Generalized Hilbert transform and its properties in 2D LCT domain

Hilbert transform plays an important role in signal processing. With the development of new transforms, one-dimensional (1D) Hilbert transform has been extended into fractional Fourier transform domain. However, the researches of two-dimensional (2D) Hilbert transform in linear canonical transform (LCT) domain become complicated for the reasons of the complexity of 2D signals and more parameters in LCT, and now they are in the infancy. In this paper, the definitions of half-planed Hilbert transform, cross-orthant Hilbert transform and single-orthant Hilbert transform are yielded in LCT domain. In addition, the relation between time domain and transformed domain is discussed. Moreover, some important properties and conclusions are obtained as well. Finally, we defined and derived 2D Bedrosian's principle in LCT domain.