傅里叶级数及其在信号处理领域的应用
Fourier Series and Its Application in Signal Processing

傅里叶级数作为一种重要的数学工具，能够将周期函数分解为一系列正弦和余弦函数的线性组合，从而揭示信号的频域特性。本文首先介绍了傅里叶级数的数学定义及其展开形式，详细推导了傅里叶系数的计算公式，并阐述了其理论基础。随后，重点探讨了傅里叶级数在信号处理领域的应用，包括频谱分析、信号滤波和信号重构等方面。通过具体实例，展示了傅里叶级数在音频信号处理、图像处理以及通信系统中的实际应用。研究结果表明，傅里叶级数不仅是理论分析的有力工具，也在实际工程问题中具有广泛的应用价值，为信号处理技术的发展提供了重要的数学支持。
As an important mathematical tool, Fourier series can decompose periodic functions into a linear combination of sine and cosine functions, thereby revealing the frequency-domain characteristics of signals. This paper first introduces the mathematical definition and expansion form of Fourier series, derives the calculation formulas for Fourier coefficients in detail, and elaborates on its theoretical foundation. Subsequently, it focuses on the application of Fourier series in the field of signal processing, including spectrum analysis, signal filtering, and signal reconstruction. Through specific examples, the practical applications of Fourier series in audio signal processing, image processing, and communication systems are demonstrated. The research results show that Fourier series is not only a powerful tool for theoretical analysis but also has extensive application value in practical engineering problems, providing crucial mathematical support for the development of signal processing technologies.