Fourier Transform Convolution Theorem | 傅里叶变换的卷积定理

📚 Fourier Transform Convolution Theorem | 傅里叶变换的卷积定理

The convolution theorem is one of the most beautiful and useful results in Fourier analysis. It tells us that convolution in the time domain corresponds to ordinary multiplication in the frequency domain. This apparently simple statement underlies much of signal processing, differential equations, probability and physics.

卷积定理是傅里叶分析中最优美且最实用的结果之一。它表明：时域中的卷积运算，等价于频域中的普通乘法。这一看似简单的结论，深刻影响着信号处理、微分方程、概率论与物理学等多个领域。

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1. Recap of the Fourier Transform | 傅里叶变换回顾

The Fourier transform of a function f(t) is defined as the integral transform

函数 f(t) 的傅里叶变换定义为如下积分变换：

F(ω) = ∫_−∞^∞ f(t) e^−iωt dt

The inverse transform recovers the original function from its frequency representation:

逆变换则从频率表示中恢复原始函数：

f(t) = (1 / (2π)) ∫_−∞^∞ F(ω) e^iωt dω

Different textbooks may use different normalization constants, such as 1/√(2π) in both formulas. In this article we use the convention above unless otherwise stated.

不同教材可能采用不同的归一化常数，例如在两个公式中均使用 1/√(2π)。本文如无特别说明，一律采用上述约定。

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2. What Is Convolution? | 什么是卷积？

Given two functions f and g, their convolution is defined by

给定两个函数 f 与 g，它们的卷积定义为

(f * g)(t) = ∫_−∞^∞ f(τ) g(t − τ) dτ

Intuitively, we reverse g, slide it across f, multiply and integrate at each overlap. For causal signals, the integral is often taken from 0 to t.

直观上，我们将 g 翻转、平移并与 f 相乘，再在每一重叠位置做积分。对于因果信号，积分区间通常取为 0 到 t。

Convolution is commutative, associative and distributive:

卷积满足交换律、结合律和分配律：

• f * g = g * f
• f * (g * h) = (f * g) * h
• f * (g + h) = f * g + f * h

These algebraic properties make convolution a natural operation for combining linear systems and probability distributions.

这些代数性质使得卷积成为组合线性系统与概率分布的天然运算。

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3. Statement of the Convolution Theorem | 卷积定理的陈述

The convolution theorem states that, under suitable convergence conditions,

卷积定理表明，在适当的收敛条件下成立：

F { f * g } (ω) = F(ω) · G(ω)

Here F(ω) and G(ω) are the Fourier transforms of f(t) and g(t). In words: convolution in the time domain becomes multiplication in the frequency domain.

其中 F(ω) 与 G(ω) 分别是 f(t) 与 g(t) 的傅里叶变换。用语言来描述：时域中的卷积，对应频域中的乘法。

The dual statement also holds:

对偶命题同样成立：

F { f(t) · g(t) } (ω) = (1 / (2π)) (F * G)(ω)

This second form says that multiplication in the time domain corresponds to convolution in the frequency domain, up to a factor 1/(2π).

这第二种形式说明：时域中的乘法，对应频域中的卷积，相差一个因子 1/(2π)。

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4. Proof of the Theorem | 定理的证明

We prove the primary form by direct substitution and interchange of the order of integration.

我们通过直接代入并交换积分顺序来证明主要形式。

F { f * g } (ω) = ∫_−∞^∞ [ ∫_−∞^∞ f(τ) g(t − τ) dτ ] e^−iωt dt

Swap the order of integration:

交换积分顺序：

= ∫_−∞^∞ f(τ) [ ∫_−∞^∞ g(t − τ) e^−iωt dt ] dτ

Let u = t − τ, so t = u + τ and dt = du. Then the inner integral becomes

令 u = t − τ，则 t = u + τ，dt = du。于是内层积分化为

∫_−∞^∞ g(u) e^{−iω(u + τ)} du = e^−iωτ G(ω)

Substituting back gives the product F(ω)G(ω). The proof requires absolute integrability and enough regularity to justify interchanging the integrals.

代回原式即得乘积 F(ω)G(ω)。证明过程需要函数绝对可积且具有足够的正则性，才能合法地交换积分次序。

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5. Intuition Behind the Theorem | 定理背后的直觉

Suppose a signal x(t) passes through a linear time-invariant system with impulse response h(t). The output is y(t) = (x * h)(t). In the frequency domain, the very same relation becomes Y(ω) = X(ω)H(ω).

假设信号 x(t) 通过一个冲激响应为 h(t) 的线性时不变系统，则输出为 y(t) = (x * h)(t)。在频域中，这同一关系变为 Y(ω) = X(ω)H(ω)。

• Time-domain operation: sliding average, echo, blur
• Frequency-domain operation: weighting each frequency component separately

• 时域操作：滑动平均、回声、模糊
• 频域操作：对各频率分量分别加权

This is why engineers often design filters in the frequency domain and then use the theorem to compute the corresponding time-domain impulse response.

这正是工程师常先在频域设计滤波器，再利用该定理求出对应时域冲激响应的原因。

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6. Example 1: Exponential Signals | 例一：指数信号

Let u(t) be the Heaviside step function, and define f(t) = e^−2tu(t), g(t) = e^−3tu(t).

设 u(t) 为阶跃函数，定义 f(t) = e^−2tu(t)，g(t) = e^−3tu(t)。

Each has a known transform:

它们各自的变换为：

F(ω) = 1 / (2 + iω), G(ω) = 1 / (3 + iω)

Direct computation of the convolution gives, for t ≥ 0,

直接计算卷积可得，当 t ≥ 0 时：

(f * g)(t) = e^−2t − e^−3t

Its Fourier transform is

其傅里叶变换为

1/(2 + iω) − 1/(3 + iω) = 1 / ((2 + iω)(3 + iω))

which is exactly F(ω)G(ω). The theorem works.

