E. Hitzer: The Quaternion Domain Fourier Transform and its Properties


E. Hitzer, The Quaternion Domain Fourier Transform and its Properties, Article in Advances in Applied Clifford Algebras, pp 1-16, First online: 02 November 2015, DOI: 10.1007/s00006-015-0620-3 . Download: http://vixra.org/abs/1511.0302

Abstract: So far quaternion Fourier transforms have been mainly defined over \({\mathbb{R}^2}\) as signal domain space. But it seems natural to define a quaternion Fourier transform for quaternion valued signals over quaternion domains. This quaternion domain Fourier transform (QDFT) transforms quaternion valued signals (for example electromagnetic scalar-vector potentials, color data, space-time data, etc.) defined over a quaternion domain (space-time or other 4D domains) from a quaternion position space to a quaternion frequency space. The QDFT uses the full potential provided by hypercomplex algebra in higher dimensions and may moreover be useful for solving quaternion partial differential equations or functional equations, and in crystallographic texture analysis. We define the QDFT and analyze its main properties, including quaternion dilation, modulation and shift properties, Plancherel and Parseval identities, covariance under orthogonal transformations, transformations of coordinate polynomials and differential operator polynomials, transformations of derivative and Dirac derivative operators, as well as signal width related to band width uncertainty relationships.

Keywords: Quaternions Fourier transform, Signal and image processing, Uncertainty, Orthogonal transformations in three and four dimensions, Quaternionic differential operators.

Mathematics Subject Classification: Primary 11R52, Secondary 42A38, 15A66

Source: http://link.springer.com/article/10.1007/s00006-015-0620-3

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