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In this paper, we develop the generalized discrete fractional Fourier transform (GDFRFT) by factorizing the generalized discrete Fourier transform (GDFT) matrix. Specifically, the eigenvalues and eigenvectors are presented and then used to define the GDFRFT. It is shown that the GDFRFT may be obtained by simple similarity transformations of the conventional DFRFT if the time and frequency shifting indexes are integers. Fast and efficient structures are presented and integer structures are also discussed.