Fourier Transforms of Some Basic Functions | 一些基本函数的傅里叶变换

📚 Fourier Transforms of Some Basic Functions | 一些基本函数的傅里叶变换

Fourier transforms provide a bridge between the time domain and the frequency domain, playing a central role in signal processing, quantum mechanics, and differential equations. For IB Mathematics students, mastering the transforms of basic functions is essential, as these form the building blocks for more complex analyses. This article derives and discusses the Fourier transforms of rectangular pulses, Gaussian functions, Dirac delta distributions, exponential decays, and other fundamental signals, using a consistent definition and clear steps.

傅里叶变换在时域与频域之间架起了一座桥梁，是信号处理、量子力学和微分方程中的核心工具。对 IB 数学的学生来说，掌握基本函数的傅里叶变换至关重要，因为它们是分析更复杂问题的基础。本文采用一致的定义，逐步推导并讨论矩形脉冲、高斯函数、狄拉克 δ 分布、指数衰减以及其他基本信号的傅里叶变换。

1. Definition of the Fourier Transform | 傅里叶变换的定义

ble or interpreted in a distributional sense, the Fourier transform is defined as F(ω) = ∫_-∞^∞ f(t) e^-iωt dt. The inverse transform recovers f(t) = (1/(2π)) ∫_-∞^∞ F(ω) e^iωt dω. Sometimes the transform is denoted by f̂(ω) or ℱ{f(t)}. This symmetric pair allows us to analyse the frequency content of a signal.

我们采用角频率约定。对于绝对可积或在分布意义下理解的函数 f(t)，傅里叶变换定义为 F(ω) = ∫_-∞^∞ f(t) e^-iωt dt。其逆变换为 f(t) = (1/(2π)) ∫_-∞^∞ F(ω) e^iωt dω。变换有时也写作 f̂(ω) 或 ℱ{f(t)}。这一对称对使我们能够分析信号的频率成分。

t follow, we compute F(ω) for several elementary signals. We assume familiarity with Euler's formula, integration by parts, and the sinc function defined as sinc(x) = sin(x)/x.

在接下来的小节中，我们将计算若干基本信号的 F(ω)。我们假定读者熟悉欧拉公式、分部积分法以及 sinc 函数的定义，即 sinc(x) = sin(x)/x。

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2. The Rectangular Pulse | 矩形脉冲

Consider the rectangular pulse of duration T and unit amplitude: f(t) = 1 for |t| written as rect(t/T). Its Fourier transform is obtained by a direct integration: F(ω) = ∫_-T/2^T/2 e^-iωt dt = (e^-iωt/(-iω)) evaluated from -T/2 to T/2.

考虑一个持续时间为 T、幅度为 1 的矩形脉冲：当 |t| ≤ T/2 时 f(t) = 1，其余处为 0。该函数常记作 rect(t/T)。其傅里叶变换可通过直接积分求得：F(ω) = ∫_-T/2^T/2 e^-iωt dt = (e^-iωt/(-iω)) 从 -T/2 到 T/2 的差值。

Simplifying gives F(ω) = (e^iωT/2 - e^-iωT/2 , we obtain sinc(ω/2) up to a constant factor depending on the definition of the Fourier transform.

化简后得到 F(ω) = (e^iωT/2 - e^-iωT/2) / (iω) = 2 sin(ωT/2) / ω = T sinc(ωT/2)。这表明时域中的矩形脉冲在频域中变成了一个 sinc 函数，且主瓣宽度与 T 成反比。对于归一化的单位矩形脉冲（T=1），根据变换定义的不同，其结果与 sinc(ω/2) 相差一个常数因子。

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3. The Gaussian Function | 高斯函数

The Gaussian function f(t) = e^-at² a > 0 is remarkable because its Fourier transform is also a Gaussian. We evaluate F(ω) = ∫_-∞^∞ e^-at² e^-iωt dt. Completing the square in the exponent yields e^{-a(t + iω/(2a))²} e^-ω²/(4a) ntegration contour does not change the value of the Gaussian integral, so we obtain F(ω) = √π/a e^-ω²/(4a).

高斯函数 f(t) = e^-at²（a > 0）的性质非常特殊，因为它的傅里叶变换仍然是一个高斯函数。我们计算 F(ω) = ∫_-∞^∞ e^-at² e^-iωt dt。在指数中进行配平方，得到 e^{-a(t + iω/(2a))²} e^-ω²/(4a)。由于平移积分路径不会改变高斯积分的值，最终可得 F(ω) = √π/a e^-ω²/(4a)。

tainty principle. In the special case a = 1/2 we obtain the standard normalised Gaussian e^-t²/2, whose transform is √2π e^-ω²/2, up to the factor convention.

