Some important fractional transformations for signal processing

The fractional Fourier transform (FRT), that is useful mathematical and optical tool for signal processing, was defined as a generalization of the conventional Fourier transform. As opposed to the Fourier transform, the Hartley transform is a real (not complex) mathematical transformation and thus might be attractive for various applications. In optics, due to the fact that it is a real operation, it can be implemented with incoherent illumination. This paper suggests a generalization of the Hartley transformation based on the fractional Fourier transform. We coined it "fractional Hartley transform (FHT)". Possible optical implementation can be easily obtained optically. Since the definition is real, there is an additional significant for digital signal processing application where the fact that the transform is real decreases the computing complexity. Additional useful transformations used for signal processing are discussed as well.