The hopping discrete fractional Fourier transform

The discrete fractional Fourier transform (DFrFT) is a powerful signal processing tool for non-stationary signals. Many types of DFrFT have been derived and successful used in different areas. However, for real-time applications that require recalculating the DFrFT at each or several samples, the existing discrete algorithms aren't the optimal. In this paper, the sliding window algoritm is used to resolve this problem. First, the sliding DFrFT (SDFrFT) algorithm with sliding step p is proposed, termed as the hopping DFrFT (HDFrFT) algorithm. Two different windowing methods which can realize windowing in the sliding process are also proposed to reduce fractional spectral leakage. Second, we apply the sliding window algorithm in computing the discrete time fractional Fourier transform (DTFrFT) and propose the hopping DTFrFT (HDTFrFT) algorithm to obtain a continuous fractional spectrum. Third, the sliding algorithm is further extended to compute the discrete fractional cosine/sine/Hartley transform (DFrCT/DFrST/DFrHT), respectively. Finally, the simulations results confirm that in a sliding process, our proposed sliding algorithms can greatly reduce the computation complexity without degrading the precision.