23 research outputs found
Relation between Quaternion Fourier Transform and Quaternion Wigner-Ville Distribution Associated with Linear Canonical Transform
The quaternion Wigner-Ville distribution associated with linear canonical transform (QWVD-LCT) is a nontrivial generalization of the quaternion Wigner-Ville distribution to the linear canonical transform (LCT) domain. In the present paper, we establish a fundamental relationship between the QWVD-LCT and the quaternion Fourier transform (QFT). Based on this fact, we provide alternative proof of the well-known properties of the QWVD-LCT such as inversion formula and Moyal formula. We also discuss in detail the relationship among the QWVD-LCT and other generalized transforms. Finally, based on the basic relation between the quaternion ambiguity function associated with the linear canonical transform (QAF-LCT) and the QFT, we present some important properties of the QAF-LCT
Hybrid Transforms
Hybrid transforms are constructed by associating the Wigner-Ville distribution (WVD) with widely-known signal processing tools, such as fractional Fourier transform, linear canonical transform, offset linear canonical transform (OLCT), and their quaternion-valued versions. We call them hybrid transforms because they combine the advantages of both transforms. Compared to classical transforms, they show better results in applications. The WVD associated with the OLCT (WVD-OLCT) is a class of hybrid transform that generalizes most hybrid transforms. This chapter summarizes research on hybrid transforms by reviewing a computationally efficient type of the WVD-OLCT, which has simplicity in marginal properties compared to WVD-OLCT and WVD
Titchmarsh Theorems and K-Functionals for the Two-Sided Quaternion Fourier Transform
The purpose of this paper is to study the Quaternion Fourier transforms of functions that satisfy Lipschitz conditions of certain orders. Thus we study the Quaternion Fourier transforms of Lipschitz function in the functions space Lr(R2; H), where H a quaternion algebra which will be specified in due course. Our investigation into the problem was motivated by a theorem proved by Titchmarsh [[29], Theorem 85] for Lipschitz functions on the real line. we will give also some results on calculation of the K-functional which have number of applications of interpolation theory. In particular some recent problems in image processing and singular integral operators require the computation of suitable K-functionals. In this paper we will give some results concerning the equivalence of a K-functional and the modulus of smoothness constructed by the Steklov function
Special Affine Stockwell Transform Theory, Uncertainty Principles and Applications
In this paper, we study the convolution structure in the special affine
Fourier transform domain to combine the advantages of the well known special
affine Fourier and Stockwell transforms into a novel integral transform coined
as special affine Stockwell transform and investigate the associated constant Q
property in the joint time frequency domain. The preliminary analysis
encompasses the derivation of the fundamental properties, Rayleighs energy
theorem, inversion formula and range theorem. Besides, we also derive a direct
relationship between the recently introduced special affine scaled Wigner
distribution and the proposed SAST. Further, we establish Heisenbergs
uncertainty principle, logarithmic uncertainty principle and Nazarovs
uncertainty principle associated with the proposed SAST. Towards the
culmination of this paper, some potential applications with simulation are
presented.Comment: arXiv admin note: text overlap with arXiv:2010.01972 by other author
Octonion special affine fourier transform: pitt's inequality and the uncertainty principles
The special affine Fourier transform (SAFT) is an extended version of the classical Fourier transform and incorporates various signal processing tools which include the Fourier transforms, the fractional Fourier transform, the linear canonical transform, and other related transforms. This paper aims to introduce a novel octonion special affine Fourier transform (O-SAFT) and establish several classes of uncertainty inequalities for the proposed transform. We begin by studying the norm split and energy conservation properties of the proposed (O-SAFT). Afterwards, we generalize several uncertainty relations for the (O-SAFT) which include Pitt's inequality, Heisenberg-Weyl inequality, logarithmic uncertainty inequality, Hausdorff-Young inequality, and local uncertainty inequalities. Finally, we provide an illustrative example and some possible applications of the proposed transform