We generalize the linear canonical transform (LCT) to quaternion-valued signals, known as the quaternionic linear canonical transform (QLCT). Using the properties of the LCT we establish an uncertainty principle for the QLCT. This uncertainty principle prescribes a lower bound on the product of the effective widths of quaternion-valued signals in the spatial and frequency domains. It is shown that only a 2D Gaussian signal minimizes the uncertainty.
"On Uncertainty Principle for Quaternionic Linear Canonical Transform." Abstr. Appl. Anal. 2013 1 - 14, 2013. https://doi.org/10.1155/2013/725952