We generalize the linear canonical transform (LCT) to quaternion-valued signals, known as the quaternionic linear canonical transform (QLCT). Using the properties of the LCT we establish an uncertainty principle for the QLCT. This uncertainty principle prescribes a lower bound on the product of the effective widths of quaternion-valued signals in the spatial and frequency domains. It is shown that only a 2D Gaussian signal minimizes the uncertainty.

https://downloads.hindawi.com/journals/aaa/2013/725952.pdf

On Uncertainty Principle for Quaternionic Linear Canonical Transform

Windowed linear canonical transform and its applications

Prolate spheroidal wave functions (PSWFs) possess many remarkable properties. They are orthogonal basis of both square integrable space of finite interval and the Paley–Wiener space of bandlimited functions on the real line. No other system of classical orthogonal functions is known to obey this unique property. This raises the question of whether they possess these properties in Clifford analysis. The aim of the article is to answer this question and extend the results to more flexible integral transforms, such as offset linear canonical transform. We also illustrate how to use the generalized Clifford PSWFs (for offset Clifford linear canonical transform) we derive to analyze the energy preservation problems. Clifford PSWFs is new in literature and has some consequences that are now under investigation. Copyright © 2012 John Wiley & Sons, Ltd.

Generalized prolate spheroidal wave functions for offset linear canonical transform in Clifford analysis

Uncertainty principles for hypercomplex signals in the linear canonical transform domains

This paper proposes a novel fractional transform, denoted as the fractional Krawtchouk transform (FrKT), a generalization of the Krawtchouk transform. The derivation of the FrKT uses the eigenvalue decomposition method. We determine the eigenvalues and the corresponding multiplicity of the Krawtchouk transform matrix. Moreover, the orthonormal eigenvectors of the transform matrix are derived. For validation purpose only and as a first illustration of the interest of FrKT, a watermarking example was chosen. Experimental results show that better watermark robustness and imperceptibility are achieved by adjusting the fractional orders in the FrKT.

Fractional Krawtchouk Transform With an Application to Image Watermarking

The main objective of this paper is to study the fractional Fourier transform (FrFT) and its some basic properties. Applications of the FrFT in solving nth order linear non-homogeneous ordinary generalized differential equations and a generalized wave equation are also given.

The fractional Fourier transform and its applications

The generalized continuous wavelet transform associated with the fractional Fourier transform

The present paper is devoted to study the propagation of Love wave in a piezoelectric layer overlying an inhomogeneous half-space. This paper deals with two different piezoelectric layers, one is an electrically open and another is an electrically short circuit. As mathematical tools, the method of separation of variables and Whittaker’s function are applied to obtain the dispersion equation of Love wave. In a particular case the dispersion equation reduces to the classical equation of Love wave when the layer is not piezoelectric and half-space is homogeneous. The numerical values of the dimensionless phase velocities are calculated and presented graphically to illustrate the effects of inhomogeneity, piezoelectricity and dielectric constants. It is observed that the phase velocities decrease with the increase of inhomogeneity parameters and electricity constant. It is also found that the phase velocity increases with the increases of the dielectric constant. Graphical user interface software has been de...

Love wave propagation in a piezoelectric layer overlying in an inhomogeneous elastic half-space:

The present paper studies the propagation of shear waves (SH-type waves) in an homogeneous isotropic medium sandwiched between two semi infinite media. The upper half-space is considered as orthotropic medium under initial stress and lower half-space considered as heterogeneous medium. We have obtained the dispersion equation of phase velocity for SH-type waves. The propagation of SH-type waves are influenced by inhomogeneity parameters and initial stress parameter. The velocity of SH-type wave has been computed for different cases. We have also obtained the dispersion equation of phase velocity in homogeneous media in the absence of initial stress. The velocities of SH-type waves are calculated numerically as a function of kH (non-dimensional wave number) and presented in a number of graphs. To study the effect of inhomogeneity parameters and initial stress parameter we have plotted the velocity of SH-type wave in several figure. We have observed that the velocity of wave increases with the increase inhomogeneity parameters. We found that in both homogeneous and inhomogeneous media the velocity of SH-type wave increases with the increase of initial stress parameter. The results may be useful for the study of seismic waves propagation during any earthquake and artificial explosions.

Santanu Manna

Papers

The fractional Fourier transform and its applications

The generalized continuous wavelet transform associated with the fractional Fourier transform

Love wave propagation in a piezoelectric layer overlying in an inhomogeneous elastic half-space:

SH-type waves dispersion in an isotropic medium sandwiched between an initially stressed orthotropic and heterogeneous semi-infinite media

Love wave dispersion in pre-stressed homogeneous medium over a porous half-space with irregular boundary surfaces