因此，时域中较窄的高斯函数（a 较大）在频域中对应较宽的高斯函数，体现了测不准原理。在特殊情形 a = 1/2 时，我们得到标准归一化高斯函数 e^-t²/2，其变换（取决于因子约定）为 √2π e^-ω²/2。

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4. The Dirac Delta Function | 狄拉克 δ 函数

The Dirac delta δ(t) is a generalised function defined by the sifting property: ∫_-∞^∞ f(t) δ(t - t₀) dt = f(t₀). Setting f(t) = e^-iωt immediately gives the Fourier transform of δ(t) as ℱ{δ(t)} = e^-iω·0 = 1. More generally, ℱ{δ(t - t₀)} = e^-iωt₀. This constant-modulus spectrum means the delta function contains all frequencies with equal weight.

狄拉克 δ 函数是一种广义函数，由筛选性质定义：∫_-∞^∞ f(t) δ(t - t₀) dt = f(t₀)。令 f(t) = e^-iωt，立即可得 δ(t) 的傅里叶变换为 ℱ{δ(t)} = e^-iω·0 = 1。更一般地，ℱ{δ(t - t₀)} = e^-iωt₀。这种模为常数的频谱意味着 δ 函数以相等的权重包含了所有频率成分。

Conversely, the inverse transform of a constant 1 recovers the delta function: (1/(2π)) ∫_-∞^∞ e^iωt t representation of δ(t). This pair underpins many results, such as the transforms of sine and cosine.

反之，常数 1 的逆变换将恢复 δ 函数：(1/(2π)) ∫_-∞^∞ e^iωt dω = δ(t)，这是 δ(t) 的一个重要积分表示。这一变换对是许多结果（如正弦和余弦的变换）的基础。

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

Let f(t) = e^-at t), where a > 0 and u(t) is the Heaviside step function. Its Fourier transform is F(ω) = ∫₀^∞ e^-at e^-iωt dt = ∫₀^∞ e^{-(a + iω)t} dt = [e^{-(a + iω)t}a + iω))] from 0 to ∞. Because a > 0, the upper limit vanishes, leaving F(ω) = 1/(a + iω).

设 f(t) = e^-at u(t)，其中 a > 0，u(t) 是单位阶跃函数。其傅里叶变换为 F(ω) = ∫₀^∞ e^-at e^-iωt dt = ∫₀^∞ e^{-(a + iω)t} dt = [e^{-(a + iω)t}/(-(a + iω))] 从 0 至 ∞ 的差值。由于 a > 0，上界趋于零，因此 F(ω) = 1/(a + iω)。

lit into real and imaginary parts: 1/(a + iω) = a/(a² + ω²) - i ω/(a² + ω²). The magnitude |F(ω)| = 1/√a² + ω² decays as 1/|ω| for large ω, showing that a sharp exponential edge generates high-frequency components.

这个复值函数可以拆分为实部和虚部：1/(a + iω) = a/(a² + ω²) - i ω/(a² + ω²)。其幅度 |F(ω)| = 1/√a² + ω² 在大 ω 时按 1/|ω| 衰减，表明陡峭的指数边缘会产生高频成分。

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6. Two-sided Exponential Function | 双边指数函数

The two-sided exponential is f(t) = e^-a|t| with a > 0. Because of the absolute value, we split the integral: F(ω) = ∫_-∞⁰ e^at e^-iωt dt + ∫₀^∞ e^-at e^-iωt becomes 1/(a - iω) and the second 1/(a + iω). Their sum simplifies to F(ω) = 2a/(a² + ω²).

双边指数函数为 f(t) = e^-a|t|，其中 a > 0。由于包含绝对值，我们将积分拆开：F(ω) = ∫_-∞⁰ e^at e^-iωt dt + ∫₀^∞ e^-at e^-iωt dt。第一个积分结果为 1/(a - iω)，第二个为 1/(a + iω)。二者相加并化简后得到 F(ω) = 2a/(a² + ω²)。

constant 1 (non-decaying), and the spectrum approaches 2π δ(ω) in the distributional sense, consistent with the delta-pair discussed earlier.

值得注意的是，该频谱为纯实数且偶对称，与 f(t) 的偶函数性质相符。当 a → 0 时，函数趋向于常数 1（无衰减），而频谱在分布意义下趋近于 2π δ(ω)，这与之前讨论的 δ 变换对一致。

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7. The Signum Function | 符号函数

nd sgn(0) = 0. It is not absolutely integrable, but we can treat it as a limit. Consider f_a(t) = e^-at for t > 0, -e^at for t < 0, which approaches sgn(t) as a → 0⁺. The Fourier transform of f_a(t) is 2iω/(a² + ω²) (up to sign). Taking the limit a → 0⁺ yields ℱ{sgn(t)} = 2/(iω), interpreted in the principal-value sense.

符号函数定义为：当 t > 0 时 sgn(t) = 1，t < 0 时 sgn(t) = -1，t = 0 时 sgn(0) = 0。它并不是绝对可积的，但我们可以将其视为极限情况。考虑 f_a(t) = e^-at（t > 0），-e^at（t < 0），当 a → 0⁺ 时趋近于 sgn(t)。f_a(t) 的傅里叶变换为 2iω/(a² + ω²)（忽略符号），取极限 a → 0⁺ 即得 ℱ{sgn(t)} = 2/(iω)，这一结果应在主值意义下理解。

hat sgn(t) contains a strong low-frequency component, but no DC term because the function has zero mean.

