Heisenberg's uncertainty principles for the 2-D nonseparable linear canonical transforms

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The uncertainty principles of the 1-D Fourier transform (FT), the 1-D fractional Fourier transform (FRFT), and the 1-D linear canonical transform (LCT) have already been derived. In this paper, we extend the previous works and derive the uncertainty principles for the two-dimensional nonseparable linear canonical transform (2-D NSLCT), including the complex input case, the real input case, and the case where det(B)=0 where B is a parameter subset of the 2-D NSLCT. Since the 2-D NSLCT is a generalization of many operations, with the derived uncertainty principles, the uncertain principles of many 2-D operations, such as the 2-D Fresnel transform, the 2-D FT, the 2-D FRFT, the 2-D LCT, and the 2-D gyrator transform, can also be found. Moreover, we find that the rotation, scaling, and chirp multiplication of the 2-D Gaussian function can minimize the product of the variances in the space and the transform domains.