164 research outputs found
Convolution products for hypercomplex Fourier transforms
Hypercomplex Fourier transforms are increasingly used in signal processing
for the analysis of higher-dimensional signals such as color images. A main
stumbling block for further applications, in particular concerning filter
design in the Fourier domain, is the lack of a proper convolution theorem. The
present paper develops and studies two conceptually new ways to define
convolution products for such transforms. As a by-product, convolution theorems
are obtained that will enable the development and fast implementation of new
filters for quaternionic signals and systems, as well as for their higher
dimensional counterparts.Comment: 18 pages, two columns, accepted in J. Math. Imaging Visio
New techniques for the two-sided quaternionic fourier transform
In this paper, it is shown that there exists a Hermite basis for the two-sided quaternionic Fourier transform. This basis is subsequently used to give an alternative proof for the inversion theorem and to give insight in translation and convolution for the quaternionic Fourier transform
Clifford algebras, Fourier transforms and quantum mechanics
In this review, an overview is given of several recent generalizations of the
Fourier transform, related to either the Lie algebra sl_2 or the Lie
superalgebra osp(1|2). In the former case, one obtains scalar generalizations
of the Fourier transform, including the fractional Fourier transform, the Dunkl
transform, the radially deformed Fourier transform and the super Fourier
transform. In the latter case, one has to use the framework of Clifford
analysis and arrives at the Clifford-Fourier transform and the radially
deformed hypercomplex Fourier transform. A detailed exposition of all these
transforms is given, with emphasis on aspects such as eigenfunctions and
spectrum of the transform, characterization of the integral kernel and
connection with various special functions.Comment: Review paper, 39 pages, to appear in Math. Methods. Appl. Sc
Two-sided Clifford Fourier transform with two square roots of -1 in Cl(p,q)
We generalize quaternion and Clifford Fourier transforms to general two-sided
Clifford Fourier transforms (CFT), and study their properties (from linearity
to convolution). Two general \textit{multivector square roots} \in \cl{p,q}
\textit{of} -1 are used to split multivector signals, and to construct the left
and right CFT kernel factors.
Keywords: Clifford Fourier transform, Clifford algebra, signal processing,
square roots of -1 .Comment: 19 pages, 1 figur
A General Geometric Fourier Transform Convolution Theorem
The large variety of Fourier transforms in geometric algebras inspired the
straight forward definition of ``A General Geometric Fourier Transform`` in
Bujack et al., Proc. of ICCA9, covering most versions in the literature. We
showed which constraints are additionally necessary to obtain certain features
like linearity, a scaling, or a shift theorem. In this paper we extend the
former results by a convolution theorem
Intelligent OFDM telecommunication system. Part 2. Examples of complex and quaternion many-parameter transforms
In this paper, we propose unified mathematical forms of many-parametric complex and quaternion Fourier transforms for novel Intelligent OFDM-telecommunication systems (OFDM-TCS). Each many-parametric transform (MPT) depends on many free angle parameters. When parameters are changed in some way, the type and form of transform are changed as well. For example, MPT may be the Fourier transform for one set of parameters, wavelet transform for other parameters and other transforms for other values of parameters. The new Intelligent-OFDM-TCS uses inverse MPT for modulation at the transmitter and direct MPT for demodulation at the receiver. © 2019 IOP Publishing Ltd. All rights reserved
Erlangen Programme at Large 3.1: Hypercomplex Representations of the Heisenberg Group and Mechanics
In the spirit of geometric quantisation we consider representations of the
Heisenberg(--Weyl) group induced by hypercomplex characters of its centre. This
allows to gather under the same framework, called p-mechanics, the three
principal cases: quantum mechanics (elliptic character), hyperbolic mechanics
and classical mechanics (parabolic character). In each case we recover the
corresponding dynamic equation as well as rules for addition of probabilities.
Notably, we are able to obtain whole classical mechanics without any kind of
semiclassical limit h->0.
Keywords: Heisenberg group, Kirillov's method of orbits, geometric
quantisation, quantum mechanics, classical mechanics, Planck constant, dual
numbers, double numbers, hypercomplex, jet spaces, hyperbolic mechanics,
interference, Segal--Bargmann representation, Schroedinger representation,
dynamics equation, harmonic and unharmonic oscillator, contextual probabilityComment: AMSLaTeX, 17 pages, 4 EPS pictures in two figures; v2, v3, v4, v5,
v6: numerous small improvement
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