📚 The Dirac Delta Function and Fourier Transforms of Periodic Functions | δ函数与周期函数的傅里叶变换
The Fourier transform is one of the most powerful tools in mathematical physics and engineering, yet it encounters a subtle difficulty when applied to periodic functions. Because periodic functions do not decay at infinity, their classical Fourier integrals fail to converge. The resolution lies in the elegant theory of distributions, centered on the Dirac delta function. This article explores how the delta function bridges the gap and provides a unified framework for transforming periodic signals.
傅里叶变换是数学物理与工程学中最强大的工具之一，但将其应用于周期函数时会遇到一个微妙的困难。由于周期函数在无穷远处不衰减，其经典傅里叶积分无法收敛。解决方案在于以狄拉克δ函数为核心的广义函数（分布）理论。本文将探讨δ函数如何架起桥梁，为周期信号的变换提供统一的框架。
1. The Dirac Delta Function: A Formal Definition | 狄拉克δ函数：形式定义
The Dirac delta function, denoted δ(x), is not a function in the ordinary sense. It is a distribution defined by two key properties. First, it is zero everywhere except at the origin: δ(x) = 0 for x ≠ 0. Second, its integral over the entire real line equals one: ∫₋∞^∞ δ(x) dx = 1.
狄拉克δ函数记作δ(x)，并非普通意义上的函数，而是一种由两条关键性质定义的分布。第一，除原点外处处为零：当x ≠ 0时，δ(x) = 0。第二，其在整个实数轴上的积分等于1：∫₋∞^∞ δ(x) dx = 1。
More rigorously, the delta function must be understood through its action on test functions. For any sufficiently smooth function φ(x) with compact support, we define δ by the sifting property: ∫₋∞^∞ δ(x) φ(x) dx = φ(0).
更严格地说，δ函数必须通过其作用于测试函数的方式来理解。对于任何具有紧支撑的充分光滑函数φ(x)，我们通过筛分性质定义δ：∫₋∞^∞ δ(x) φ(x) dx = φ(0)。
Physically, one may think of δ(x) as the idealised limit of a sequence of narrow, tall pulses, such as Gaussian functions with width approaching zero and height approaching infinity, while always maintaining unit area. This is intuitive but must be treated carefully in rigorous mathematics.
在物理上，可以将δ(x)想象为一族又窄又高的脉冲序列的理想化极限，例如宽度趋于零、高度趋于无穷而面积恒为1的高斯函数。这一直观理解在严谨的数学处理中需格外谨慎。
2. The Sifting Property and Its Significance | 筛分性质及其重要性
The sifting property is the most operationally important aspect of the delta function. It states that multiplying any function by δ(x – a) and integrating extracts the value of the function at x = a:
筛分性质是δ函数在运算中最重要的特性。它表明，将任意函数乘以δ(x – a)并积分，即可提取该函数在x = a处的值：
∫₋∞^∞ f(x) δ(x – a) dx = f(a)
This property allows us to treat point evaluations as integrals, which is indispensable in Fourier analysis. For instance, the Fourier transform of a pure sinusoid can be obtained by recognising that the integral that defines it reduces, via Euler’s formula, to a sifting integral involving δ.
这一性质使我们能够将点取值表达为积分形式，这在傅里叶分析中不可或缺。例如，纯正弦函数的傅里叶变换可以通过欧拉公式将定义积分化为涉及δ的筛分积分来求得。
In discrete signal processing, the sifting property is the continuous analogue of the Kronecker delta δₙ, which selects a single term from a sum. This analogy clarifies why δ(x) appears so frequently in sampling theory and in the analysis of periodic phenomena.
在离散信号处理中，筛分性质是克罗内克δ（δₙ）的连续类比，后者从求和中选中单独一项。这一类比阐明了为何δ(x)在采样理论和周期现象分析中如此频繁出现。
3. The Delta Function as a Distribution | δ函数作为分布
To place the delta function on firm mathematical ground, we turn to the theory of distributions developed by Laurent Schwartz. A distribution is a continuous linear functional on the space of test functions. The delta function is the distribution that maps a test function φ(x) to its value at zero: δ[φ] = φ(0).