这恰好是 F(ω)G(ω)。定理成立。

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7. Example 2: Rectangular Function and Sinc | 例二：矩形函数与 Sinc 函数

Let Π(t) be the rectangular pulse equal to 1 on [−1/2, 1/2] and 0 elsewhere. Its Fourier transform is

设 Π(t) 为在 [−1/2, 1/2] 上取 1、其余取 0 的矩形脉冲。其傅里叶变换为

F(ω) = sin(ω/2) / (ω/2) = sinc(ω/2)

The convolution of two identical rectangles is a triangular pulse:

两个相同矩形的卷积是一个三角脉冲：

(Π * Π)(t) = 1 − |t| for |t| ≤ 1, and 0 otherwise

By the convolution theorem, the Fourier transform of this triangle must be sinc²(ω/2). Direct integration confirms the result. This example appears frequently in diffraction and sampling theory.

根据卷积定理，该三角波的傅里叶变换必为 sinc²(ω/2)。直接积分可以验证这一结论。此例在衍射与采样理论中频繁出现。

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8. Example 3: Gaussian Functions | 例三：高斯函数

The Gaussian function f(t) = e^−at² has a Gaussian transform:

高斯函数 f(t) = e^−at² 的傅里叶变换仍是高斯函数：

F(ω) = √(π / a) · e^{−ω² / (4a)}

Convolving two Gaussians yields another Gaussian whose squared width is the sum of the two squared widths. We can verify this quickly:

两个高斯函数卷积后仍是高斯函数，其宽度平方等于两者宽度平方之和。我们可快速验证：

e^−at² * e^−bt² = √(π / (a + b)) · e^{−ab t² / (a + b)}

In the frequency domain, the theorem says the transformed product is simply the product of two Gaussian transforms, which is again Gaussian. This property is central to the probability density of sums of independent normal random variables.

在频域中，卷积定理表明，变换后的结果只是两个高斯变换的乘积，而这仍然是高斯函数。这一性质对于独立正态随机变量和的概率密度至关重要。

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9. Application in Signal Processing: Filtering | 信号处理中的应用：滤波

An ideal low-pass filter keeps frequencies below a cutoff W and removes all others. In the frequency domain, we multiply the input spectrum by a rectangular window.

理想低通滤波器保留低于截止频率 W 的分量并滤除其余部分。在频域中，我们将输入频谱乘以一个矩形窗函数。

By the convolution theorem, the filtered output in the time domain is the input convolved with the inverse transform of the window, which is a sinc function:

由卷积定理，时域中滤波后的输出等于输入信号与该窗函数逆变换的卷积，而逆变换是一个 sinc 函数：

h(t) = (W / π) · sinc(W t)

The sinc impulse response causes ripple, known as the Gibbs phenomenon, because the rectangular window is discontinuous. In practice, smoother windows are used.

Sinc 冲激响应会引起振铃，即吉布斯现象，因为矩形窗不连续。实际应用中常改用更平滑的窗函数。

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10. Application in Probability | 概率论中的应用

If X and Y are independent continuous random variables with probability density functions f_X and f_Y, then the density of Z = X + Y is the convolution f_X * f_Y.

若 X 与 Y 是相互独立的连续型随机变量，其概率密度函数分别为 f_X 与 f_Y，则 Z = X + Y 的密度函数为 f_X * f_Y。

The convolution theorem then relates the characteristic functions:

卷积定理进一步将特征函数联系起来：

φ_Z(ω) = φ_X(ω) · φ_Y(ω)

This neatly explains why sums of independent normals remain normal: the product of Gaussian characteristic functions is again Gaussian. The same method helps prove the Central Limit Theorem.

这简洁地解释了为何独立正态变量之和仍为正态：高斯特征函数相乘仍是高斯函数。同样的方法还可以用于证明中心极限定理。

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11. Common Pitfalls and Exam Tips | 常见误区与考试提示

Students often confuse the different conventions for the Fourier transform. Always state which definition you are using.

学生常常混淆傅里叶变换的不同约定。解题时务必说明你所采用的分布定义。

Pitfall | 常见误区	Correction | 改正方法
Forgetting the 1/(2π) factor in the inverse transform	Always include 1/(2π) for the chosen convention
Using wrong limits on causal convolution	If f and g vanish for t < 0, integrate from 0 to t
Swapping integrals without checking convergence	Assume absolute integrability or cite a density argument
Confusing f * g with ordinary product	Convolution produces a function of t, not a scalar

In IB examinations, the convolution theorem may appear in paper-based questions involving Fourier transforms. Practise with exponential and rectangular functions to build speed and accuracy.

在 IB 考试中，卷积定理可能出现在涉及傅里叶变换的笔试题中。建议多用指数函数和矩形函数进行练习，以提高速度与准确率。

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12. Conclusion | 总结

The convolution theorem is a bridge between two domains: it turns a complicated sliding integration into a simple product. Its applications range from linear systems to probability and optics.

卷积定理架起了两座领域之间的桥梁：它将复杂的滑动积分转化为简单的乘积。其应用涵盖了线性系统、概率论与光学等领域。

By mastering the definition, proof and examples, you gain not only a practical computational tool but also deeper insight into the structure of linear transformations and Fourier analysis.

掌握了定义、证明与典型例子，你不仅获得了实用的计算工具，更能深入理解线性变换与傅里叶分析的结构本质。

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