📚 Z-Transform Behavior in the Complex Plane | z变换在复平面中的行为特性
The z-transform is a fundamental tool in discrete-time signal processing and system analysis. Its behavior in the complex plane — particularly the location of poles, zeros, and the region of convergence — determines the stability, causality, and frequency response of discrete-time systems. This article explores the geometric and analytic properties of the z-transform in the complex plane.
z变换是离散时间信号处理与系统分析中的基础工具。它在复平面中的行为特性——尤其是极点、零点的位置以及收敛域——决定了离散时间系统的稳定性、因果性和频率响应。本文将深入探讨z变换在复平面中的几何与分析性质。
1. Definition of the Z-Transform | z变换的定义
For a discrete-time sequence x[n], the bilateral (two-sided) z-transform is defined as:
X(z) = Σₙ₌₋∞ᐞ∞ x[n]·z⁻ⁿ
where z = reʲω is a complex variable. The unilateral (one-sided) form, used for causal signals, sums from n = 0 to ∞. The transform maps a time-domain sequence into a function of the complex variable z.
对于离散时间序列 x[n]，双边z变换定义为：
X(z) = Σₙ₌₋∞ᐞ∞ x[n]·z⁻ⁿ
其中 z = reʲω 为复变量。单边z变换（用于因果信号）从 n = 0 求和到 ∞。该变换将时域序列映射为复变量 z 的函数。
2. The Region of Convergence (ROC) | 收敛域
The region of convergence is the set of all z in the complex plane for which the defining sum converges absolutely. For a finite-power signal, the ROC is an annular region in the z-plane:
ROC = { z ∈ ℂ : R₁ < |z| < R₂ }
with 0 ≤ R₁ < R₂ ≤ ∞. The ROC never contains poles, and its shape strongly constrains the possible time-domain signals associated with a given X(z).
收敛域是复平面中使定义级数绝对收敛的所有 z 的集合。对于有限功率信号，收敛域是z平面中的环形区域：
ROC = { z ∈ ℂ : R₁ < |z| < R₂ }
其中 0 ≤ R₁ < R₂ ≤ ∞。收敛域内不包含任何极点，并且其形状对给定 X(z) 所能对应的时域信号种类施加了强烈约束。
3. Poles, Zeros, and the Rational Z-Transform | 极点、零点与有理z变换
Most practical z-transforms are rational functions of z, expressed as:
X(z) = N(z) / D(z) = K · Πᵢ₌₁ᴹ (z − zᵢ) / Πⱼ₌₁ᴺ (z − pⱼ)
The zeros zᵢ are the roots of the numerator polynomial N(z); the poles pⱼ are the roots of the denominator polynomial D(z). The constant K is a gain factor. Pole-zero plots provide an immediate visual summary of the system’s behavior.
大多数实际z变换是z的有理函数，可表示为：
X(z) = N(z) / D(z) = K · Πᵢ₌₁ᴹ (z − zᵢ) / Πⱼ₌₁ᴺ (z − pⱼ)
零点 zᵢ 是分子多项式 N(z) 的根；极点 pⱼ 是分母多项式 D(z) 的根。常数 K 为增益因子。极点-零点图提供了系统行为的直观视觉总结。
4. ROC and Time-Domain Direction | 收敛域与时域方向
The ROC determines which time-domain sequence corresponds to a given algebraic expression. For a right-sided (causal) sequence, the ROC extends outward from the outermost pole:
ROC: |z| > R_max (right-sided)
For a left-sided (anti-causal) sequence, the ROC extends inward toward the origin:
ROC: |z| < R_min (left-sided)
For a two-sided sequence, the ROC is a ring: R₁ < |z| < R₂. A finite-length sequence has an ROC that is the entire z-plane except possibly z = 0 and z = ∞.
