Generalized sampling expansion for the quaternion linear canonical transform

The theory of quaternions has gained firm ground in recent times and is being widely explored, with the field of signal and image processing being no exception. However, many important aspects of quaternionic signals are yet to be explored, particularly the formulation of generalized sampling expansions (GSE). In the present article, our aim is to formulate the GSE in the realm of a one-dimensional quaternion linear canonical transform. To facilitate the intent, we construct a set of quaternionic filter functions which are used to construct a system of equations determining the synthesis functions for the process of reconstruction. Besides, as a special case, another sampling formula involving the derivatives of the quaternionic signal is also obtained in the sequel. The proposed method not only expands our understanding of quaternionic signal processing but also holds promising implications for various applications in the field. As an endorsement of the obtained results, an example with simulations demonstrating the signal reconstruction is presented at the end.