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 mathbb {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.
Highlights
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, where it is represented by amplitudes of the samples, 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)
We will focus on using the octonion Fourier transform (OFT) and notion of quadruple-complex numbers in the analysis of 3-D linear time-invariant (LTI) systems of linear partial differential equations (PDEs) with constant coefficients
Summary
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). In our research we derived some properties of the OFT, analogous to the properties of the classical (complex) and quaternion Fourier transform, e.g. symmetry properties (analogue to the Hermitian symmetry properties), shift theorem, Plancherel and Parseval theorems, and Wiener-Khintchine theorem Proofs of the those theorems were based on the previous research of Hahn and Snopek, who used the fact that real–valued functions can be expressed as a sum of components of different parity (Hahn and Snopek 2011). Despite these works, the state of modern knowledge about octonion Fourier transform is negligible and requires a thorough extension.
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