收敛域决定了给定的代数表达式对应哪一个时域序列。对于右边（因果）序列，收敛域从最外层极点向外延伸：
ROC: |z| > R_max（右边序列）
对于左边（反因果）序列，收敛域从最内层极点向内延伸至原点：
ROC: |z| < R_min（左边序列）
对于双边序列，收敛域为环形：R₁ < |z| < R₂。有限长序列的收敛域为整个z平面，可能除 z = 0 和 z = ∞ 之外。
5. Stability and the Unit Circle | 稳定性与单位圆
A discrete-time linear time-invariant (LTI) system is bounded-input bounded-output (BIBO) stable if and only if its ROC includes the unit circle, |z| = 1. This condition is equivalent to the absolute summability of the impulse response:
Σₙ₌₋∞ᐞ∞ |h[n]| < ∞ ⇔ unit circle ⊂ ROC
Causality plus stability requires all poles to lie strictly inside the unit circle, i.e., |pⱼ| < 1 for all j. A pole on the unit circle indicates marginal stability, often producing oscillatory or constant steady-state components.
一个离散时间线性时不变（LTI）系统是有界输入有界输出（BIBO）稳定的，当且仅当其收敛域包含单位圆 |z| = 1。该条件等价于脉冲响应绝对可求和：
Σₙ₌₋∞ᐞ∞ |h[n]| < ∞ ⇔ 单位圆 ⊂ 收敛域
因果性加稳定性要求所有极点严格位于单位圆内，即对所有 j 有 |pⱼ| < 1。极点在单位圆上表示临界稳定，通常会产生振荡或恒定的稳态分量。
6. Frequency Response and the Unit Circle | 频率响应与单位圆
The frequency response of a discrete-time system is obtained by evaluating the z-transform on the unit circle:
H(eʲω) = H(z) | z = eʲω
Since z = eʲω = cos ω + j sin ω, moving along the unit circle corresponds to sweeping the continuous frequency variable ω from 0 to 2π. The magnitude |H(eʲω)| and phase ∠H(eʲω) are the gain and phase shift experienced by a complex exponential input eʲωⁿ.
离散时间系统的频率响应通过在单位圆上计算z变换得到：
H(eʲω) = H(z) | z = eʲω
由于 z = eʲω = cos ω + j sin ω，沿单位圆移动相当于将连续频率变量 ω 从 0 扫到 2π。幅值 |H(eʲω)| 和相位 ∠H(eʲω) 分别是复指数输入 eʲωⁿ 所经历增益与相移。
7. Geometric Interpretation of Magnitude Response | 幅值响应的几何解释
The magnitude of H(z) on the unit circle can be expressed as the product of distances from the evaluation point z = eʲω to zeros, divided by distances to poles:
|H(eʲω)| = |K| · Πᵢ |eʲω − zᵢ| / Πⱼ |eʲω − pⱼ|
This geometric view is powerful: a pole close to the unit circle creates a peak in |H(eʲω)| at the frequency closest to that pole; a zero close to the unit circle creates a dip or notch. This insight directly links pole-zero geometry to filtering behavior.
单位圆上 H(z) 的幅值可表示为从求值点 z = eʲω 到各零点的距离之积除以到各极点的距离之积：
|H(eʲω)| = |K| · Πᵢ |eʲω − zᵢ| / Πⱼ |eʲω − pⱼ|
这种几何视图非常有力：靠近单位圆的极点在与其最接近的频率处产生幅值峰；靠近单位圆的零点则产生凹陷或陷波。这一见解直接将极点-零点几何与滤波行为联系起来。
8. Relationship with the Laplace Transform | 与拉普拉斯变换的关系
The z-transform is related to the (bilateral) Laplace transform through the substitution z = eˢᵀ, where T is the sampling period. Under this mapping:
X(z) = Xₐ(s) | s = (1/T)·ln z
This relationship maps the imaginary axis s = jω to the unit circle z = eʲωᵀ, and the left half-plane Re(s) < 0 to the interior of the unit circle |z| < 1 (for T > 0). Consequently, continuous-time stability (all poles in LHP) translates to discrete-time stability (all poles inside the unit circle).
