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Analytic signal in many dimensions

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Abstract

In this paper we extend analytic signal to the multidimensional case. First it is shown how to obtain separate phase-shifted components and how to combine them into instantaneous amplitude and phase. Secondly we define the proper hypercomplex analytic signal as a holo-morphic hypercomplex function on the boundary of polydisk in the hypercomplex space. Next it is shown that the correct phase-shifted components can be obtained by positive frequency restriction of the Scheffers-Fourier transform based on the commutative and associative algebra generated by the set of elliptic hypercomplex numbers. Moreover we demonstrate that for d > 2 there is no corresponding Clifford-Fourier transform that allows to recover phase-shifted components correctly. Finally the euclidean-domain construction of instantaneous amplitude is extended to manifold and manifold-like graphs and point clouds.
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hal-01877403 , version 1 (19-09-2018)

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Mikhail Tsitsvero, Pierre Borgnat, Paulo Gonçalves. Analytic signal in many dimensions. 2018. ⟨hal-01877403⟩
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