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 new anti-jamming technique against interrupted sampling jamming of self-protection repeater jammer is proposed without the knowledge of the jamming parameters, which is based on fractional Fourier transform that can separate the overlapping true target echo and jamming pulses in the fractional domain.
Abstract:
Interrupted sampling repeater jammer generates multiple false targets to confuse chirp radar systems. In practical situations, maintaining separation between true target echo and jamming signal is not possible because the jamming pulses and the true target echo are overlapping in both time and frequency domains. A new anti-jamming technique against interrupted sampling jamming of self-protection repeater jammer is proposed without the knowledge of the jamming parameters. The proposed technique is based on fractional Fourier transform that can separate the overlapping true target echo and jamming pulses in the fractional domain, and then the resulting pulses are returned to the time domain, the true target can be easily distinguished from the false ones because the jamming pulses lag behind the true target echo by the jammer's delay. The theoretical analysis and simulation results demonstrate the efficiency and validity of the proposed technique.
Journal Article•
TL;DR: This paper proposes a robust video watermarking algorithm that makes use of Fractional Fourier Transform, Singular Value Decomposition and Visual Cryptography and generates two shares namely the Master share and the Ownership share, for each selected frame of the video that is to be watermarked.
Abstract: When the digital world was at its infant stage, it was quite complicated and required a high level of expertise to duplicate digital contents so that they look like the original. However, in the present digital world, this is not true. Today, it is made so easy that almost anyone can duplicate, reproduce or manipulate digital data, with no degradation in data quality. As a consequence there was a search for a mechanism for providing the copyright protection scheme for providing the authentication of digital data. Digital watermarking is one of the most widely used technologies to perform this task. The basic idea of watermarking is to hide some information (the watermark) into a digital media, so that it can be later extracted or detected for a variety of purposes like identification and authentication. In this paper we propose a robust video watermarking algorithm that makes use of Fractional Fourier Transform (FrFt), Singular Value Decomposition (SVD) and Visual Cryptography (VC) and generates two shares namely the Master share and the Ownership share, for each selected frame of the video that is to be watermarked. The two shares of a particular frame will give no information individually about the embedded watermark. Visual cryptography is used to reveal the embedded watermark by combining the two shares of a frame. The properties of FrFt and SVD make the watermarking more robust to common signal processing attacks. The probability of proving the ownership is also more in case of attacks like frame insertion and frame deletion.
31 Jan 2020
TL;DR: In this article, the kernel of N-dimensional fractional Fourier transform up to n-dimensional was developed by extending the definition of first dimensional FF transform and the properties of kernel up to N-dimensions were also presented.
Abstract: In this paper we have developed the kernel of N-dimensional fractional Fourier transform by extending the definition of first dimensional fractional Fourier transform. The properties of kernel up to N- dimensional are also presented here which is missing in the literature of fractional Fourier transform. The properties of kernel of fractional Fourier transforms up to N- dimensional will help the researcher to extend their research in this aspect.
Cites background from "The fractional Fourier transform: t..."
...1-Dimensional Fractional Fourier Transform The operator of FRFT has an order parameter α that is an arbitrary angle a π 2 , and it will reduce to Fourier transform whenever the rotation of angle α is π 2 a detailed discussion is found in [1-4] The one-dimensional fractional Fourier transform is defined as...
[...]
Posted Content•
TL;DR: A double encryption algorithm that uses the lack of invertibility of the fractional Fourier transform (FRFT) on L^{1} to recover the encrypted signal on the FRFT domain.
Abstract: We provide a double encryption algorithm that uses the lack of invertibility of the fractional Fourier transform (FRFT) on $L^{1}$. One encryption key is a function, which maps a "good" $L^{2}(\mathbb{R})$ signal to a "bad" $L^{1}(\mathbb{R})$ signal. The FRFT parameter which describes the rotation associated with this operator on the time-frequency plane provides the other encryption key. With the help of approximate identities, such as of the Abel and Gauss means of the FRFT established in \cite{CFGW}, we recover the encrypted signal on the FRFT domain. This design of an encryption algorithm seems new even when using the classical Fourier transform.
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]....
[...]