摘 要：分数阶微积分作为传统整数阶微积分的推广，近年来在信号处理领域展现出独特优势和广阔应用前景本文旨在探讨分数阶微积分理论在信号建模、分析与处理中的应用价值，通过引入分数阶导数和积分算子，构建更加灵活的信号描述框架研究发现，分数阶微积分能够有效刻画信号的非局部特性和记忆效应，尤其在复杂系统建模和奇异信号分析中表现出显著优越性本研究基于分数阶微分方程建立了新型信号处理模型，并提出了一种高效的数值求解算法，该算法显著提升了信号特征提取的精度和效率实验结果表明，相比传统方法，分数阶微积分方法在噪声抑制、边缘检测及信号重构等方面具有更强适应性和鲁棒性这一创新性工作不仅拓展了分数阶微积分的应用范围，还为信号处理领域的理论发展提供了新思路
关键词：分数阶微积分；信号处理；非局部特性；记忆效应；数值求解算法
Application of Fractional Calculus in Signal Processing
英文人名
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Abstract：Fractional calculus, as a generalization of traditional integer-order calculus, has demonstrated unique advantages and broad application prospects in the field of signal processing in recent years. This study aims to explore the application value of fractional calculus theory in signal modeling, analysis, and processing by introducing fractional-order derivative and integral operators to construct a more flexible fr amework for signal desc ription. Research findings indicate that fractional calculus can effectively characterize the non-local properties and memory effects of signals, showing significant superiority particularly in the modeling of complex systems and the analysis of singular signals. Based on fractional differential equations, this study establishes a novel signal processing model and proposes an efficient numerical solution algorithm, which significantly enhances the accuracy and efficiency of signal feature extraction. Experimental results show that compared with traditional methods, fractional calculus-based approaches exhibit stronger adaptability and robustness in aspects such as noise suppression, edge detection, and signal reconstruction. This innovative work not only expands the application scope of fractional calculus but also provides new insights into the theoretical development of signal processing.
Keywords: Fractional Calculus；Signal Processing；Non-Local Property；Memory Effect；Numerical Solution Algorithm
目 录
引言 1
一、分数阶微积分基础理论 1
(一)分数阶微积分定义与性质 1
(二)常用分数阶算子介绍 2
(三)分数阶微积分计算方法 2
二、信号处理中的分数阶模型构建 3
(一)分数阶模型的基本形式 3
(二)模型参数的优化设计 3
(三)模型在实际信号中的应用 4
三、分数阶滤波器的设计与实现 4
(一)分数阶滤波器的理论基础 4
(二)滤波器性能分析与改进 5
(三)实验验证与结果分析 5
四、分数阶微积分在复杂信号处理中的应用 6
(一)复杂信号特征提取方法 6
(二)分数阶算法在降噪中的应用 6
(三)非线性信号处理案例研究 7
结论 7
参考文献 9
致谢 9
