Hyperbolic linear canonical transforms of quaternion signals and uncertainty

This paper is concerned with Linear Canonical Transforms (LCTs) associated with two-dimensional quaternion-valued signals defined in an open rectangle of the Euclidean plane endowed with a hyperbolic measure, which we call Quaternion Hyperbolic Linear Canonical Transforms (QHLCTs). These transforms are defined by replacing the Euclidean plane wave with a corresponding hyperbolic relativistic plane wave in one dimension multiplied by quadratic modulations in both the hyperbolic spatial and frequency domains, giving the hyperbolic counterpart of the Euclidean LCTs. We prove the fundamental properties of the partial QHLCTs and the right-sided QHLCT by employing hyperbolic geometry tools and establish main results such as the Riemann–Lebesgue Lemma, the Plancherel and Parseval Theorems, and inversion formulas. The analysis is carried out in terms of novel hyperbolic derivative and hyperbolic primitive concepts, which lead to the differentiation and integration properties of the QHLCTs. The results are applied to establish two quaternionic versions of the Heisenberg uncertainty principle for the right-sided QHLCT. These uncertainty principles prescribe a lower bound on the product of the effective widths of quaternion-valued signals in the hyperbolic spatial and frequency domains. It is shown that only hyperbolic Gaussian quaternion functions minimize the uncertainty relations.