A novel method for discrete fractional Fourier transform computation

Fractional Fourier transform (FRFT) is a generalization of Fourier transform, and it indicates a rotation of signal in the time-frequency plane[l). The FRFT has been widely successfully used in the many areas[l][2]. Because of the importance of FRFT, discrete fractional Fourier transform (DFRFT) becomes an important issue in recent years[3][4] [5][6]. In the development of DFRFT, the DFRFT has been considered as the combination of four parts[3]: the original signal, its DFT, circular flipped of signal and circular flipped of its DFT. This method can be realized by the DFT fast computation aJgorithm[7], but it can not have the similar results as continuous case. In 1996, we have found that the DFRFT with DFT Hermite eigenvectors can have similar outputs as those of the continuous case[5][6]. These DFRFTs use the DFT Hermite eigenvectors as their eigenvectors and have similar eigen decomposition form as continuous FRFT kernel. They can perform a rotation of discrete signals in the time-frequency plane, and have the mixed time and frequency characteristics of discrete signals. The eigen decomposition method have been proved and justified in [SI, and it is successfully used in many applications. The DFRFT discussed in this paper is for the eigen decomposition method[5][6]. Although the eigen decomposition method can have similar results as continuous case, its computation cost is very large. The goal of this paper is to introduce a novel computation method for the DFRFTs