Fourier Series Transformations | IB数学：傅里叶级数间的相互变换

📚 Fourier Series Transformations | IB数学：傅里叶级数间的相互变换

Fourier series can be written in several equivalent forms. Converting between these forms is a core skill in IB Mathematics, especially when analysing periodic functions, solving differential equations, or preparing for university-level physics and engineering.

傅里叶级数可以写成若干等价的形式。在 IB 数学中，学会在这些形式之间相互转换是一项核心技能，尤其是在分析周期函数、求解微分方程或为大学阶段的物理与工程学习做准备时。

1. The Trigonometric Fourier Series | 三角傅里叶级数

For a periodic function f(x) with period T, the angular frequency is ω₀ = 2π/T. The trigonometric Fourier series expresses f(x) as an infinite sum of sines and cosines.

对于周期为 T 的周期函数 f(x)，角频率为 ω₀ = 2π/T。三角傅里叶级数将 f(x) 表示成正弦函数与余弦函数的无穷级数。

f(x) = a₀/2 + ∑ (aₙ cos(nω₀x) + bₙ sin(nω₀x))

The coefficients are obtained by exploiting the orthogonality of trigonometric functions.

系数通过利用三角函数的正交性求得。

aₙ = (2/T) ∫₀ᵀ f(x) cos(nω₀x) dx, bₙ = (2/T) ∫₀ᵀ f(x) sin(nω₀x) dx

Here n is a non-negative integer for aₙ and a positive integer for bₙ. The term a₀/2 is the constant or DC component.

其中 n 对 aₙ 取非负整数，对 bₙ 取正整数。项 a₀/2 是常数项，也称直流分量。

2. Amplitude-Phase Form | 幅度-相位形式

Each pair of sine and cosine terms at the same frequency can be combined into a single cosine with an amplitude Aₙ and a phase shift φₙ.

在同一频率下的正弦项与余弦项可以合并为一个带有幅度 Aₙ 和相位偏移 φₙ 的余弦项。

aₙ cos(nω₀x) + bₙ sin(nω₀x) = Aₙ cos(nω₀x – φₙ)

The transformation rules are simple but must respect the correct quadrant for φₙ.

变换规则并不复杂，但计算 φₙ 时必须注意所在的象限。

Aₙ = √(aₙ² + bₙ²), φₙ = arctan(bₙ/aₙ)

If aₙ is negative, add π to φₙ. This form is particularly useful for sketching frequency spectra.

如果 aₙ 为负，则 φₙ 需要加上 π。这种形式在绘制频谱时尤其方便。

3. Complex Exponential Form | 复指数形式

Using Euler’s formula, the Fourier series can be rewritten as a sum of complex exponentials. This form is often more compact and algebraically convenient.

借助欧拉公式，傅里叶级数可以被改写为复指数函数的和。这种形式通常更紧凑，代数处理也更方便。

f(x) = ∑ cₙ exp(i n ω₀ x), n = -∞ … ∞

The complex coefficients are calculated by a single integral.

复系数通过一个积分即可计算。

cₙ = (1/T) ∫₀ᵀ f(x) exp(-i n ω₀ x) dx

For a real-valued function f(x), the coefficients satisfy c₋ₙ = cₙ*, where * means complex conjugate.

对于实值函数 f(x)，系数满足 c₋ₙ = cₙ*，其中 * 表示复共轭。

4. Converting Between Trigonometric and Exponential Forms | 三角形式与指数形式的相互转换

The two forms are linked by Euler’s identity: exp(iθ) = cosθ + i sinθ. This yields a direct algebraic correspondence between aₙ, bₙ and cₙ.

这两种形式通过欧拉恒等式 exp(iθ) = cosθ + i sinθ 联系。由此可以得到 aₙ、bₙ

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