Laplace and z transforms: applications | 拉普拉斯变换与Z变换的应用

📚 Laplace and z transforms: applications | 拉普拉斯变换与Z变换的应用

Integral transforms are powerful mathematical tools that convert differential or difference equations into algebraic equations, making them much easier to solve. The Laplace transform, defined for continuous-time signals, is widely used in engineering to analyse linear time-invariant systems, while the z-transform extends similar ideas to discrete-time signals. This article explores the core applications of both transforms, from solving ordinary differential equations to digital signal processing, control theory, and probability.

积分变换是一种强大的数学工具，它将微分方程或差分方程转换为代数方程，从而让求解过程变得简单。拉普拉斯变换针对连续时间信号，广泛用于分析线性时不变系统，而Z变换则将类似思想推广到离散时间信号。本文探讨这两种变换的核心应用，涵盖常微分方程求解、数字信号处理、控制理论和概率等领域。

---

1. Solving Linear ODEs with Laplace Transforms | 用拉普拉斯变换求解线性常微分方程

nsform converts derivatives into multiplications by s, turning the differential equation into an algebraic equation in the complex frequency domain. For example, given y'' + 3y' + 2y = eᵗ with y(0)=1, y'(0)=0, we take the Laplace transform: L{y''} = s²Y(s) - s·1 - 0, L{y' sed by partial fractions and inverted term-by-term to yield the time-domain solution y(t).

拉普拉斯变换最直接的应用之一是求解带初始条件的线性常微分方程。变换将导数转换为乘以 s，把微分方程转化为复频率域中的代数方程。例如，给定 y'' + 3y' + 2y = eᵗ，初始条件 y(0)=1, y'(0)=0，取拉普拉斯变换：L{y''} = s²Y(s) - s·1 - 0，L{y'} = sY(s) - 1，L{eᵗ} = 1/(s-1)。得到 (s²+3s+2)Y(s) - s - 3 = 1/(s-1)，解出 Y(s)，再通过部分分式分解并逐项反演，得到时域解 y(t)。

This technique avoids the need to find homogeneous and particular solutions separately, automatically incorporating initial conditions. It is especially effective for systems with forcing functions that are piecewise continuous or impulsive, where classical methods become cumbersome.

这种方法无需分别寻找齐次解和特解，能自动包含初始条件。对于具有分段连续或脉冲激励函数的系统尤其有效，传统解法往往十分繁琐。

Y(s) = (s+3)/((s+1)(s+2)) + 1/((s-1)(s+1)(s+2))

---

2. Circuit Analysis in the s-Domain | s域电路分析

In electrical engineering, the Laplace transform provides a unified approach to analysing circuits containing resistors, capacitors, and inductors. By transforming each element into its s-domain impedance — R for a resistor, sL for an inductor, and 1/(sC) for a capacitor — the whole circuit can be treated with algebraic techniques analogous to resistive networks. Kirchhoff's laws hold identically in the s-domain, and initial conditions on capacitors and inductors appear as independent sources, simplifying transient analysis.

在电气工程中，拉普拉斯变换为分析含电阻、电容和电感的电路提供了统一方法。将每个元件转换为其 s 域阻抗 —— 电阻为 R，电感为 sL，电容为 1/(sC) —— 整个电路就可以像纯电阻网络那样用代数方法处理。基尔霍夫定律在 s 域完全相同，电容和电感的初始条件表现为独立源，使瞬态分析大为简化。

. Solving for I(s) and taking the inverse transform yields the complete current response including both forced and transient components. The poles of the impedance function reveal the natural frequencies and damping characteristics.

对于由电压 v(t) 驱动的串联 RLC 电路，s 域回路方程为 I(s)[R + sL + 1/(sC)] = V(s)。求解 I(s) 并取逆变换即可获得包含强制和暂态分量的完整电流响应。阻抗函数的极点揭示了固有频率和阻尼特性。

• Resistor: Z(s) = R

  电阻：Z(s) = R

• Inductor: Z(s) = sL

  电感：Z(s) = sL

• Capacitor: Z(s) = 1/(sC)

  电容：Z(s) = 1/(sC)

---

3. Transfer Functions and Control Systems | 传递函数与控制系统

nsfer function G(s) = Y(s)/U(s) is central to control engineering. It represents the input– nd denominator, zeros and poles provide insights into stability, transient response, and frequency behaviour.

