Octonion Spectrum of 3D Octonion- Valued Signals - Properties and Possible Applications

Błaszczyk, Łukasz

doi:10.23919/eusipco.2018.8553228

Cited by 13 publications

(17 citation statements)

References 4 publications

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“…In particular, the following theorem holds, which is the generalization of the result of [31], where it was proved for real-valued functions. This result was originally stated in [2], here we complete the details of the proof.…”

Section: Octonion Fourier Transformsupporting

confidence: 59%

“…Some of the results presented in this paper have been signaled in earlier works [3,2], here we present a broader view of these issues and give details of the proofs. We also provide some new results, mainly regarding the use of classical transformation techniques for calculating the OFT, as well as regarding the differentiation of the octonion transform.…”

Section: Introductionmentioning

confidence: 63%

“…This seems important especially in the context of new applications that have appeared in recent years (and which we described earlier in this section)there is a tendency to describe the results of practical experiments, but without adequate theoretical justification. Some of the results presented in this paper have been signaled in earlier works [3,2], here we present a broader view of these issues and give details of the proofs. We also provide some new results, mainly regarding the use of classical transformation techniques for calculating the OFT, as well as regarding the differentiation of the octonion transform.…”

Section: Introductionmentioning

confidence: 63%

“…shift theorem stated in Theorem 13 and Hermitian symmetry analogue) and in this section we will derive equivalents of other classical properties such as argument scaling and modulation theorem. Sketches of some proofs can be found in [2], here we will present all previously omitted details, but also present previously unpublished results. The summary of the content of this section is shown in Table 3.…”

Section: Properties Of the Octonion Fourier Transformmentioning

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%

See 4 more Smart Citations

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: Octonion Fourier Transformsupporting

confidence: 59%

Section: Introductionmentioning

confidence: 63%

Section: Introductionmentioning

confidence: 63%

Section: Properties Of the Octonion Fourier Transformmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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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Real Paley‐Wiener theorem for the octonion Fourier transform

Ren

2021

Math Methods in App Sciences

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Octonion short‐time linear canonical transform

Gao

2022

Math Methods in App Sciences

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We investigate the octonion short-time linear canonical transform (OCSTLCT) in this paper. First, we propose the new definition of the OCSTLCT, and then several important properties of newly defined OCSTLCT, such as bounded, shift, modulation, time-frequency shift, inversion formula, and orthogonality relation, are derived based on the spectral representation of the octonion linear canonical transform (OCLCT). Second, by the Heisenberg uncertainty principle for the OCLCT and the orthogonality relation property for the OCSTLCT, the Heisenberg uncertainty principle for the OCSTLCT is established. Finally, we give an example of the OCSTLCT.

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Octonion Spectrum of 3D Octonion- Valued Signals - Properties and Possible Applications

Cited by 13 publications

References 4 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

Real Paley‐Wiener theorem for the octonion Fourier transform

Octonion short‐time linear canonical transform

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