Uncertainty Principle for the Two-Sided Quaternion Windowed Linear Canonical Transform

In this paper, we investigate the (two-sided) quaternion windowed linear canonical transform (QWLCT) and study the uncertainty principles associated with the QWLCT. First, several important properties of the QWLCT such as bounded, shift, modulation and orthogonality relations are presented based on the spectral representation of the quaternionic linear canonical transform (QLCT). Second, Pitt’s inequality and the Lieb inequality for the QWLCT are explored. Moreover, we study different kinds of uncertainty principles for the QWLCT, such as the logarithmic uncertainty principle, the entropic uncertainty principle, the Lieb uncertainty principle and Donoho–Stark’s uncertainty principle. Finally, we provide a numerical example and a potential application to signal recovery by using Donoho–Stark’s uncertainty principle associated with the QWLCT.