scholarly journals Quasiprobability Distribution Functions from Fractional Fourier Transforms

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 344 ◽  
Author(s):  
Jorge Anaya-Contreras ◽  
Arturo Zúñiga-Segundo ◽  
Héctor Moya-Cessa
Keyword(s):  
Fourier Transform ◽  
Characteristic Function ◽  
Fourier Transforms ◽  
Fractional Fourier Transform ◽  
Distribution Functions ◽  
Fractional Fourier Transforms ◽  
Fresnel Transform ◽  
Quasiprobability Distribution

We show, in a formal way, how a class of complex quasiprobability distribution functions may be introduced by using the fractional Fourier transform. This leads to the Fresnel transform of a characteristic function instead of the usual Fourier transform. We end the manuscript by showing a way in which the distribution we are introducing may be reconstructed by using atom-field interactions.

Wavelet packets associated with linear canonical transform on spectrum

2021 ◽  
pp. 2150030
Author(s):  
M. Younus Bhat ◽  
Aamir H. Dar
Keyword(s):  
Fourier Transform ◽  
Multiresolution Analysis ◽  
Fourier Transforms ◽  
Fractional Fourier Transform ◽  
Integral Transforms ◽  
Wavelet Packets ◽  
Linear Canonical Transform ◽  
Fresnel Transform ◽  
The Fourier Transform ◽  
Vector Valued

The linear canonical transform (LCT) provides a unified treatment of the generalized Fourier transforms in the sense that it is an embodiment of several well-known integral transforms including the Fourier transform, fractional Fourier transform, Fresnel transform. Using this fascinating property of LCT, we, in this paper, constructed associated wavelet packets. First, we construct wavelet packets corresponding to nonuniform Multiresolution analysis (MRA) associated with LCT and then those corresponding to vector-valued nonuniform MRA associated with LCT. We investigate their various properties by means of LCT.

scholarly journals The Weighted Fractional Fourier Transform and Its Application in Image Encryption

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Tieyu Zhao ◽  
Qiwen Ran
Keyword(s):  
Fourier Transform ◽  
Theoretical Analysis ◽  
Image Encryption ◽  
Fourier Transforms ◽  
Fractional Fourier Transform ◽  
Rapid Development ◽  
Matrix Group ◽  
Future Developments ◽  
Security And Reliability ◽  
Fractional Fourier Transforms

With the rapid development of information, the requirements for the security and reliability of cryptosystems have become increasingly difficult to meet, which promotes the development of the theory of a class of fractional Fourier transforms. In this paper, we present a review of the development and applications of the weighted fractional Fourier transform (WFRFT) in image encryption. Relationships between the algorithms are established using the generalized permutation matrix group in theoretical analysis. In addition, the advantages and potential weaknesses of each algorithm in image encryption are analyzed and discussed. It is expected that this review will provide a clear picture of the current developments of the WFRFT in image encryption and may shed some light on future developments.

scholarly journals Uncertainty principles invariant under the fractional Fourier transform

1991 ◽  
Vol 33 (2) ◽  
pp. 180-191 ◽  
Author(s):  
David Mustard
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Fractional Fourier Transform ◽  
Intrinsic Property ◽  
Continuous Group ◽  
Uncertainty Principles ◽  
New Family ◽  
Special Cases ◽  
Fractional Fourier Transforms ◽  
The Fourier Transform

AbstractUncertainty principles like Heisenberg's assert an inequality obeyed by some measure of joint uncertainty associated with a function and its Fourier transform. The more groups under which that measure is invariant, the more that measure represents an intrinsic property of the underlying object represented by the given function. The Fourier transform is imbedded in a continuous group of operators, the fractional Fourier transforms, but the Heisenberg measure of overall spread turns out not to be invariant under that group. A new family is developed of measures that are invariant under the group of fractional Fourier transforms and that obey associated uncertainty principles. The first member corresponds to Heisenberg's measure but is generally smaller than his although equal to it in special cases.

scholarly journals Some Structures of Parallel VLSI-Oriented Processing Units for Implementation of Small Size Discrete Fractional Fourier Transforms

Electronics ◽  
2019 ◽  
Vol 8 (5) ◽  
pp. 509 ◽  
Author(s):  
Aleksandr Cariow ◽  
Janusz Papliński ◽  
Dorota Majorkowska-Mech
Keyword(s):  
Fourier Transform ◽  
Parallel Algorithms ◽  
Fourier Transforms ◽  
Fractional Fourier Transform ◽  
Processing Unit ◽  
Discrete Hartley Transform ◽  
Discrete Fractional Fourier Transform ◽  
Discrete Orthogonal ◽  
Fractional Fourier Transforms ◽  
Control Circuits

