The octonionic Fourier transform: Uncertainty relations and convolution

Lian, Pan

doi:10.1016/j.sigpro.2019.06.015

Cited by 25 publications

(22 citation statements)

References 8 publications

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“…In [2] we presented a detailed commentary on these assertions, including indicating the significance of the assumption in Theorem 17 that the considered functions are realvalued -for the octonion-valued functions the claim of Theorem 17 doesn't hold. In case of real-valued functions Theorem 18 (also known in classical theory as Rayleigh Theorem) is direct corollary of Theorem 17, but (as we proved in [2] and was shown independently in [23]) is valid also in the general case of octonion-valued functions.…”

Section: )mentioning

confidence: 56%

“…It is worth noting that the above theorem was independently proved also in a recent article [23], in which the author used other methods.…”

Section: Octonion Fourier Transformmentioning

confidence: 74%

“…Of course, the above theorem shows that OFT preserves the energy of octonion-valued functions. However, it is worth mentioning the recent result in [23], where the author argues that OFT of octonion-valued function also satisfies the Hausdorff-Young inequality.…”

Section: )mentioning

confidence: 99%

“…The main goal of this paper is further development of such generalization based on the Cayley-Dickson algebra of order 8 (octonions). Analysis of the current state of knowledge on applications of octonions in the signal processing shows some areas previously unexplored or requiring thorough theoretical and experimental studies, although some gaps have recently been filled [3,2,4,23].…”

Section: Introductionmentioning

confidence: 99%

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A generalization of the octonion Fourier transform to 3-D octonion-valued signals: properties and possible applications to 3-D LTI partial differential systems

Błaszczyk

2020

Multidim Syst Sign Process

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The paper is devoted to the development of the octonion Fourier transform (OFT) theory initiated in 2011 in articles by Hahn and Snopek. It is also a continuation and generalization of earlier work by Błaszczyk and Snopek, where they proved few essential properties of the OFT of real-valued functions, e.g. symmetry properties. The results of this article focus on proving that the OFT is well-defined for octonion-valued functions and almost all well-known properties of classical (complex) Fourier transform (e.g. argument scaling, modulation and shift theorems) have their direct equivalents in octonion setup. Those theorems, illustrated with some examples, lead to the generalization of another result presented in earlier work, i.e. Parseval and Plancherel Theorems, important from the signal and system processing point of view. Moreover, results presented in this paper associate the OFT with 3-D LTI systems of linear PDEs with constant coefficients. Properties of the OFT in context of signal-domain operations such as derivation and convolution of Rvalued functions will be stated. There are known results for the QFT, but they use the notion of other hypercomplex algebra, i.e. double-complex numbers. Considerations presented here require defining other higher-order hypercomplex structure, i.e. quadruple-complex numbers. This hypercomplex generalization of the Fourier transformation provides an excellent tool for the analysis of 3-D LTI systems.

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Section: )mentioning

confidence: 56%

“…It is worth noting that the above theorem was independently proved also in a recent article [23], in which the author used other methods.…”

Section: Octonion Fourier Transformmentioning

confidence: 74%

Section: )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A generalization of the octonion Fourier transform to 3-D octonion-valued signals: properties and possible applications to 3-D LTI partial differential systems

Błaszczyk

2020

Multidim Syst Sign Process

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“…As in the case of classical signal processing, so the discrete counterpart of this theory has so far mainly focused on signals with real and complex values, as well as their complex spectra. In recent years, however, more and more works have started to appear, which authors use in their research hypercomplex algebras, among others quaternions and octonions (Brackx et al 2013;Hahn and Snopek 2016;Lian 2019;Snopek 2015;Wang et al 2017). The area of applications is focused so far on the study of neural networks (Popa 2016(Popa , 2018, analysis of color and multispectral images (Ell et a.…”

Section: Introductionmentioning

confidence: 99%

Discrete octonion Fourier transform and the analysis of discrete 3-D data

Błaszczyk

2020

Comp. Appl. Math.

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The purpose of this article is to further develop the theory of octonion Fourier transformations (OFT), but from a different perspective than before. It follows the earlier work by Błaszczyk and Snopek, where they proved a few essential properties of the OFT of real-valued functions of three continuous variables. The research described in this article applies to discrete transformations, i.e. discrete-space octonion Fourier transform (DSOFT) and discrete octonion Fourier transform (DOFT). The described results combine the theory of Fourier transform with the analysis of solutions for difference equations, using for this purpose previous research on algebra of quadruple-complex numbers. This hypercomplex generalization of the discrete Fourier transformation provides an excellent tool for the analysis of 3-D discrete linear time-invariant (LTI) systems and 3-D discrete data.

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Real Paley‐Wiener theorem for the octonion Fourier transform

Ren

2021

Math Methods in App Sciences

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The octonionic Fourier transform: Uncertainty relations and convolution

Cited by 25 publications

References 8 publications

A generalization of the octonion Fourier transform to 3-D octonion-valued signals: properties and possible applications to 3-D LTI partial differential systems

A generalization of the octonion Fourier transform to 3-D octonion-valued signals: properties and possible applications to 3-D LTI partial differential systems

Discrete octonion Fourier transform and the analysis of discrete 3-D data

Real Paley‐Wiener theorem for the octonion Fourier transform

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