scholarly journals Poisson Summation Formulae Associated with the Special Affine Fourier Transform and Offset Hilbert Transform

2017 ◽  
Vol 2017 ◽  
pp. 1-5 ◽  
Author(s):  
Zhi-Hai Zhuo
Keyword(s):  
Fourier Transform ◽  
Hilbert Transform ◽  
Fractional Fourier Transform ◽  
The Novel ◽  
Generalized Poisson ◽  
Summation Formulae ◽  
Points Of View ◽  
The Relationship ◽  
Transform Domains ◽  
Poisson Summation

This paper investigates the generalized pattern of Poisson summation formulae from the special affine Fourier transform (SAFT) and offset Hilbert transform (OHT) points of view. Several novel summation formulae are derived accordingly. Firstly, the relationship between SAFT (or OHT) and Fourier transform (FT) is obtained. Then, the generalized Poisson sum formulae are obtained based on above relationships. The novel results can be regarded as the generalizations of the classical results in several transform domains such as FT, fractional Fourier transform, and the linear canonical transform.

scholarly journals The Solvability of a Class of Convolution Equations Associated with 2D FRFT

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1928
Author(s):  
Zhen-Wei Li ◽  
Wen-Biao Gao ◽  
Bing-Zhao Li
Keyword(s):  
Fourier Transform ◽  
Convolution Operator ◽  
Fractional Fourier Transform ◽  
Hankel Transform ◽  
Polar Coordinates ◽  
Two Dimensional ◽  
Spatial Shift ◽  
Fractional Hankel Transform ◽  
The Relationship ◽  
Convolution Equations

In this paper, the solvability of a class of convolution equations is discussed by using two-dimensional (2D) fractional Fourier transform (FRFT) in polar coordinates. Firstly, we generalize the 2D FRFT to the polar coordinates setting. The relationship between 2D FRFT and fractional Hankel transform (FRHT) is derived. Secondly, the spatial shift and multiplication theorems for 2D FRFT are proposed by using this relationship. Thirdly, in order to analyze the solvability of the convolution equations, a novel convolution operator for 2D FRFT is proposed, and the corresponding convolution theorem is investigated. Finally, based on the proposed theorems, the solvability of the convolution equations is studied.

scholarly journals Weil type representations and automorphic forms

1980 ◽  
Vol 77 ◽  
pp. 145-166 ◽  
Author(s):  
Toshiaki Suzuki
Keyword(s):  
Dirichlet Series ◽  
Functional Equations ◽  
Theta Series ◽  
Type Representation ◽  
Generalized Poisson ◽  
Summation Formulae ◽  
Reciprocity Law ◽  
Weil Type ◽  
Generalized Theta Series ◽  
Poisson Summation

During 1934-1936, W. L. Ferrar investigated the relation between summation formulae and Dirichlet series with functional equations, inspired by Voronoi’s works (1904) on summation formula with respect to the numbers of divisors. In [11] A. Weil showed that the automorphic properties of theta series are expressed by generalized Poisson summation formulae with respect to the so-called Weil representation. On the other hand, T. Kubota, in his study of the reciprocity law in a number field, defined a generalized theta series and a generalized Weil type representation of SL(2, C) and obtained similar results to those of A. Weil (1970-1976, [5], [6], [7]). The methods, used by W. L. Ferrar and T. Kubota, to obtain (generalized Poisson) summation formulae depend similarly on functional equations of Dirichlet series (attached to the generalized theta series).

scholarly journals Discrete Transforms and Matrix Rotation Based Cancelable Face and Fingerprint Recognition for Biometric Security Applications

Entropy ◽  
2020 ◽  
Vol 22 (12) ◽  
pp. 1361
Author(s):  
Abeer D. Algarni ◽  
Ghada El Banby ◽  
Sahar Ismail ◽  
Walid El-Shafai ◽  
Fathi E. Abd El-Samie ◽  
...  
Keyword(s):  
Fourier Transform ◽  
Fractional Fourier Transform ◽  
Fingerprint Recognition ◽  
Discrete Wavelet ◽  
Security Level ◽  
Cancelable Biometrics ◽  
Security Applications ◽  
Discrete Transforms ◽  
Transform Domains ◽  
Biometric Templates

The security of information is necessary for the success of any system. So, there is a need to have a robust mechanism to ensure the verification of any person before allowing him to access the stored data. So, for purposes of increasing the security level and privacy of users against attacks, cancelable biometrics can be utilized. The principal objective of cancelable biometrics is to generate new distorted biometric templates to be stored in biometric databases instead of the original ones. This paper presents effective methods based on different discrete transforms, such as Discrete Fourier Transform (DFT), Fractional Fourier Transform (FrFT), Discrete Cosine Transform (DCT), and Discrete Wavelet Transform (DWT), in addition to matrix rotation to generate cancelable biometric templates, in order to meet revocability and prevent the restoration of the original templates from the generated cancelable ones. Rotated versions of the images are generated in either spatial or transform domains and added together to eliminate the ability to recover the original biometric templates. The cancelability performance is evaluated and tested through extensive simulation results for all proposed methods on a different face and fingerprint datasets. Low Equal Error Rate (EER) values with high AROC values reflect the efficiency of the proposed methods, especially those dependent on DCT and DFrFT. Moreover, a comparative study is performed to evaluate the proposed method with all transformations to select the best one from the security perspective. Furthermore, a comparative analysis is carried out to test the performance of the proposed schemes with the existing schemes. The obtained outcomes reveal the efficiency of the proposed cancelable biometric schemes by introducing an average AROC of 0.998, EER of 0.0023, FAR of 0.008, and FRR of 0.003.

Triple image encryption scheme in fractional Fourier transform domains

2009 ◽  
Vol 282 (4) ◽  
pp. 518-522 ◽  
Author(s):  
Zhengjun Liu ◽  
Jingmin Dai ◽  
Xiaogang Sun ◽  
Shutian Liu
Keyword(s):  
Fourier Transform ◽  
Image Encryption ◽  
Fractional Fourier Transform ◽  
Encryption Scheme ◽  
Transform Domains

scholarly journals Three Term Weighted Type Fractional Fourier Transform Based Generalized Hybrid Carrier and Its Application into PAPR Suppression

2021 ◽  
Author(s):  
Yong Li ◽  
Zhiqun Song ◽  
Teng Sun ◽  
Bin Wang
Keyword(s):  
Fourier Transform ◽  
Orthogonal Frequency Division Multiplexing ◽  
Fractional Fourier Transform ◽  
Average Power ◽  
The Novel ◽  
Cumulative Density Function ◽  
Ofdm System ◽  
Single Carrier ◽  
Definition Of ◽  
Weighted Type

To suppress the peak to average power ratio (PAPR) of wireless communication based upon multi-carrier system. We, in this paper, proposed the three term weighted type fractional Fourier transform (3-WFRFT) based generalized hybrid carrier (GHC) system. We first provide the definition of 3-WFRFT. Moreover, some useful properties of 3-WFRFT have been presented, in this paper, which will helpful to comprehend the novel 3-WFRFT transform. Furthermore, we take PAPR of the proposed algorithm, in comparison with orthogonal frequency division multiplexing (OFDM) system and single carrier modulation (SC) system under typical complementary cumulative density function (CCDF) level. It would be demonstrated that, from some numerical simulations, the proposed 3-WFRFT based GHC performs better than OFDM system and will be useful to reduce the PAPR level.

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