【摘要】 研究目的是在计算机上数值实现信号的分数阶微积分。首先,分析比较分数阶微积分常用的3种时域定义,以及其在傅立叶变换域和子波变换域中的两种频域定义;然后,推导比较信号分数阶微分的幂级数数值算法、傅里叶级数数值算法、基于Grümwald-Letnikov定义的数值算法之间的优劣;进而,推导具有较高精度和计算速度的基于子波变换的分数阶微积分快速数值算法;最后,以计算精度为代价进一步提高计算速度,推导基于子波变换和连续内插的快速工程算法。理论推导和实验结果均证明基于子波变换的数值算法具有较高精度和运算速度,其改进的快速工程算法运算速度最高,但精度下降。这两种算法都具有较强的实用价值。更多还原

【Abstract】 This paper is to do numerical fractional calculus of modern signal on computers. In the first place, it compares three common time domain definitions of fractional calculus, the two frequency domain definitions in Fourier transformation domain and wavelet transformation domain respectively. Then, it deduces and compares the advantages and disadvantages of three numerical algorithms of power series, Fourier series and Grümwald-Letnikov-based of signal’s fractional calculus. Furthermore, it deduces the fast numerical algorithm, which is based on wavelet transform and has higher calculating precision and faster speed. In the last, it gets a fast engineering algorithm based on wavelet transforms and continuous interpolation, which dramatically improves the calculating speed and reduces the precision. Computer simulation is well confirmed with theoretical conclusions. So, we can get from the paper that wavelet-based numerical algorithm has higher calculating precision and faster calculating speed, and the improved engineering algorithm has the fastest calculating speed and the lowest precision. These two algorithms, however, both have great values in practice.更多还原