scholarly journals Wave splitting for first-order systems of equations

2002 ◽  
Vol 32 (6) ◽  
pp. 371-381 ◽  
Author(s):  
G. Caviglia ◽  
A. Morro
Keyword(s):  
Fourier Transform ◽  
Partial Differential Equations ◽  
Time Domain ◽  
Space Variable ◽  
Transform Domain ◽  
Systems Of Equations ◽  
First Order ◽  
Wave Splitting ◽  
The Fourier Transform ◽  
Fourier Transform Domain

Systems of first-order partial differential equations are considered and the possible decomposition of the solutions in forward and backward propagating is investigated. After a review of a customary procedure in the space-time domain (wave splitting), attention is addressed to systems in the Fourier-transform domain, thus considering frequency-dependent functions of the space variable. The characterization is given for the direction of propagation and applications are developed to some cases of physical interest.

Seismic trace interpolation in the Fourier transform domain

Geophysics ◽  
2003 ◽  
Vol 68 (1) ◽  
pp. 355-369 ◽  
Author(s):  
Necati Gülünay
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Interpolation Method ◽  
Original Data ◽  
Transform Domain ◽  
3D Data ◽  
Adaptive Interpolation ◽  
Data Adaptive ◽  
The Fourier Transform ◽  
Fourier Transform Domain

A data adaptive interpolation method is designed and applied in the Fourier transform domain (f‐k or f‐kx‐ky for spatially aliased data. The method makes use of fast Fourier transforms and their cyclic properties, thereby offering a significant cost advantage over other techniques that interpolate aliased data. The algorithm designs and applies interpolation operators in the f‐k (or f‐kx‐ky domain to fill zero traces inserted in the data in the t‐x (or t‐x‐y) domain at locations where interpolated traces are needed. The interpolation operator is designed by manipulating the lower frequency components of the stretched transforms of the original data. This operator is derived assuming that it is the same operator that fills periodically zeroed traces of the original data but at the lower frequencies, and corresponds to the f‐k (or f‐kx‐ky domain version of the well‐known f‐x (or f‐x‐y) domain trace interpolators. The method is applicable to 2D and 3D data recorded sparsely in a horizontal plane. The most common prestack applications of the algorithm are common‐mid‐point domain shot interpolation, source‐receiver domain shot interpolation, and cable interpolation.

scholarly journals The N-Dimensional Uncertainty Principle for the Free Metaplectic Transformation

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1685
Author(s):  
Rui Jing ◽  
Bei Liu ◽  
Rui Li ◽  
Rui Liu
Keyword(s):  
Fourier Transform ◽  
Quantum Mechanics ◽  
Uncertainty Principle ◽  
Transformation Operator ◽  
Local Analysis ◽  
Transform Domain ◽  
Uncertainty Principles ◽  
The Fourier Transform ◽  
Heisenberg's Uncertainty Principle ◽  
Fourier Transform Domain

The free metaplectic transformation is an N-dimensional linear canonical transformation. This transformation operator is useful, especially for signal processing applications. In this paper, in order to characterize simultaneously local analysis of a function (or signal) and its free metaplectic transformation, we extend some different uncertainty principles (UP) from quantum mechanics including Classical Heisenberg’s uncertainty principle, Nazarov’s UP, Donoho and Stark’s UP, Hardy’s UP, Beurling’s UP, Logarithmic UP, and Entropic UP, which have already been well studied in the Fourier transform domain.

Fault impedance and earthquake energy in the Fourier transform domain

1980 ◽  
Vol 70 (5) ◽  
pp. 1683-1698
Author(s):  
D. J. Andrews
Keyword(s):  
Elastic Medium ◽  
Time Domain ◽  
Fourier Transformation ◽  
Slip Rate ◽  
Transform Domain ◽  
Impedance Function ◽  
Stress Functions ◽  
Two Space Dimensions ◽  
The Fourier Transform ◽  
Fourier Transform Domain

abstract The response of the elastic medium in which a fault is contained determines one relation between the slip function and the stress function on the fault. In the space-time domain, the slip and stress functions are related by a singular integral equation. If a Fourier transformation is performed over the two space dimensions on the fault and over time, the stress transform equals the slip rate transform times an impedance function. Using this impedance function, we may determine the energy radiated or stored in the elastic medium from either the slip or the stress transform.

scholarly journals Compressive Sensing in Signal Processing: Algorithms and Transform Domain Formulations

2016 ◽  
Vol 2016 ◽  
pp. 1-16 ◽  
Author(s):  
Irena Orović ◽  
Vladan Papić ◽  
Cornel Ioana ◽  
Xiumei Li ◽  
Srdjan Stanković
Keyword(s):  
Signal Processing ◽  
Fourier Transform ◽  
Compressive Sensing ◽  
Signal Reconstruction ◽  
Signal Acquisition ◽  
Transform Domain ◽  
Hermite Transform ◽  
Time Frequency ◽  
Reconstruction Methods ◽  
Fourier Transform Domain

Compressive sensing has emerged as an area that opens new perspectives in signal acquisition and processing. It appears as an alternative to the traditional sampling theory, endeavoring to reduce the required number of samples for successful signal reconstruction. In practice, compressive sensing aims to provide saving in sensing resources, transmission, and storage capacities and to facilitate signal processing in the circumstances when certain data are unavailable. To that end, compressive sensing relies on the mathematical algorithms solving the problem of data reconstruction from a greatly reduced number of measurements by exploring the properties of sparsity and incoherence. Therefore, this concept includes the optimization procedures aiming to provide the sparsest solution in a suitable representation domain. This work, therefore, offers a survey of the compressive sensing idea and prerequisites, together with the commonly used reconstruction methods. Moreover, the compressive sensing problem formulation is considered in signal processing applications assuming some of the commonly used transformation domains, namely, the Fourier transform domain, the polynomial Fourier transform domain, Hermite transform domain, and combined time-frequency domain.

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