Fourier Series

The Significance of Fourier Series

The Fourier series represents a periodic function as a weighted sum of sine and cosine functions with different frequencies. Thus, the Fourier series can be seen as a frequency analysis of periodic functions. Fourier series have wide applications in mathematics, physics, engineering, and other fields.

The Formula of Fourier Series

The frequency components of a periodic signal will have a lowest frequency component called the fundamental frequency, and other frequency components will be multiples of this frequency, also known as the related harmonics. The coefficients of these frequency components can be calculated using the Fourier series. If the signal is not periodic but has a limited interval, we can extend this interval repeatedly, treating it as a periodic function.

Assuming a function $s(x)$ is defined in the interval $[x_0, x_0+P]$, its Fourier series is

$$\frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n\cos(\frac{2\pi n x}{P}) + b_n\sin(\frac{2\pi n x}{P})),$$

where

$$a_n = \frac{2}{P}\int_{x_0}^{x_0+P} s(x)\cos(\frac{2\pi n x}{P})\ dx$$

$$b_n = \frac{2}{P}\int_{x_0}^{x_0+P} s(x)\sin(\frac{2\pi n x}{P})\ dx$$

where $a_n$ and $b_n$ are the $n$-th harmonic coefficients.

Applications of Fourier Series

Fourier series have broad applications in fields like mathematics, physics, and engineering. For example:

• In mathematics, Fourier series can be used to solve differential equations, integral equations, etc.
• In physics, Fourier series can be used to analyze periodic wave phenomena such as sound waves and light waves.
• In engineering, Fourier series can be applied in signal processing, image processing, and more.

Here are some specific application examples:

• In audio processing, the Fourier series can be used to decompose audio signals into components of different frequencies, thereby achieving functions such as audio filtering and compression.
• In image processing, the Fourier series can be used to decompose images into components of different frequencies, thereby achieving functions such as image denoising and enhancement.

Limitations of Fourier Series

The Fourier series is only applicable to periodic functions. For non-periodic functions, Fourier transform can be used for analysis.