为将δ函数置于坚实的数学基础之上，我们需要借助洛朗·施瓦茨建立的分布理论。分布是测试函数空间上的连续线性泛函。δ函数就是将测试函数φ(x)映射到其在零处取值的分布：δ[φ] = φ(0)。
Distributions have the advantage that they can be differentiated arbitrarily many times. The derivative of the delta function, δ'(x), acts on a test function according to the rule:
分布的优点在于它可以被任意次求导。δ函数的导数δ'(x)作用于测试函数时遵循如下规则：
∫₋∞^∞ δ'(x) φ(x) dx = -φ'(0)
This generalised calculus is essential for solving differential equations with impulsive forcing terms. For example, the Green’s function of the harmonic oscillator involves derivatives of δ in its construction.
这种广义微积分对于求解含脉冲激励项的微分方程至关重要。例如，谐振子的格林函数构造中就涉及δ的导数。
4. The Fourier Transform of the Delta Function | δ函数的傅里叶变换
We adopt the convention for the Fourier transform commonly used in physics: F(ω) = ∫₋∞^∞ f(t) e^(-iωt) dt, with the inverse transform given by f(t) = (1/2π) ∫₋∞^∞ F(ω) e^(iωt) dω.
我们采用物理学中常用的傅里叶变换约定：F(ω) = ∫₋∞^∞ f(t) e^(-iωt) dt，相应的逆变换为f(t) = (1/2π) ∫₋∞^∞ F(ω) e^(iωt) dω。
Applying the sifting property directly, the Fourier transform of δ(t) is simply unity:
直接应用筛分性质，δ(t)的傅里叶变换就是1：
∫₋∞^∞ δ(t) e^(-iωt) dt = e^0 = 1
Similarly, the transform of a shifted delta function δ(t – t₀) is e^(-iωt₀). This single result is remarkably powerful: it tells us that a spike at a single instant contains all frequencies with equal amplitude and a linear phase shift.
类似地，平移后的δ函数δ(t – t₀)的变换为e^(-iωt₀)。这一结果极为强大：它告诉我们，单一时刻的尖峰包含了所有频率，且各频率振幅相等、相位线性移动。
5. The Inverse Transform: A Key Identity | 逆变换：一个关键恒等式
By symmetry of the Fourier transform pair, the constant function f(t) = 1 has its transform equal to 2π δ(ω). To verify this, consider the inverse transform of δ(ω):
由傅里叶变换对的对称性，常数函数f(t) = 1的变换等于2π δ(ω)。为验证这一点，考虑δ(ω)的逆变换：
(1/2π) ∫₋∞^∞ δ(ω) e^(iωt) dω = 1/2π
Multiplying both sides by 2π gives the integral representation of the delta function:
两边乘以2π即得到δ函数的积分表示：
∫₋∞^∞ e^(iωt) dω = 2π δ(t)
This identity is the cornerstone of Fourier analysis of periodic functions. It expresses the fact that an integral over all frequencies of a pure phase can only construct a localised impulse, a direct manifestation of the uncertainty principle between time and frequency.
该恒等式是周期函数傅里叶分析的基石。它表达了这样的事实：对所有频率的纯相位积分只能构造局部化的脉冲，这是时间与频率之间不确定原理的直接体现。
6. The Problem of Periodic Functions | 周期函数面临的难题
Consider a periodic function f(t) with period T, so that f(t + T) = f(t). Under the classical definition, its Fourier transform integral ∫₋∞^∞ f(t) e^(-iωt) dt does not converge in the ordinary sense, because the integrand does not tend to zero as |t| → ∞.
考虑周期为T的周期函数f(t)，满足f(t + T) = f(t)。在经典定义下，其傅里叶变换积分∫₋∞^∞ f(t) e^(-iωt) dt在普通意义下不收敛，因为被积函数在|t| → ∞时并不趋于零。
This failure is not merely a technical inconvenience. It reflects the physical fact that an ideal periodic signal, extending forever in time, carries infinite energy. The total energy ∫₋∞^∞ |f(t)|² dt diverges, so f(t) is not square-integrable and does not belong to L²(R).
这一失败不仅是技术上的不便。它反映了物理事实：理想的周期信号在时间上无限延伸，携带无限能量。总能量∫₋∞^∞ |f(t)|² dt发散，因此f(t)不是平方可积的，不属于L²(R)。
The classical Fourier series handles periodic functions by decomposing them into discrete harmonics: f(t) = Σₙ cₙ e^(i2πnt/T). But this does not provide a spectral density function F(ω) in the usual sense. To obtain a unified picture, we extend the transform to include distributions.
