The generalized fractional fourier transform

A new transform, called the generalized fractional Fourier transform (gFrFT), is proposed. Originally, the eigenfunctions of the fractional Fourier transform (FrFT) are known as the Hermite Gaussian functions (HGFs). Besides, in optics, the HGFs are generalized to be the generalized Hermite Gaussian functions (gHGFs) and their adjoint functions (AgHGFs). Therefore, we can define the gFrFT by the eigenvalues of the FrFT and the eigenfunctions (gHGFs/AgHGFs) in the analysis or synthesis step. Four types of the gFrFT are defined and discussed. The integral forms of the gFrFTs are derived and they are closely related to some popular transforms, such as the Fourier transform (FT), the FrFT, and the complex linear canonical transform (CLCT). We can also extend the FT and the FrFT to the standard and elegant versions. Finally, some properties of the gFrFT are discussed.