Then, a particle optimization method is used to optimize SVM parameters, and the feature vectors as training samples are input into SVM for training while the test samples are input for fault recognition. Experimental results show that the gear fault analysis method proposed in this paper is able to effectively extract the weak fault signal. The accuracy rate for identification of the type of gear fault reached 96.7%.
11 citations
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]....
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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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