Relations Between Gabor Transforms and Fractional Fourier Transforms and Their Applications for Signal Processing

Many useful relations between the Gabor transform (GT) and the fractional Fourier transform (FRFT) have been derived. First, we find that, like the Wigner distribution function (WDF), the FRFT is also equivalent to the rotation operation of the GT. Then, we show that performing the scaled inverse Fourier transform (IFT) along an oblique line of the GT of f(t) can yield its FRFT. Since the GT is closely related to the FRFT, we can use it for analyzing the characteristics of the FRFT. Compared with the WDF, the GT does not have the cross-term problem. This advantage is important for the applications of filter design, sampling, and multiplexing in the FRFT domain. Moreover, we find that if the GT is combined with the WDF, the resultant operation [called the Gabor-Wigner transform (GWT)] also has rotation relation with the FRFT. We also derive the general form of the linear distribution that has rotation relation with the FRFT.