A Fourier series is a representation of a function in terms of a summation of an infinite number of harmonically-related sinusoids with different amplitudes and phases. The amplitude and phase of a sinusoid can be combined into a single complex number, called a Fourier coefficient. The Fourier series is a periodic function. So it cannot represent any arbitrary function. It can represent either:
![\left\{ a_n\right\}](/Images/godic/202501/09/a6e60769526e4938ce4ebf07c6cd432a0314.png")
，其离散傅里叶级数为![\left\{ x_k \right\}](/Images/godic/202501/09/e8c1b32be4f5da07e5f052664224ae6c0314.png")
：![x[k]=\sum_{n=<N>} a_n\cdot e^{-in(\frac{2\pi}{N})k}](https://wiki-gateway.eudic.net/wikipedia_zh/I/m/bab1bf09dc143052dc4e89a9ea1139d1.png)
![a_n=\frac{1}{N}\sum_{k=<N>} x[k]\cdot e^{in(\frac{2\pi}{N})k}](https://wiki-gateway.eudic.net/wikipedia_zh/I/m/d659d96289589a707e551c7c0ad49578.png)
（k=<N>表示对一个周期N内的值求和）