经典傅里叶级数通过将周期函数分解为离散谐波来处理：f(t) = Σₙ cₙ e^(i2πnt/T)。但这并未给出通常意义下的谱密度函数F(ω)。为获得统一的图景，我们将变换扩展以包含分布。
7. The Dirac Comb (Shah Function) | 狄拉克梳状函数（Shah函数）
The key tool for periodic functions is the Dirac comb, or Shah function, denoted Ш(t). It is an infinite train of equally spaced delta functions:
处理周期函数的关键工具是狄拉克梳状函数，也称Shah函数，记作Ш(t)。它是等间距δ函数的无限序列：
Ш(t) = Σₙ₌₋∞^∞ δ(t – nT)
The comb is itself a periodic distribution with period T. Its action on a test function φ(t) produces the sum of samples of φ at all integer multiples of T.
梳状函数本身是以T为周期的周期分布。它作用于测试函数φ(t)时，产生φ在所有T的整数倍处的采样值之和。
In signal processing, the comb is the mathematical model for ideal sampling: multiplying a continuous signal by Ш(t) extracts instantaneous samples at discrete instants. This makes the comb indispensable in the theory of analog-to-digital conversion.
在信号处理中，梳状函数是理想采样的数学模型：将连续信号乘以Ш(t)即可在离散时刻提取瞬时样本。这使得梳状函数在模数转换理论中不可或缺。
8. The Fourier Transform of the Dirac Comb | 狄拉克梳状函数的傅里叶变换
We now derive the Fourier transform of the comb. Using the integral representation of δ from Section 5, we write:
现在我们推导梳状函数的傅里叶变换。利用第5节的δ积分表示，我们写出：
Ш(t) = Σₙ δ(t – nT) = (1/2π) Σₙ ∫₋∞^∞ e^(iω(t – nT)) dω
Taking the Fourier transform of Ш(t) and interchanging the order of summation and integration yields a second comb in the frequency domain:
对Ш(t)进行傅里叶变换并交换求和与积分次序，得到频域中的另一个梳状函数：
∫₋∞^∞ Ш(t) e^(-iωt) dt = (2π/T) Σₙ δ(ω – n·2π/T)
Thus, the Fourier transform of a Dirac comb with period T in time is a Dirac comb in frequency with spacing 2π/T, multiplied by the factor 2π/T. This duality is visually striking: a periodic train of impulses transforms into another periodic train of impulses, with the spacing reciprocally related.
因此，时域中周期为T的狄拉克梳状函数，其傅里叶变换是频域中间距为2π/T的狄拉克梳状函数，并乘以因子2π/T。这种对偶性令人惊叹：周期脉冲串变换为另一个周期脉冲串，且间距互为倒数关系。
9. Fourier Transform of a General Periodic Function | 一般周期函数的傅里叶变换
Any reasonable periodic function f(t) with period T can be expressed as a convolution of a fundamental period with the comb: f(t) = (f₀ * Ш)(t), where f₀ is the restriction of f to one period and * denotes convolution.
任何合理的周期函数f(t)（周期为T）都可以表示为基本周期函数与梳状函数的卷积：f(t) = (f₀ * Ш)(t)，其中f₀是f在一个周期内的限制，*表示卷积。
By the convolution theorem, the Fourier transform of a convolution is the product of the individual transforms. The transform of f₀ is a continuous function F₀(ω), while the transform of Ш is a frequency comb. Hence:
由卷积定理，卷积的傅里叶变换等于各自变换的乘积。f₀的变换是连续函数F₀(ω)，而Ш的变换是频域梳状函数。因此：
F(ω) = F₀(ω) · (2π/T) Σₙ δ(ω – nω₀), ω₀ = 2π/T
Since the delta functions sample F₀ at the discrete frequencies nω₀, this reduces to an infinite sum of weighted delta functions located at the harmonic frequencies. The weights are precisely the Fourier series coefficients of f(t).
由于δ函数在离散频率nω₀处对F₀采样，上式化简为位于各谐波频率处的加权δ函数无穷和。这些权值正是f(t)的傅里叶级数系数。
10. An Illustrative Example: The Cosine Function | 实例说明：余弦函数
Let us compute the Fourier transform of the cosine function f(t) = cos(ω₀t). Using Euler’s formula, we write:
我们来计算余弦函数f(t) = cos(ω₀t)的傅里叶变换。利用欧拉公式，我们写出：
cos(ω₀t) = ½(e^(iω₀t) + e^(-iω₀t))
Taking the Fourier transform term by term, we recall from Section 5 that the transform of e^(iω₀t) is 2π δ(ω – ω₀). Therefore:
逐项取傅里叶变换，我们回顾第5节的结论：e^(iω₀t)的变换为2π δ(ω – ω₀)。因此：
F(ω) = π[δ(ω – ω₀) + δ(ω + ω₀)]
This beautiful result shows that a pure cosine contains exactly two frequencies: +ω₀ and -ω₀, each with spectral weight π. The spectrum is entirely discrete or “line-like,” with no continuous component.