等价地，2/(iω) 可以写作 -2i/ω。该频谱在 ω = 0 处的极点表明 sgn(t) 具有强烈的低频分量，但由于均值为零，因此没有直流分量。

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8. The Unit Step Function | 单位阶跃函数

1 for t > 0. We can express u(t) = ½ + ½ sgn(t). Using linearity, its Fourier transform is ℱ{u(t)} = ½ ℱ{1} + ½ ℱ{sgn(t)}. We have ℱ{1} = 2π δ(ω) and ℱ{sgn(t)} = 2/(iω). Hence, ℱ{u(t)} = π δ(ω) + 1/(iω).

单位阶跃函数 u(t) 在 t < 0 时为 0，t > 0 时为 1。我们可以将其写作 u(t) = ½ + ½ sgn(t)。利用线性性质，其傅里叶变换为 ℱ{u(t)} = ½ ℱ{1} + ½ ℱ{sgn(t)}。已知 ℱ{1} = 2π δ(ω)，ℱ{sgn(t)} = 2/(iω)，因此 ℱ{u(t)} = π δ(ω) + 1/(iω)。

This combination of a delta impulse at ω = 0 and a term decaying as 1/ω reflects the step' . In many practical contexts, the term 1/(iω) is understood as a principal-value integral.

这种在 ω = 0 处的 δ 冲激与 1/ω 衰减项的组合，反映了阶跃函数的直流偏移以及在 t = 0 处的突变。在许多实际情境中，1/(iω) 这一项理解为柯西主值积分。

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

For a pure cosine wave, f(t) = cos(ω₀t) = ½(e^iω₀t + e^-iω₀t). Using the shift property of the delta transform, we obtain ℱ ectrum consists of two symmetric impulses at ±ω₀, indicating a single frequency component.

对于纯余弦波 f(t) = cos(ω₀t) = ½(e^iω₀t + e^-iω₀t)，利用 δ 变换的平移性质，得到 ℱ{cos(ω₀t)} = π[δ(ω - ω₀) + δ(ω + ω₀)]。频谱由位于 ±ω₀ 的两个对称冲激组成，对应单一的频率分量。

For the sine function, f(t) = sin(ω₀t) = (1/(2i))(e^iω₀t - e^-iω₀t). Its transform is ℱ nary impulses reflect the odd nature of the sine wave. These pairs are foundational for modulation and communications theory.

对于正弦函数 f(t) = sin(ω₀t) = (1/(2i))(e^iω₀t - e^-iω₀t)，其变换为 ℱ{sin(ω₀t)} = (π/i)[δ(ω - ω₀) - δ(ω + ω₀)] = iπ[δ(ω + ω₀) - δ(ω - ω₀)]。虚数冲激反映了正弦波的奇函数特性。这些变换对是调制和通信理论的基础。

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10. The Impulse Train (Dirac Comb) | 脉冲序列（狄拉克梳状函数）

The Dirac comb with period T is defined as III_T(t) = Σ_k=-∞^∞ δ(t - kT). Because it is periodic, we can expand it in a Fourier series: III_T(t) = (1/T) Σ_m=-∞^∞ e^{i 2π m t / T}. Taking the Fourier transform term by term and using ℱ{e^{i 2π m t / T}} = 2π δ(ω - 2π m / T) yields ℱ{III_T(t)} = (2π/T) Σ_m=-∞^∞ 2π m / T). This is another impulse train in the frequency domain, with spacing Δω = 2π/T.

周期为 T 的狄拉克梳状函数定义为 III_T(t) = Σ_k=-∞^∞ δ(t - kT)。由于它是周期函数，我们可以将其展开为傅里叶级数：III_T(t) = (1/T) Σ_m=-∞^∞ e^{i 2π m t / T}。逐项进行傅里叶变换，并利用 ℱ{e^{i 2π m t / T}} = 2π δ(ω - 2π m / T)，可得 ℱ{III_T(t)} = (2π/T) Σ_m=-∞^∞ δ(ω - 2π m / T)。这正是频域中的另一个脉冲序列，其间隔为 Δω = 2π/T。

This elegant duality is the mathematical foundation of sampling theory. A signal sampled by multiplication with a comb in time leads to a periodic replication of its spectrum in frequency. To avoid aliasing, the sampling rate must be high enough so that these replicas do not overlap, a principle captured by the Nyquist–Shannon sampling theorem.

这种优美的对偶性是采样理论的数学基础。在时域中用梳状函数进行混频采样，会导致其频谱在频域中作周期性复制。为避免混叠，采样频率必须足够高以确保这些复制品不会重叠，这正是奈奎斯特–香农采样定理所阐述的原理。

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11. Summary Table of Basic Transforms | 基本变换汇总表

The following table collects the Fourier transform pairs discussed. Conventions may vary by a constant factor depending on the definition, but the functional forms remain consistent.

下表

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