Fourier Transforms of Basic Functions | 基本函数的傅里叶变换

📚 Fourier Transforms of Basic Functions | 基本函数的傅里叶变换

The Fourier transform is one of the most powerful tools in mathematics and the physical sciences. It decomposes a function of time (or space) into its constituent frequencies, revealing the spectrum hidden inside the signal. For IB students exploring further mathematics, physics, or engineering, a solid grasp of the Fourier transforms of basic functions is essential for building intuition and solving problems confidently.

傅里叶变换是数学和物理科学中最强大的工具之一。它把时间（或空间）的函数分解为组成它的各个频率成分，揭示隐藏在信号内部的频谱。对于学习进阶数学、物理或工程的 IB 学生来说，牢固掌握基本函数的傅里叶变换是建立直觉、自信解题的关键。

1. Definition and Notation | 定义与记号

We adopt the following convention for the Fourier transform of a function f(t):

F(ω) = ∫_−∞^∞ f(t) e^−iωt dt

and the corresponding inverse transform:

f(t) = (1/2π) ∫_−∞^∞ F(ω) e^iωt dω

Here f(t) is the time-domain signal and F(ω) is its frequency-domain spectrum, with ω representing angular frequency in radians per second. The factor 1/2π appears in the inverse transform; other textbooks may place factors differently, so always state your convention clearly.

这里 f(t) 是时域信号，F(ω) 是它的频域频谱，ω 表示以弧度每秒为单位的角频率。1/2π 出现在逆变换中；其他教材可能采用不同的因子放置方式，因此务必明确说明你所使用的约定。

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2. The Rectangular Pulse | 矩形脉冲（箱函数）

Consider a rectangular pulse of width 2a centered at the origin:

f(t) = 1 for |t| < a, and f(t) = 0 otherwise.

Its Fourier transform is straightforward to compute:

F(ω) = ∫_−a^a e^−iωt dt = [e^−iωt/(−iω)]_−a^a = 2 sin(ωa) / ω

Thus F(ω) = 2a·sinc(ωa/π), where the sinc function is sine divided by its argument. The spectrum has a main lobe and decaying side lobes, with zeros at ωa = π, 2π, 3π, … . Notice that a shorter pulse (smaller a) gives a wider spread of frequencies, illustrating the uncertainty principle between time and frequency.

因此 F(ω) = 2a·sinc(ωa/π)，其中 sinc 函数是正弦函数除以其自变量。频谱有一个主瓣和衰减的旁瓣，零点出现在 ωa = π, 2π, 3π, …。注意，脉冲越窄（a 越小），频谱越宽，这体现了时间与频率之间的不确定性原理。

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3. The Gaussian Function | 高斯函数

Now take a Gaussian function f(t) = e^−at² with a > 0. Completing the square in the exponent gives:

F(ω) = ∫_−∞^∞ e^−at² e^−iωt dt = √(π/a) e^−ω²/(4a)

A remarkable result: the Fourier transform of a Gaussian is also a Gaussian. If a = 1/2, then f(t) = e^−t²/2 and F(ω) = √(2π) e^−ω²/2. This self-similar property makes the Gaussian vital in probability, quantum mechanics, and signal processing, where it represents a state of minimum uncertainty.

一个惊人的结论是：高斯函数的傅里叶变换仍然是高斯函数。若 a = 1/2，则 f(t) = e^−t²/2，F(ω) = √(2π) e^−ω²/2。这种自相似性使高斯函数在概率论、量子力学和信号处理中至关重要，它代表着最小不确定性的状态。

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4. The Dirac Delta Function | 狄拉克 δ 函数

The Dirac delta function δ(t) is defined by the sifting property: for any well-behaved f(t),

∫_−∞^∞ f(t) δ(t) dt = f(0)

and extended to δ(t − t₀) by shifting. Its Fourier transform is beautifully simple:

δ(t) ⇌ 1, and δ(t − t₀) ⇌ e^−iωt₀

Conversely, the transform of a constant function is a delta in the frequency domain:

1 ⇌ 2π δ(ω)

This pair defines a pure direct-current (DC) component. The delta function allows us to treat impulses and periodic signals rigorously within the Fourier framework.

狄拉克 δ 函数由筛选性质定义：对任意性质良好的 f(t)，∫_−∞^∞ f(t) δ(t) dt = f(0)，并可推广到 δ(t − t₀)。它的傅里叶变换极为简洁：δ(t) ⇌ 1，δ(t − t₀) ⇌ e^−iωt₀。反过来，常数的变换是频域中的 δ 函数：1 ⇌ 2π δ(ω)。这一变换对刻画了纯直流分量。δ 函数使我们能够在傅里叶框架中严格处理冲激信号和周期信号。

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5. One-Sided Exponential Decay | 单边指数衰减

Consider the causal exponential:

f(t) = e^−at for t ≥ 0, and f(t) = 0 for t < 0, with a > 0

Then

F(ω) = ∫₀^∞ e^−at e^−iωt dt = ∫₀^∞ e^−(a+iω)t dt = 1/(a + iω)

