分数阶数字FIR微分器的快速WSLD设计算法

一阶逼近格林瓦尔-莱特尼科夫(G-L)加权系数的计算具有准确快速的递推公式，而高阶逼近鲁比希加权系数的求解则复杂度高，计算消耗时间长。本文通过傅里叶变换证明了鲁比希算子的逼近阶，并基于移位鲁比希算子提出一类四阶逼近的加权移位鲁比希差分(WSLD)算子。从数字信号处理角度分析WSLD算子滤波特性，设计基于WSLD算子的分数阶数字FIR微分滤波器并进行数值仿真验证。对比Al-Alaoui、鲁比希2种典型分数阶算子，结果表明，利用WSLD算子求解分数阶数字FIR滤波器滤波系数的算法简单、高效，且相对其他算子能有效减小吉布斯效应影响。

Fast WSLD design algorithm for digital FIR fractional differentiator

The calculation of Grünwald-Letnikov''s(G-L) weighted coefficient of first-order approximation has an accurate and fast recursive formula, while the calculation of Lubich''s weighted coefficient of high-order approximation has a high complexity and a long calculation time. The approximation order of Lubich operator is proved by Fourier transform, and a class of Weighted Shifted Lubich Difference(WSLD) operators of fourth order approximation is proposed based on shifted Lubich operators. Then, the filtering characteristics of WSLD operator are analyzed from the perspective of digital signal processing, and a fractional digital Finite Impulse Response(FIR) differential filter based on WSLD operator is designed and verified by numerical simulation. Compared with Al-Alaoui and Lubich,the results show that the algorithm using WSLD operator to solve the filter coefficient of fractional digital FIR filter is simple and efficient, and can effectively reduce the influence of Gibbs effect compared with other operators.