这一优美结果表明，纯余弦信号恰好包含两个频率：+ω₀和-ω₀，各带有谱权重π。频谱完全离散，呈”线状谱”，不含连续分量。
11. Physical Applications | 物理应用
The framework developed above has profound applications in physics and engineering. In X-ray crystallography, the diffraction pattern of a crystal is the Fourier transform of its electron density. A perfect crystal, being periodic, produces a discrete pattern of bright spots — the Laue peaks — exactly as predicted by the delta-comb transform.
上述理论框架在物理学和工程学中有着深远应用。在X射线晶体学中，晶体的衍射图样是其电子密度的傅里叶变换。理想晶体是周期性的，因此产生离散的亮点图案——劳厄峰——这正是梳状函数变换所预测的。
In quantum mechanics, a particle in a periodic potential (described by Bloch’s theorem) has a quasi-momentum that is only defined modulo 2π/T. The Dirac comb naturally encodes this periodicity in reciprocal space, providing a rigorous basis for understanding electronic band structure.
在量子力学中，周期势场中的粒子（由布洛赫定理描述）的准动量仅在模2π/T意义下确定。狄拉克梳状函数自然地在倒空间中编码了这种周期性，为理解电子能带结构提供了严谨基础。
In communication engineering, the modulation theorem relies on the δ-function shift property. When a baseband signal is multiplied by a carrier wave at frequency f_c, its spectrum shifts to ±f_c — an operation described by convolution with δ(f – f_c) + δ(f + f_c).
在通信工程中，调制定理依赖于δ函数的平移性质。当基带信号乘以频率为f_c的载波时，其频谱移动到±f_c处——这一操作可用与δ(f – f_c) + δ(f + f_c)的卷积来描述。
12. Summary and Key Exam Points | 总结与考试要点
The following results are essential for examinations and should be memorised with care. First, the Fourier transform of δ(t) is 1, and conversely the transform of the constant 1 is 2π δ(ω). Second, the transform of the Dirac comb with period T is a frequency comb with spacing 2π/T and amplitude 2π/T.
以下结论对考试至关重要，务必牢记。第一，δ(t)的傅里叶变换为1，反之常数1的变换为2π δ(ω)。第二，周期为T的狄拉克梳状函数的变换是间距为2π/T、幅度为2π/T的频域梳状函数。
Third, the Fourier transform of any periodic function consists of weighted delta functions at the discrete harmonic frequencies, with weights equal to the Fourier series coefficients. Fourth, the transform of cos(ω₀t) is π[δ(ω – ω₀) + δ(ω + ω₀)], and similarly for sin(ω₀t) the transform is iπ[δ(ω + ω₀) – δ(ω – ω₀)].
第三，任何周期函数的傅里叶变换都是位于离散谐波频率处的加权δ函数，权值等于傅里叶级数系数。第四，cos(ω₀t)的变换为π[δ(ω – ω₀) + δ(ω + ω₀)]，类似地sin(ω₀t)的变换为iπ[δ(ω + ω₀) – δ(ω – ω₀)]。
Always remember to include the correct normalisation factor 2π when working with the inverse transform, as conventions vary across textbooks. A common exam trap is to forget the 2π in the transform of the constant function or in the comb transform — the factor is unavoidable and must appear exactly once in every transform pair.
在处理逆变换时，务必记住包含正确的归一化因子2π，因为不同教材的约定可能不同。一个常见的考试陷阱是忘记常数函数变换或梳状函数变换中的2π——该因子不可省略，且在每个变换对中恰好出现一次。
Finally, be aware that the δ-function approach is valid only within the theory of distributions. When asked to “derive” the transform of a periodic function, always begin by writing the periodicity condition, then apply the comb representation, and finally invoke the convolution theorem. This systematic method earns full marks.
最后，请注意δ函数方法仅在分布理论框架内有效。当被要求”推导”周期函数的变换时，始终先写出周期性条件，然后应用梳状函数表示，最后引用卷积定理。这种系统性方法能获得满分。
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