Discrete orthogonal transforms such as the discrete Fourier transform, discrete cosine transform, discrete Hartley transform, etc., are important tools in numerical analysis, signal processing, and statistical methods. The successful application of transform techniques relies on the existence of efficient fast algorithms for their implementation. A special place in the list of transformations is occupied by the discrete fractional Fourier transform (DFrFT). In this paper, some parallel algorithms and processing unit structures for fast DFrFT implementation are proposed. The approach is based on the resourceful factorization of DFrFT matrices. Some parallel algorithms and processing unit structures for small size DFrFTs such as N = 2, 3, 4, 5, 6, and 7 are presented. In each case, we describe only the most important part of the structures of the processing units, neglecting the description of the auxiliary units and the control circuits.

scholarly journals THEKERNEL OF N- DIMENSIONAL FRACTIONAL FOURIER TRANSFORM

2020 ◽  
Vol 7 (1) ◽  
pp. 36-41
Author(s):  
Zakia Abdul Wahid ◽  
Saleem Iqbal ◽  
Farhana Sarwar ◽  
Abdul Rehman
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Fractional Fourier Transform ◽  
Fractional Fourier Transforms ◽  
Definition Of

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.

scholarly journals Discrete Pseudo-Fractional Fourier Transform and Its Fast Algorithm

Electronics ◽  
2021 ◽  
Vol 10 (17) ◽  
pp. 2145
Author(s):  
Dorota Majorkowska-Mech ◽  
Aleksandr Cariow
Keyword(s):  
Fourier Transform ◽  
Fast Algorithm ◽  
Fourier Transforms ◽  
Fractional Fourier Transform ◽  
Composite Number ◽  
Fractional Fourier Transforms ◽  
Fractional Transform

In this article, we introduce a new discrete fractional transform for data sequences whose size is a composite number. The main kernels of the introduced transform are small-size discrete fractional Fourier transforms. Since the introduced transformation is not, in the generally known sense, a classical discrete fractional transform, we call it discrete pseudo-fractional Fourier transform. We also provide a generalization of this new transform, which depends on many fractional parameters. A fast algorithm for computing the introduced transform is developed and described.

scholarly journals Transformasi Fourier Fraksional dari Fungsi Gaussian

2019 ◽  
Vol 16 (1) ◽  
pp. 19
Author(s):  
IIN SUTRISNA ◽  
Asriadi Nasrun ◽  
Mawardi Bahri ◽  
Syamsuddin Toaha
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Fractional Fourier Transform ◽  
Gaussian Function ◽  
Fractional Fourier Transforms

The fractional Fourier transform is one of the generalizations of ordinary Fourier transform that depend on a particular angle . In this paper we will derive the fractional Fourier transforms of a function that is well known in the field of analysis, namely Gaussian function.

Generalized signal-tuned Gabor approach for signal representation and analysis

2017 ◽  
Vol 28 (01) ◽  
pp. 1750001 ◽  
Author(s):  
José R. A. Torreão
Keyword(s):  
Fourier Transform ◽  
Visual Cortex ◽  
Fourier Transforms ◽  
Fractional Fourier Transform ◽  
Receptive Fields ◽  
Gabor Functions ◽  
Plausible Model ◽  
Spectral Signal ◽  
Potential Applications ◽  
Space Frequency

The signal-tuned Gabor approach is based on spatial or spectral Gabor functions whose parameters are determined, respectively, by the Fourier and inverse Fourier transforms of a given “tuning” signal. The sets of spatial and spectral signal-tuned functions, for all possible frequencies and positions, yield exact representations of the tuning signal. Moreover, such functions can be used as kernels for space-frequency transforms which are tuned to the specific features of their inputs, thus allowing analysis with high conjoint spatio-spectral resolution. Based on the signal-tuned Gabor functions and the associated transforms, a plausible model for the receptive fields and responses of cells in the primary visual cortex has been proposed. Here, we present a generalization of the signal-tuned Gabor approach which extends it to the representation and analysis of the tuning signal’s fractional Fourier transform of any order. This significantly broadens the scope and the potential applications of the approach.

Certain Fourier Transforms of Distributions

1951 ◽  
Vol 3 ◽  
pp. 140-144 ◽  
Author(s):  
Eugene Lukacs ◽  
Otto Szasz
Keyword(s):  
Probability Distribution ◽  
Characteristic Function ◽  
Fourier Transforms ◽  
Probability Distributions ◽  
Distribution Functions ◽  
Characteristic Functions

Fourier transforms of distribution functions are frequently studied in the theory of probability. In this connection they are called characteristic functions of probability distributions. It is often of interest to decide whether a given function φ(t) can be the characteristic function of a probability distribution, that is, whether it admits the representation

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