Discrete linear canonical Wigner distribution function

Recently, numerous novel phase-space distributions have been proposed by combining the linear canonical transform (LCT) and the Wigner distribution function (WDF), known as the linear canonical Wigner distributions (LCWDs). Among them, the closed-form instantaneous cross-correlation function type of Wigner distribution (CICFWD) has been proven to provide better flexibility for non-stationary signal processing and detection, due to its additional free parameters. Currently, the discrete CICFWD is only defined under a specific LCT parameter constraint and lacks a sampling theorem. This limits its applicability, since in numerical simulations, experiments, and practical optical systems, continuous fields or signals are discretized through digital sampling processes. In this paper, we will develop a general discrete CICFWD formulation without limitations, propose a numerical calculation algorithm, and a corresponding sampling theorem. We will also demonstrate their correctness by comparing the analytical CICFWDs of continuous test signals with their numerically approximated counterparts. Our results will enable more efficient and accurate numerical simulations using CICFWDs, which will be useful in applying them to problems, including optical measurements based on Newton’s rings, numerical optical signal processing, partial coherence, broadband communication and radar systems.