Wigner–Ville Distribution Associated with Clifford Geometric Algebra <i>Cl</i>_n,0, <i>n</i>=3(mod 4) Based on Clifford–Fourier Transform

Abstract

In this study, the Wigner–Ville distribution is associated with the one sided Clifford–Fourier transform over Rn, n = 3(mod 4). Accordingly, several fundamental properties of the WVD-CFT have been established, including non-linearity, the shift property, dilation, the vector differential, the vector derivative, and the powers of τ∈Rn. Moreover, powerful results on the WVD-CFT have been derived such as Parseval’s theorem, convolution theorem, Moyal’s formula, and reconstruction formula. Eventually, we deduce a directional uncertainty principle associated with WVD-CFT. These types of results, as well as methodologies for solving them, have applications in a wide range of fields where symmetry is crucial.

Keywords

• Fourier transform
• Clifford–Fourier transform
• Wigner–Ville distribution
• Moyal’s formula
• uncertainty principle