A revisit of the unique polar representation of the Vector-valued Hyperanalytic Signal

In this paper, we extend the classic analytic signal to the Vector-valued Hyperanalytic Signal (VHaS) that is denoted to distinguish from the multivariate hypercomplex data. The 2d-Dimensional (2d-D) VHaS, S(t) : [0,1] → C_2d, is defined by a complexification of two d-D Vector-valued Hypercomplex Signals (VHcS), S(t) := G(t)e₀ + H^C2d_ed[G](t) e_d, where H^C2d_ed and e_i represent the Hilbert transform and the i_th unit axis, and G(t) ϵ C_d, e_i G C_2d. Inspired by the unique polar form of a classic analytic signal and the one of a 4-D VHaS proposed in the work of Huang and Kunoth (2014), we provide a theoretical explanation of the unique polar representation of a 6-D or 8-D VHaS by replacing the quaternion with octonion, which further implies the possible extension for d-D VHaS with d> 8. Moreover, the derived continuous VHcS envelope and phase from the polar form lead to a unified definition of the time-frequency-amplitude spectrum of the given VHcS G(t).