IB Mathematics: Bilateral Fourier Series — Concepts and Applications | IB数学：双边傅里叶级数的概念与应用

📚 IB Mathematics: Bilateral Fourier Series — Concepts and Applications | IB数学：双边傅里叶级数的概念与应用

Fourier series is one of the most powerful tools in applied mathematics. It represents a periodic function as a sum of simple oscillating terms, allowing us to analyse signals, solve differential equations, and understand resonance.

傅里叶级数是应用数学中最强大的工具之一。它将周期函数表示为简单振荡项之和，使我们能够分析信号、求解微分方程并理解共振现象。

The bilateral, or two-sided, Fourier series uses complex exponentials rather than separate sine and cosine terms. Its key feature is that the index n runs over all integers, from −∞ to ∞, so positive and negative frequencies are treated on exactly the same footing.

双边傅里叶级数（又称两态傅里叶级数）使用复指数而不是单独的 sin 和 cos 项。其关键特点是求和指标 n 取遍所有整数，从 −∞ 到 ∞，因此正频率与负频率被完全平等地处理。

1. Definition and Notation | 定义与记号

Let f(t) be periodic with period T and fundamental angular frequency ω₀ = 2π/T. The bilateral Fourier series is written as

设 f(t) 是周期为 T 的函数，基波角频率为 ω₀ = 2π/T。双边傅里叶级数写为

f(t) = Σ_n=−∞^∞ cₙ e^{i n ω₀ t}

The coefficients cₙ are found by projecting f(t) onto each basis function e^{i n ω₀ t}.

系数 cₙ 通过将 f(t) 向每个基函数 e^{i n ω₀ t} 上投影来获得。

cₙ