Papoulis' sampling theorem: Revisited

Multi-dimensional sampling approaches are often faced with certain intricacies as we need to deal with matrices and the non-commutativity of matrices precludes straightforward extensions of the usual one-dimensional results. In this article, we reformulate the Papoulis' sampling theorem for the reconstruction of higher-dimensional signals that are band-limited in the sense of free metaplectic transformation, which encapsulates the classical Fourier transform as well as the ensuing generalized phase-space transforms. The key idea is to invoke the arbitrary lattice sampling of the higher-dimensional Euclidean space based on general separable or non-separable matrices. Besides, a newly formulated convolution operation satisfying the fundamental properties of commutativity and associativity is employed for the construction of a set of filters allowing data acquisition on a per-channel basis, which is an intrinsic feature of the Papoulis' sampling theorem. Towards the culmination, it is demonstrated that the proposed sampling procedure is also of substantial importance in the context of fan beam tomography .

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