Relations between Gabor transforms and fractional Fourier transforms and their applications for signal processing

Many wonderful relations between the Gabor transform and the fractional Fourier transform (FRFT), which is a generalization of the Fourier transform, are derived. First, we find that, as the Wigner distribution function (WDF), the FRFT is also equivalent to the rotation operation of the Gabor transform. We also derive the shifting, the projection, the power integration, and the energy sum relations between the Gabor transform and the FRFT. Since the Gabor transform is closely related to the FRFT, we can use it for analyzing the effect of the FRFT. Compared with the WDF, the Gabor transform does not have the problem of cross terms. It makes the Gabor transform a very powerful assistant tool for fractional sampling and designing the filter in the FRFT domain. Moreover, we show that any combination of the WDF and the Gabor transform also has the rotation relation with the FRFT.