A generalization of the octonion Fourier transform to 3-D octonion-valued signals: properties and possible applications
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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 Łukasz Błaszczyk1 Received: 5 February 2019 / Revised: 20 January 2020 / Accepted: 27 January 2020 © The Author(s) 2020
Abstract 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 R-valued 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. Keywords Octonion Fourier transform · Cayley–Dickson numbers · Hypercomplex algebras · Multidimensional linear time-invariant systems Mathematics Subject Classification 30G35 (primary) · 42B10 · 94A12
The research was supported by National Science Centre (Poland) Grant No. 2016/23/N/ST7/00131.
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Łukasz Błaszczyk [email protected] Faculty of Mathematics and Information Science, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warszawa, Poland
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Multidimensional Systems and Signal Processing
1 Introduction Fourier analysis is one of the fundamental tools in signal and image processing. Fourier series and Fourier transform enable us to look at the concept of signal in a dual manner— by studying its properties in the time domain (or in the space domain in case of images), where it is represented by amplitudes of the samples (or pixels), or by investigating it in the frequency domain, where the signal can be represented by the infinite sums of complex harmonic functions, each with different frequency and amplitude (Allen and Mills 2003). The classical signal theory deals with real- or complex-valued time series (or images). However, in some practical applications, signals are represented by more abstract structures, e.g. hypercomplex algebras (Ell et al. 2014; Grigoryan
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