传递函数 G(s) = Y(s)/U(s) 的概念是控制工程的核心。它表示线性时不变系统在 s 域的输入–输出关系，与输入信号的性质无关。通过对分子分母因式分解，零点和极点可以揭示系统的稳定性、暂态响应和频率特性。

Feedback systems are designed by shaping the loop transfer function. Standard controllers — proportional (P), integral (I), derivative (D), or combinations — are easily analysed using Laplace transforms. The stability of a closed-loop system is assessed via pole locations: all poles must lie in the left-half s-plane for bounded-input bounded-output (BIBO) stability.

反馈系统通过塑造回路传递函数来设计。标准控制器 —— 比例 (P)、积分 (I)、微分 (D) 及其组合 —— 可用拉普拉斯变换轻松分析。闭环系统的稳定性通过极点位置判断：所有极点必须位于 s 左半平面以保证有界输入有界输出稳定。

Closed-loop: T(s) = G(s)/(1+G(s)H(s))

---

4. Mechanical Vibrations and Damping | 机械振动与阻尼

Mechanical oscillatory systems, such as mass–spring–damper setups, are modelled by second-order differential equations m x'' + c x' tion X(s)/F(s) = 1/(m s² + c s + k). The denominator determines the natural frequency ωₙ = √k/m and the damping ratio ζ = c/(2√km), which classify the system as underdamped, critically damped, or overdamped.

机械振动系统，如质量–弹簧–阻尼器装置，通常用二阶微分方程 m x'' + c x' + k x = F(t) 建模。在零初始条件下应用拉普拉斯变换，得到传递函数 X(s)/F(s) = 1/(m s² + c s + k)。分母决定了固有频率 ωₙ = √k/m 和阻尼比 ζ = c/(2√km)，并据此将系统划分为欠阻尼、临界阻尼或过阻尼。

The impulse response, step response, and frequency response can be obtained directly from the inverse transform, offering a complete characterisation of the vibration behaviour. Engineers use these results to design shock absorbers, building structures, and vibration isolation systems.

脉冲响应、阶跃响应和频率响应均可通过逆变换直接获得，从而对振动行为进行全面表征。工程师利用这些结果设计减震器、建筑结构和隔振系统。

---

广告 / Sponsored

5. The z-Transform and Difference Equations | Z变换与差分方程

he z-transform plays a role analogous to the Laplace transform. The definition Z{x[n]} = Σ_n=0^∞ x[n] z^-n -1], allowing linear constant-coefficient difference equations to be solved algebraically.

对于离散时间信号，Z变换扮演着与拉普拉斯变换类似的角色。定义 Z{x[n]} = Σ_n=0^∞ x[n] z^-n 将序列转化为复变量 z 的函数。其关键性质是时移对应乘以 z 的幂次：Z{x[n-1]} = z⁻¹ X(z) + x[-1]，从而能用代数方法求解线性常系数差分方程。

e transform gives the closed-form solution y[n] = 2ⁿ - 1, avoiding iterative substitution.

考虑 y[n] - 3y[n-1] + 2y[n-2] = 0，初始条件 y[-1]=0, y[-2]=1。应用变换得到关于 Y(z) 的代数方程，整理并部分分式分解后，逆变换给出闭式解 y[n] = 2ⁿ - 1，无需迭代代入。

---

6. Digital Filters and Signal Processing | 数字滤波器与信号处理

ion in z⁻¹. The frequency response is obtained by evaluating H(z) on the unit circle z = e^jω, revealing the filter' both poles and zeros, offering sharper roll-offs at the expense of potential instability.

数字滤波器由线性差分方程描述。Z变换给出传递函数 H(z) = Y(z)/X(z)，它是 z⁻¹ 的有理函数。频率响应可通过在单位圆 z = e^jω 上求值获得，展示滤波器的幅度和相位特性。FIR 滤波器没有分母极点（除 z=0 外），而 IIR 滤波器同时具有极点和零点，可提供更陡峭的滚降，但可能带来不稳定问题。

Common filter structures, such as the biquadratic section, are derived directly from the z-domain transfer function. The bilinear transform method is used to convert continuous-time filter prototypes (e.g. Butterworth, Chebyshev) into their discrete equivalents, preserving key properties like passband ripple and stopband attenuation.

常见的滤波器结构，如双二阶节，可直接由 z 域传递函数导出。双线性变换法用于将连续时间滤波器原型（如巴特沃斯、切比雪夫）转换为离散等效形式，同时保持通带纹波和阻带衰减等关键特性。

---

7. Stability of Discrete-Time Systems | 离散时间系统的稳定性

O stable if and only if all poles of its transfer function H(z) lie strictly inside the unit circle |z| rginal stability, while poles outside make the system unstable. This stability criterion is the discrete counterpart of the left-half s-plane condition for continuous systems.