z变换通过代换 z = eˢᵀ 与（双边）拉普拉斯变换相联系，其中 T 为采样周期。在该映射下：
X(z) = Xₐ(s) | s = (1/T)·ln z
该关系将虚轴 s = jω 映射到单位圆 z = eʲωᵀ，将左半平面 Re(s) < 0 映射到单位圆内部 |z| < 1（当 T > 0 时）。因此，连续时间稳定性（所有极点位于左半平面）转化为离散时间稳定性（所有极点位于单位圆内）。
9. Worked Example: First-Order System | 实例：一阶系统
Consider a causal first-order system with:
H(z) = 1 / (1 − 0.9 z⁻¹) = z / (z − 0.9)
There is one pole at p₁ = 0.9 and one zero at z₁ = 0. The ROC is |z| > 0.9. Since the unit circle |z| = 1 lies within the ROC, the system is stable. The impulse response is h[n] = (0.9)ⁿ u[n], which decays to zero as n → ∞.
考虑一个因果一阶系统：
H(z) = 1 / (1 − 0.9 z⁻¹) = z / (z − 0.9)
系统中在 p₁ = 0.9 处有一个极点，在 z₁ = 0 处有一个零点。收敛域为 |z| > 0.9。由于单位圆 |z| = 1 位于收敛域内，系统是稳定的。冲激响应为 h[n] = (0.9)ⁿ u[n]，当 n → ∞ 时衰减至零。
10. Higher-Order Poles and Repeated Roots | 高阶极点与重根
When the denominator D(z) has repeated roots, the partial fraction expansion includes terms of the form:
X(z) = Σₖ Aₖ / (z − pₖ)ᵐₖ
where mₖ is the multiplicity of pole pₖ. The inverse z-transform then involves sequences of the form n^{mₖ−1}·(pₖ)ⁿ. A repeated pole on the unit circle yields polynomial growth in the time domain, which is unstable in the BIBO sense.
当分母 D(z) 具有重根时，部分分式展开包含以下形式的项：
X(z) = Σₖ Aₖ / (z − pₖ)ᵐₖ
其中 mₖ 是极点 pₖ 的重数。逆z变换此时涉及形如 n^{mₖ−1}·(pₖ)ⁿ 的序列。单位圆上的重极点会导致时域中的多项式增长，这在BIBO意义下是不稳定的。
11. Complex Conjugate Poles | 共轭复数极点
For real-coefficient systems, poles and zeros occur in complex conjugate pairs. A pair of poles at p = r·eʲω₀ and p* = r·e⁻ʲω₀ (0 < r < 1) produces oscillatory temporal behavior:
h[n] = A·rⁿ·cos(ω₀n + φ)·u[n]
The angular position ω₀ determines the oscillation frequency, while the radius r controls the decay rate. As r → 1 the oscillations persist longer; as r → 0 they die out almost immediately.
对于实系数系统，极点和零点成共轭复数对出现。位于 p = r·eʲω₀ 和 p* = r·e⁻ʲω₀（0 < r < 1）的一对共轭极点产生振荡时域行为：
h[n] = A·rⁿ·cos(ω₀n + φ)·u[n]
角度位置 ω₀ 决定振荡频率，半径 r 控制衰减速率。当 r → 1，振荡持续更久；当 r → 0，振荡几乎立即消失。
12. Practical Interpretation | 实际解读
In practical engineering contexts, the pole-zero plot of the z-transform is used to design digital filters and analyze control systems. Designers place roots to achieve desired frequency selectivity, ensuring all poles lie inside the unit circle for stability. The ROC concept also plays a central role in deconvolution, inverse filtering, and solving difference equations.
在实际工程背景下，z变换的极点-零点图用于设计数字滤波器和分析控制系统。设计者通过配置根的位置来实现所需的频率选择性，同时确保所有极点位于单位圆内以保证稳定性。收敛域的概念在反卷积、逆滤波以及求解差分方程中也发挥着核心作用。
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