The magnitude is |F(ω)| = 1/√(a² + ω²) and the phase is arg F(ω) = −arctan(ω/a). This is a classic Lorentzian (or Cauchy) form. If we instead use the two-sided exponential f(t) = e^−a|t|, the transform is real and even:

e^−a|t| ⇌ 2a/(a² + ω²)

单边指数衰减函数的变换是 F(ω) = 1/(a + iω)，其模为 |F(ω)| = 1/√(a² + ω²)，相位为 arg F(ω) = −arctan(ω/a)。这是典型的洛伦兹（柯西）形式。若改用双边指数衰减 f(t) = e^−a|t|，则变换为实偶函数：e^−a|t| ⇌ 2a/(a² + ω²)。

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6. Sine and Cosine Functions | 正弦与余弦函数

Sine and cosine are not absolutely integrable, but using the delta distribution we can still obtain their Fourier transforms. Recall e^iω₀t ⇌ 2πδ(ω − ω₀). With Euler’s formula:

cos(ω₀t) ⇌ π[δ(ω − ω₀) + δ(ω + ω₀)]

sin(ω₀t) ⇌ −iπ[δ(ω − ω₀) − δ(ω + ω₀)]

These spectra consist of two impulses located at ±ω₀, with cosine giving a real even spectrum and sine giving an imaginary odd spectrum. This result underlies Fourier series and the analysis of periodic signals, where each frequency component appears as a pair of symmetric peaks.

正弦和余弦函数并非绝对可积，但借助 δ 分布仍可得到其傅里叶变换。由欧拉公式可得：cos(ω₀t) ⇌ π[δ(ω − ω₀) + δ(ω + ω₀)]，sin(ω₀t) ⇌ −iπ[δ(ω − ω₀) − δ(ω + ω₀)]。它们的频谱由位于 ±ω₀ 的两个冲激组成：余弦的频谱是实偶函数，正弦的频谱是虚奇函数。这一结论是傅里叶级数和周期信号分析的基础，每个频率分量都表现为一对对称的尖峰。

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7. Heaviside Step and Sign Functions | 海维赛德阶跃函数与符号函数

The Heaviside step function u(t) = 1 for t > 0, 0 for t < 0, has a distributional Fourier transform:

u(t) ⇌ πδ(ω) + 1/(iω)

Since the sign function sgn(t) = 2u(t) − 1, we obtain:

sgn(t) ⇌ 2/(iω)

These transforms are indispensable in control theory and signal analysis. They show that step-like discontinuities produce a 1/ω frequency tail plus a DC delta component; the sign function, being purely odd, has no delta term.

海维赛德阶跃函数 u(t)（t > 0 时为 1，t < 0 时为 0）的分布傅里叶变换为 u(t) ⇌ πδ(ω) + 1/(iω)。由于符号函数 sgn(t) = 2u(t) − 1，因此 sgn(t) ⇌ 2/(iω)。这些变换在控制理论和信号分析中不可或缺。它们表明，阶跃型不连续会产生 1/ω 的频谱尾部加上直流 δ 分量；而符号函数是纯奇函数，不含 δ 项。

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8. Key Properties | 重要性质

The following properties allow us to transform complicated functions by manipulating known transforms. Here a, b are constants and t₀, ω₀, a (in scaling) are real constants.

Property / 性质	Time domain / 时域	Frequency domain / 频域
Linearity / 线性	af(t) + bg(t)	aF(ω) + bG(ω)
Time shift / 时移	f(t − t₀)	e−iωt₀F(ω)
Frequency shift / 频移	eiω₀t f(t)	F(ω − ω₀)
Time scaling / 时间缩放	f(at), a ≠ 0	(1/|a|)F(ω/a)
Time reversal / 时间反转	f(−t)	F(−ω)
Differentiation / 微分	f′(t)	iωF(ω)
Convolution / 卷积	(f * g)(t)	F(ω)G(ω)

These properties are extremely useful in differential equations and system analysis. For example, the differentiation property converts an ordinary differential equation into an algebraic equation, which can be solved for F(ω) and then transformed back.

这些性质在微分方程和系统分析中极其有用。例如，微分性质把常微分方程转化为代数方程，可先求解 F(ω)，再逆变换回时域。

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9. Applications and Exam Tips | 应用与考试要点

Fourier transforms appear throughout science and engineering. In optics, the diffraction pattern of a single slit is exactly the Fourier transform of the slit’s aperture function, as derived in Section 2. In quantum mechanics, the momentum-space wavefunction is the Fourier transform of the position-space wavefunction. In audio processing, removing unwanted frequency components relies on the spectral representation discussed here.

傅里叶变换贯穿科学与工程领域。在光学中，单缝衍射图样正是缝孔径函数的傅里叶变换，正如第 2 节所推导。在量子力学中

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