Warsaw University Technology Faculty of Electronics and Information Technology Institute of Radioelectronics

The paper is devoted to the polar representation of n-D complex and hypercomplex analytic signals with emphasis on the 3-dimensional (3-D) case. Their definition is based on the proposed general form of the Cauchy integral. The definitions of complex/hypercomplex signals are presented in signaland frequency domains. The new notion of lower rank signals is introduced. It is shown that starting with the 3-D analytic hypercomplex signals and decreasing their rank by extending the support in the frequencyspace to a so called space quadrant, we get a signal having the quaternionic structure. The advantage of this procedure is demonstrated in the context of the polar representation of 3-D hypercomplex signals. Some new reconstruction formulas are presented. Their validation has been confirmed using two 3-D test signals: a Gaussian signal and a spherical signal. Keywords—complex/hypercomplex analytic signal, hypercomplex Fourier transform, hypercomplex delta distribution, polar representation Introduction The theory of hypercomplex signals is a subject of many publications involving either mathematicians or engineers working in different fields [1]-[4]. It is based on the theory of hypercomplex numbers belonging to different algebras, e.g., to the Cayley-Dickson [5] or Clifford algebras [6]. Different definitions of complex/hypercomplex signals have appeared recently. The most interesting approach is the Clifford hypercomplex signal defined by Bülow and Sommer in [7]. For n = 2, it is identical to the quaternionic analytic signal A very detailed comparison of different definitions has been presented in [9]. Especially the case of 2-D analytic signals has been studied in detail and formulas relating analytic, quaternionic and monogenic signals have been derived. This paper is devoted to the study and comparison of properties of n-D complex/hypercomplex signals with emphasis on the 3-D case. Especially, it will be shown that there are closed formulae enabling calculation of hypercomplex amplitude and phase functions in terms of the corresponding polar representation of complex functions. As well, it is noticed that the total number of amplitudes and phases of n-D complex and hypercomplex signals is equal to 2 n . The Complex and Hypercomplex Multidimensional Analytic Functions Defined by the Cauchy Integral Consider the n-D hypercomplex space  of hypercomplex variables:   1 2 , , , : n k k k k z z z z x e y    z where ek are imaginary units (in the domain of complex numbers they are usually denoted as k k k z x jy   ). The space  n is a Cartesian product of complex planes , 1,2, , k k n  , that is, 1 2 n n     . We define a complex-valued n-D function   f z , analytic (holomorphic) in the interior of a region 1 2 n n D D D D    , n n D  , k k D  .