Fast Numerical Computation of Two-Dimensional Non-Separable Linear Canonical Transform Based on Matrix Decomposition

Two-dimensional non-separable linear canonical transform (2D NSLCT), as a generalized form of linear canonical transform (LCT) and 2D separable linear canonical transform (2D SLCT), has important applications in many engineering fields. Therefore, accurate and efficient digital implementation is urgent. In this paper, a new four-stage parameter matrix decomposition method is proposed for the efficient implementation of 2D NSLCT. According to the theory of matrix decomposition, we construct a set of decomposition forms which are composed of 2D chirp multiplication (2D CM), 2D Fourier transform (2D FT), 2D affine transformation (2D AT) and 2D CM. The decomposition not only keeps the form simple, but also achieves the ideal effect in a larger range of conditions. Compared with previous methods, the proposed method shows much smaller error and higher accuracy level when the adjustable sampling interval is greater than or equal to 0.05 in the intermediate stage of the algorithm. What's more, simulation results show that our method has conspicuous advantages when changing different parameters and input functions. In addition, the additivity and reversibility of 2D NSLCT also have good performances. Finally, we construct a new optical system and apply our method to it. Simulation results confirm that the proposed method is still more efficient and exact in optical system analysis.