Uncertainty Principles for Wigner–Ville Distribution Associated with the Quaternion Offset Linear Canonical Transform

Wigner–Ville distribution (WVD) associated with the quaternion offset linear canonical transform (QOLCT) (WVD–QOLCT) is the known furthest generalization of the WVD in quaternion algebra. WVD–QOLCT is a hybrid transform that combines the flexibility and results of both WVD and QOLCT. Recently, some properties and classical Heisenberg's uncertainty principle (UP) have been derived for the two-dimensional (2D) two-sided WVD–QOLCT. This paper complements it by presenting conjugation symmetry and nonlinearity properties. Then, to characterize the simultaneous localization of a signal and its WVD–QOLCT, we establish different UPs for the 2D WVD–QOLCT, such as logarithmic UP, Hardy's UP, and Beurling's UP. In the end, by using the nonlinearity property, the applications of the 2D WVD–QOLCT in the linear frequency modulated signal detection are proposed.