Clifford-valued linear canonical transform: Convolution and uncertainty principles

Abstract

Linear canonical transform is a three-parameter class of linear integral transformation, which has found many applications in optics, filter design, speech, image, video, and signal processing due to its flexibility in representing information within the linear canonical domain. In this article, we extend the linear canonical transform to the class of Clifford-valued signals and introduce a novel transform coined as Clifford-valued linear canonical transform. The primary analysis encompasses the derivation of fundamental properties of the proposed transform such as translation and scaling covariance, inner product relation and inversion formula. Subsequently, we investigate the convolution, continuity, and differentiation theorems for the Clifford-valued linear canonical transform. A non-homogeneous partial differential equation has also been solved in the Clifford framework as an application of the differentiation theorem. Finally, we culminate our investigation by deriving several uncertainty principles for the proposed transform.