离散时间线性系统是有界输入有界输出稳定的，当且仅当其传递函数 H(z) 的所有极点都严格位于单位圆 |z| = 1 的内部。极点在单位圆上导致临界稳定，而极点在圆外则使系统不稳定。该稳定判据是连续系统左半 s 平面条件的离散对应。

The Jury stability test and the bilinear transformation allow stability assessment without factoring high-degree polynomials. By mapping the unit disc to the left-half s-plane, classic Routh–Hurwitz criteria can be applied. Understanding pole locations helps design stable digital controllers and adaptive filters.

Jury 稳定判据和双线性变换无需因式分解高次多项式即可评估稳定性。通过将单位圆盘映射到左半 s 平面，可以应用经典劳斯–赫尔维茨判据。理解极点位置有助于设计稳定的数字控制器和自适应滤波器。

---

8. Inverse Transforms via Partial Fractions | 通过部分分式求逆变换

Both Laplace and z transforms rely heavily on partial fraction expansion to obtain inverse transforms. A rational function X(s) or X(z) is expressed as a sum of simpler terms whose inverses are known from standard tables. Proper fractions, distinct poles, repeated poles, and complex conjugate pairs each follow specific decomposition rules.

拉普拉斯变换和 Z 变换都高度依赖部分分式展开来求逆变换。将有理函数 X(s) 或 X(z) 表为若干简单项之和，这些项的逆变换可从标准表格查得。真分式、单极点、重极点和复共轭极点对各自遵循特定的分解规则。

For Laplace, the inverse of 1/(s+a) is e^-atu(t). For z-transforms, the inverse of z/(z-a) is aⁿu[n]. Repeated factors introduce polynomial multipliers, and algorithms like Heaviside's cover-up method speed up the process. Mastery of this technique is essential for practical computation.

对拉普拉斯变换，1/(s+a) 的逆变换为 e^-atu(t)。对 Z 变换，z/(z-a) 的逆变换为 aⁿu[n]。重因子会引入多项式乘子，而海维赛德遮盖法等算法可加速求解过程。掌握这一技术对实际计算至关重要。

---

9. Probability: Moment Generating Functions | 概率：矩母函数

eory as the moment generating function (MGF) of a non-negative random variable: M(t) = E[e^tX] = ∫ e^tx f_X(x) dx, which is essentially the two-sided Laplace transform of the pdf evaluated at -t. Moments are obtained by differentiating the MGF at zero. The cumulant generating function and characteristic function are closely related transforms.

拉普拉斯变换在概率论中以非负随机变量的矩母函数 (MGF) 形式出现：M(t) = E[e^tX] = ∫ e^tx f_X(x) dx，它本质上是在 -t 处求值的双侧拉普拉斯变换。矩可通过在零点对 MGF 求导获得。累积量生成函数和特征函数是与之密切相关的变换。

riables taking non-negative integer values, the probability generating function G(z) = E[z^X] = Σ p_k z^k is exactly the z-transform of the probability mass sequence p_k, evaluated at z instead of z⁻¹. This connection allows the use of transform techniques to solve recurrence relations in Markov chains and queuing theory.

类似地，对于取非负整数值的离散随机变量，概率生成函数 G(z) = E[z^X] = Σ p_k z^k 正好是概率质量序列 p_k 的 Z 变换在 z 而非 z⁻¹ 处的求值。这种联系使得变换技术可用于求解马尔可夫链和排队论中的递推关系。

---

10. Bridging Continuous and Discrete: The Bilinear Transform | 桥接连续与离散：双线性变换

z-1)/(z+1), where T is the sampling period. This transformation maps the entire left-half s-plane onto the unit disc, preserving stability. It is a cornerstone of digital filter design from analogue prototypes, though it introduces frequency warping that must be compensated for by pre-warping critical frequencies.

双线性变换通过代换 s = (2/T)·(z-1)/(z+1) 提供 s 平面与 z 平面之间的映射，其中 T 为采样周期。该变换将整个左半 s 平面映射到单位圆盘上，从而保持稳定性。它是从模拟原型设计数字滤波器的基石，但会引入频率畸变，必须通过对关键频率进行预畸变来补偿。

This connection highlights the deep structural relationship between the Laplace and z transforms. Many concepts — pole-zero diagrams, frequency responses, and convolution theorems — have direct parallels in the two domains, unifying the analysis of continuous- and discrete-time systems.

这一联系彰显了拉普拉斯变换与 Z 变换之间的深层结构关系。许多概念 —— 零极点图、频率响应和卷积定理 —— 在两个域中都有直接对应，统一了连续与离散时间系统的分析。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）