299 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
The Study of Properties of n-D Analytic Signals and Their Spectra in Complex and Hypercomplex Domains
In the paper, two various representations of a n-dimensional (n-D) real signal u(x1,x2,…,xn) are investigated. The first one is the n-D complex analytic signal with a single-orthant spectrum defined by Hahn in 1992 as the extension of the 1-D Gabor’s analytic signal. It is compared with two hypercomplex approaches: the known n-D Clifford analytic signal and the Cayley-Dickson analytic signal defined by the Author in 2009. The signal-domain and frequency-domain definitions of these signals are presented and compared in 2-D and 3-D. Some new relations between the spectra in 2-D and 3-D hypercomplex domains are presented. The paper is illustrated with the example of a 2-D separable Cauchy pulse
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
Fourier transforms of hypercomplex signals
An overview is given to a new approach for obtaining generalized Fourier transforms in the context of hypercomplex analysis (or Clifford analysis). These transforms are applicable to higher-dimensional signals with several components and are different from the classical Fourier transform in that they mix the components of the signal. Subsequently, attention is focused on the special case of the so-called Clifford-Fourier transform where recently a lot of progress has been made. A fractional version of this transform is introduced and a series expansion for its integral kernel is obtained
Instantaneous frequency and amplitude of complex signals based on quaternion Fourier transform
The ideas of instantaneous amplitude and phase are well understood for
signals with real-valued samples, based on the analytic signal which is a
complex signal with one-sided Fourier transform. We extend these ideas to
signals with complex-valued samples, using a quaternion-valued equivalent of
the analytic signal obtained from a one-sided quaternion Fourier transform
which we refer to as the hypercomplex representation of the complex signal. We
present the necessary properties of the quaternion Fourier transform,
particularly its symmetries in the frequency domain and formulae for
convolution and the quaternion Fourier transform of the Hilbert transform. The
hypercomplex representation may be interpreted as an ordered pair of complex
signals or as a quaternion signal. We discuss its derivation and properties and
show that its quaternion Fourier transform is one-sided. It is shown how to
derive from the hypercomplex representation a complex envelope and a phase.
A classical result in the case of real signals is that an amplitude modulated
signal may be analysed into its envelope and carrier using the analytic signal
provided that the modulating signal has frequency content not overlapping with
that of the carrier. We show that this idea extends to the complex case,
provided that the complex signal modulates an orthonormal complex exponential.
Orthonormal complex modulation can be represented mathematically by a polar
representation of quaternions previously derived by the authors. As in the
classical case, there is a restriction of non-overlapping frequency content
between the modulating complex signal and the orthonormal complex exponential.
We show that, under these conditions, modulation in the time domain is
equivalent to a frequency shift in the quaternion Fourier domain. Examples are
presented to demonstrate these concepts
Complex and Hypercomplex Discrete Fourier Transforms Based on Matrix Exponential Form of Euler's Formula
We show that the discrete complex, and numerous hypercomplex, Fourier
transforms defined and used so far by a number of researchers can be unified
into a single framework based on a matrix exponential version of Euler's
formula , and a matrix root of -1
isomorphic to the imaginary root . The transforms thus defined can be
computed using standard matrix multiplications and additions with no
hypercomplex code, the complex or hypercomplex algebra being represented by the
form of the matrix root of -1, so that the matrix multiplications are
equivalent to multiplications in the appropriate algebra. We present examples
from the complex, quaternion and biquaternion algebras, and from Clifford
algebras Cl1,1 and Cl2,0. The significance of this result is both in the
theoretical unification, and also in the scope it affords for insight into the
structure of the various transforms, since the formulation is such a simple
generalization of the classic complex case. It also shows that hypercomplex
discrete Fourier transforms may be computed using standard matrix arithmetic
packages without the need for a hypercomplex library, which is of importance in
providing a reference implementation for verifying implementations based on
hypercomplex code.Comment: The paper has been revised since the second version to make some of
the reasons for the paper clearer, to include reviews of prior hypercomplex
transforms, and to clarify some points in the conclusion
Fractional fourier transforms of hypercomplex signals
An overview is given to a new approach for obtaining generalized Fourier transforms in the context of hypercomplex analysis (or Clifford analysis). These transforms are applicable to higher-dimensional signals with several components and are different from the classical Fourier transform in that they mix the components of the signal. Subsequently, attention is focused on the special case of the so-called Clifford-Fourier transform where recently a lot of progress has been made. A fractional version of this transform is introduced and a series expansion for its integral kernel is obtained. For the case of dimension 2, also an explicit expression for the kernel is given
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