分数阶微积分运算数字滤波器设计与电路实现及其应用
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摘要
分数阶微积分运算包括分数阶微分运算和分数阶积分运算,它的含义就是将普通意义下的微积分运算的运算阶次从整数阶推广到分数和复数的情况。从1695年Leibniz与Hospital的最早提出开始,到现在已经有三百多年历史,由于实现计算复杂度比较高的原因,因此一直只能局限于理论研究领域。近年来,随着计算机科学的发展,计算能力的提高,分数阶微积分的计算和实现成为可行,分数阶微积分运算才被工程研究人员所认识和研究。
分数阶微积分由于独特的对信号分析和处理的性质,其实现的阶次灵活性,自由度也更大,因此被逐渐应用于工程实践中,并取得很好的应用效果。目前分数阶微积分应用在多个领域中:控制理论、信号处理、机械力学、电子学、化学、生物学、经济学、流变力学、机器人、材料科学、岩石力学、地震信息处理、分形理论、电磁场理论等。特别是在信息科学领域中,一些新颖的应用被相继地实现和提出,如系统建模、曲线拟合、信号滤波、模式识别、图像边界提取、系统辨识、系统稳定性分析等等。
本文从工程的角度出发,研究了分数阶微积分运算的实现,包括分数阶微积分数字滤波器实现和模拟电路实现。本文的主要工作有:
1、较为系统地分析和总结了分数阶微积分的基本理论,包括分数阶微积分运算的提出与发展历程、研究和应用现状、分数阶微积分的各种定义及其之间的转换、具有的性质、已提出的物理意义和几何意义解释、分数阶微分方程概念、自然界存在的材料实现以及几种分数阶微积分运算电路实现方案。
Fractional calculus includes fractional derivatives and fractional integrals. It means to generalize the differentiation and integration into fractional and complex order. It is a more than 300-years-old topic from the first raised by Leibniz and Hospital in 1695. Because of the high computation complexity, it was restricted in the theoretical research field by mathematicians. With the high speed development of computer science and the increasing ability of calculation in recent years, the realization of fractional calculus becomes feasible and fractional calculus was noticed and researched by more and more engineers.
Fractional calculus has special capabilities for signal processing and analyzing. It has more degrees of freedom to be used and gets good performance in many fields, such as control theory, signal processing, mechanics, electronics, chemistry, biology, economy, rheology,robotics, materials,rock mechanics, fractal,seismic data processing,electromagnetic theory. Especially in information technology science domain, several new applications have been developed, for example, system modeling, curve fitting, filtering, pattern recognition, contour detection, system identification, stability analysis, and etc.
From the point of technology view, this dissertation does some research on the implementation of fractional calculus, including the digital filter design and analog
分数阶微积分由于独特的对信号分析和处理的性质,其实现的阶次灵活性,自由度也更大,因此被逐渐应用于工程实践中,并取得很好的应用效果。目前分数阶微积分应用在多个领域中:控制理论、信号处理、机械力学、电子学、化学、生物学、经济学、流变力学、机器人、材料科学、岩石力学、地震信息处理、分形理论、电磁场理论等。特别是在信息科学领域中,一些新颖的应用被相继地实现和提出,如系统建模、曲线拟合、信号滤波、模式识别、图像边界提取、系统辨识、系统稳定性分析等等。
本文从工程的角度出发,研究了分数阶微积分运算的实现,包括分数阶微积分数字滤波器实现和模拟电路实现。本文的主要工作有:
1、较为系统地分析和总结了分数阶微积分的基本理论,包括分数阶微积分运算的提出与发展历程、研究和应用现状、分数阶微积分的各种定义及其之间的转换、具有的性质、已提出的物理意义和几何意义解释、分数阶微分方程概念、自然界存在的材料实现以及几种分数阶微积分运算电路实现方案。
Fractional calculus includes fractional derivatives and fractional integrals. It means to generalize the differentiation and integration into fractional and complex order. It is a more than 300-years-old topic from the first raised by Leibniz and Hospital in 1695. Because of the high computation complexity, it was restricted in the theoretical research field by mathematicians. With the high speed development of computer science and the increasing ability of calculation in recent years, the realization of fractional calculus becomes feasible and fractional calculus was noticed and researched by more and more engineers.
Fractional calculus has special capabilities for signal processing and analyzing. It has more degrees of freedom to be used and gets good performance in many fields, such as control theory, signal processing, mechanics, electronics, chemistry, biology, economy, rheology,robotics, materials,rock mechanics, fractal,seismic data processing,electromagnetic theory. Especially in information technology science domain, several new applications have been developed, for example, system modeling, curve fitting, filtering, pattern recognition, contour detection, system identification, stability analysis, and etc.
From the point of technology view, this dissertation does some research on the implementation of fractional calculus, including the digital filter design and analog
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