理想的分数阶数字微分器系数的解析表示  

Analysis representation of the coefficients of an ideal fractional-order digital differentiator

作　　者：杨劲松 杨紫怡 袁晓[1] YANG Jin-Song;YANG Zi-Yi;YUAN Xiao(College of Electronics and Information Engineering,Sichuan University,Chengdu 610065,China)

机构地区：[1]四川大学电子信息学院,成都610065

出　　处：《四川大学学报(自然科学版)》2025年第2期425-432,共8页Journal of Sichuan University(Natural Science Edition)

基　　金：国家自然科学基金(62171303)。

摘　　要：本文旨在论证理想的分数阶数字微分器系数等于Sinc函数分数阶导数整数点的取值,利用不完全伽马函数和超几何函数分别表示出理想分数阶数字微分器的系数和Sinc函数分数阶导数.根据不完全伽马函数的定义与性质,从理论上推导出理想的分数阶数字微分器系数的解析表示.根据正余弦函数的泰勒级数展开与超几何函数定义,从理论上推导出理想的分数阶数字微分器系数的超几何函数表示.通过与Sinc函数分数阶导数的高精度数值算法结果比较,验证了利用不完全伽马函数与超几何函数精确表示理想的分数阶数字微分器系数与Sinc函数分数阶导数的可行性.This paper demonstrates that the coefficients of an ideal fractional-order digital differentiator are equal to the values of the fractional-order derivatives of the Sinc function at integer points.The coefficients of the ideal fractional-order digital differentiator and the fractional-order derivatives of the Sinc function are represented using the incomplete gamma function and the hypergeometric function,respectively.Based on the definition and properties of the incomplete gamma function,the analytical representation of the coefficients of the ideal fractional-order digital differentiator is theoretically derived.Furthermore,by utilizing the Taylor series expansion of sine and cosine functions and the definition of the hypergeometric function,the hypergeometric function representation of the coefficients of the ideal fractional-order digital differentiator is theoretically derived.The feasibility of accurately representing the coefficients of the ideal fractional-order digital differentiator and the fractional-order derivatives of the Sinc function using the incomplete gamma function and the hypergeometric function is verified by comparison with high-precision numerical algorithm results for the fractional-order derivatives of the Sinc function.

关 键 词：分数阶微积分 单位冲激响应 Sinc函数 不完全伽马函数 超几何函数 解析表示 

分 类 号：TP911.72[自动化与计算机技术]