Abstract 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.

中文翻译：

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跳跃离散分数傅立叶变换

摘要 离散分数阶傅里叶变换（DFrFT）是一种强大的非平稳信号处理工具。许多类型的 DFrFT 已被派生并成功应用于不同领域。但是，对于需要在每个或多个样本处重新计算 DFrFT 的实时应用程序，现有的离散算法并不是最佳选择。本文采用滑动窗口算法来解决这个问题。首先，提出了滑动步长为 p 的滑动 DFrFT (SDFrFT) 算法，称为跳跃 DFrFT (HDFrFT) 算法。还提出了两种不同的加窗方法，可以在滑动过程中实现加窗，以减少分数谱泄漏。第二，我们将滑动窗口算法应用于计算离散时间分数傅里叶变换 (DTFrFT) 并提出跳频 DTFrFT (HDTFrFT) 算法以获得连续分数谱。第三，滑动算法进一步扩展到分别计算离散分数余弦/正弦/哈特利变换 (DFrCT/DFrST/DFrHT)。最后，仿真结果证实，在滑动过程中，我们提出的滑动算法可以在不降低精度的情况下大大降低计算复杂度。