Offset quaternion linear canonical transform: Properties, uncertainty inequalities and application

In this present work, we first establish some basic properties of the offset quaternion linear canonical transform such as shifting and modulation, which are missed in the existing literature. We then present the relation of the quaternion Fourier transform to the quaternion linear canonical transform and the offset quaternion linear canonical transform. We also make a direct connection between the quaternion linear canonical transform and the offset quaternion linear canonical transform. By means of the properties and relations, we derive an analogue of sharp Hausdorff–Young inequality, Matolcsi-Szücs uncertainty principle, logarithmic Sobolev-type uncertainty inequality and Benedicks–Amrein–Berthier uncertainty inequality in the framework of the offset quaternion linear canonical transform. Additionally, we implement the quaternionic Gabor filter to verify sharp Hausdorff–Young inequality concerning the considered transformation. Finally, the utility of the proposed offset quaternion linear canonical transform in the quaternion linear frequency modulated (QLFM) signal is studied.