Generalized Hilbert transform and its applications in geophysics

2003 ◽  
Vol 22 (3) ◽  
pp. 198-202 ◽  
Author(s):  
Yi Luo ◽  
Saleh Al-Dossary ◽  
Maher Marhoon ◽  
Mohammad Alfaraj
Keyword(s):  
Hilbert Transform ◽  
Generalized Hilbert Transform

Unification of Euler and Werner deconvolution in three dimensions via the generalized Hilbert transform

Geophysics ◽  
2001 ◽  
Vol 66 (6) ◽  
pp. 1805-1810 ◽  
Author(s):  
Misac N. Nabighian ◽  
R. O. Hansen
Keyword(s):  
Magnetic Dipole ◽  
Hilbert Transform ◽  
Natural Generalization ◽  
Three Dimensions ◽  
Explicit Calculation ◽  
Euler Deconvolution ◽  
Hilbert Transforms ◽  
Vertical Magnetic Dipole ◽  
Generalized Hilbert Transform ◽  
Practical Level

The extended Euler deconvolution algorithm is shown to be a generalization and unification of 2‐D Euler deconvolution and Werner deconvolution. After recasting the extended Euler algorithm in a way that suggests a natural generalization to three dimensions, we show that the 3‐D extension can be realized using generalized Hilbert transforms. The resulting algorithm is both a generalization of extended Euler deconvolution to three dimensions and a 3‐D extension of Werner deconvolution. At a practical level, the new algorithm helps stabilize the Euler algorithm by providing at each point three equations rather than one. We illustrate the algorithm by explicit calculation for the potential of a vertical magnetic dipole.

The properties of generalized offset linear canonical Hilbert transform and its applications

2017 ◽  
Vol 15 (04) ◽  
pp. 1750031 ◽  
Author(s):  
Shuiqing Xu ◽  
Li Feng ◽  
Yi Chai ◽  
Youqiang Hu ◽  
Lei Huang
Keyword(s):  
Signal Processing ◽  
Fourier Transform ◽  
Hilbert Transform ◽  
Analytic Signal ◽  
Linear Canonical Transform ◽  
Single Sideband ◽  
Generalized Hilbert Transform ◽  
Offset Linear Canonical Transform ◽  
The Fourier Transform ◽  
The Hilbert Transform

The Hilbert transform is tightly associated with the Fourier transform. As the offset linear canonical transform (OLCT) has been shown to be useful and powerful in signal processing and optics, the concept of generalized Hilbert transform associated with the OLCT has been proposed in the literature. However, some basic results for the generalized Hilbert transform still remain unknown. Therefore, in this paper, theories and properties of the generalized Hilbert transform have been considered. First, we introduce some basic properties of the generalized Hilbert transform. Then, an important theorem for the generalized analytic signal is presented. Subsequently, the generalized Bedrosian theorem for the generalized Hilbert transform is formulated. In addition, a generalized secure single-sideband (SSB) modulation system is also presented. Finally, the simulations are carried out to verify the validity and correctness of the proposed results.

scholarly journals Sampling in the Linear Canonical Transform Domain

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Bing-Zhao Li ◽  
Tian-Zhou Xu
Keyword(s):  
Hilbert Transform ◽  
Sampling Rate ◽  
Sampling Theorem ◽  
Fourier Domain ◽  
Transform Domain ◽  
Linear Canonical Transform ◽  
Generalized Hilbert Transform ◽  
Simulation Results ◽  
Bandpass Signals

This paper investigates the interpolation formulae and the sampling theorem for bandpass signals in the linear canonical transform (LCT) domain. Firstly, one of the important relationships between the bandpass signals in the Fourier domain and the bandpass signals in the LCT domain is derived. Secondly, two interpolation formulae from uniformly sampled points at half of the sampling rate associated with the bandpass signals and their generalized Hilbert transform or the derivatives in the LCT domain are obtained. Thirdly, the interpolation formulae from nonuniform samples are investigated. The simulation results are also proposed to verify the correctness of the derived results.

scholarly journals Invisibility on demand based on a generalized Hilbert transform

2018 ◽  
Vol 98 (1) ◽  
Author(s):  
Zeki Hayran ◽  
Ramon Herrero ◽  
Muriel Botey ◽  
Hamza Kurt ◽  
Kestutis Staliunas
Keyword(s):  
Hilbert Transform ◽  
On Demand ◽  
Generalized Hilbert Transform

Generalized Hilbert transform and its properties in 2D LCT domain

2009 ◽  
Vol 89 (7) ◽  
pp. 1395-1402 ◽  
Author(s):  
Xu Guanlei ◽  
Wang Xiaotong ◽  
Xu Xiaogang
Keyword(s):  
Hilbert Transform ◽  
Generalized Hilbert Transform

Related singular problems and the generalized Hilbert transform

1977 ◽  
Vol 79 (1-2) ◽  
pp. 173-182 ◽  
Author(s):  
John W. Dettman
Keyword(s):  
Half Space ◽  
Hilbert Transform ◽  
Integral Transformation ◽  
Axially Symmetric ◽  
Singular Problems ◽  
Poisson Formula ◽  
Special Cases ◽  
The Cauchy Problem ◽  
Generalized Hilbert Transform ◽  
Darboux Equation

SynopsisAbstract versions of the Cauchy problem for the Euler-Poisson-Darboux equation and the Dirichlet problem for the equation of generalized axially symmetric potential theory are related by an integral transformation. In certain special cases, this leads to abstract versions of (1) the Poisson formula for the solution of G.A.S.P.T. in a half-space, (2) pseudo-analytic functions in a half-space, and (3) a generalized Hilbert transform related to the work of Heywood, Kober, and Okikiolu. Some properties of this generalized Hilbert transform are studied including an inversion theorem.

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