The n-D analytic signals and Fourier spectra in complex and hypercomplex domains

In the paper, two various representations of a n-D real signal are investigated. The first one is the n-D complex analytic signal with a single-orthant spectrum defined as the extension of the 1-D Gabor's analytic signal. The second one is the n-D hypercomplex analytic signal defined in Clifford and Cayley-Dickson algebras. Their signal-domain and frequency-domain definitions are presented and compared. Some relations between the spectra in 2-D and 3-D hypercomplex domains are presented. The paper is illustrated with the example of a 2-D separable Cauchy pulse.