7.1 z-Transform of a Discrete-Time Signal

离散时间信号的 z 变换

Z-Pair Explorer — z 变换对照表

Pick a canonical sequence and see four things in sync: the closed-form , the region of convergence (ROC, shaded green), the pole/zero map on the z-plane, and the time-domain stem.
选一种典型序列，可一键看到它的解析式、收敛域 ROC（绿色阴影）、z 平面零极点图与时域 stem。 对比第 3、4 项：解析式完全相同，但 ROC 不同，也不同—— 这正是 ROC 不可省略的原因。

Definition:

0.60

q (delay)

Ω₀ / π

0.30

L (length)

X(z) =

ROC:

Poles:

Zeros:

z-plane: × pole 极点 zero 零点 ROC 收敛域 unit circle 单位圆

Time domain

z-plane (poles ×, zeros ○, ROC ▒)

Try this 试试看: switch between aⁿ·u[n] (right) and −aⁿ·u[−n−1] (left). The closed-form is literally identical in both rows, but the ROC flips from outside the pole () to inside the pole (), and the stem plot becomes a completely different sequence. The pair (X(z), ROC) is what is invertible — X(z) alone is not.

s ↔ z Mapping —

Why is the z-plane "circular" while the s-plane is "flat"? Because , with . Drag the sliders to a point in the s-plane and watch the image dot in the z-plane. Every -shift of brings you back to the same z — this is the geometric origin of spectral aliasing.
s 平面上每偏移， z 平面上的像点回到同一位置；这就是谱混叠的几何根源。

Map:

σT (real part)

-0.30

Ω = ωT / π

0.50

Show: Ghost points at Ω ± 2π (aliasing copies) Principal strip |Ω| ≤ π highlighted

s-plane (σT, Ω)

z-plane (image of )

Three observations 三点观察: (1) (imaginary axis of s) maps to the unit circle of z; (2) (left half-plane, stable LT) maps inside the unit circle, maps outside; (3) Every horizontal strip of height in wraps onto the same ring in the z-plane — which is exactly why DT spectra are -periodic.

Think About It 想一想

Q1: Why do and share the same ?两个完全不同的 x[n] 为何对应相同的 X(z)？

The closed form is an analytic continuation of the geometric series. The algebraic identity is only meaningful inside a particular ROC: right-sided sums converge for ; left-sided sums converge for . So the pair (X(z), ROC) is what uniquely determines — not alone.

闭式是几何级数的解析延拓—— 单独看代数表达式不可逆，必须配上 ROC：右边序列对应，左边序列对应。

Q2: For a right-sided sequence, what does "pole inside / on / outside the unit circle" mean physically?右边序列的极点在单位圆内/上/外，分别意味着什么？

For a causal (right-sided) sequence, the ROC is "outside the outermost pole". The DTFT exists iff the unit circle lies inside the ROC, i.e. iff all poles are strictly inside the unit circle:

Pole inside (): decaying sequence — stable, DTFT exists.
Pole on (): marginally stable — bounded but not absolutely summable; DTFT exists only as a distribution (e.g. has a δ-spike at DC).
Pole outside (): unbounded growth — unstable; no DTFT.

右边序列的 ROC 是"最外极点之外"；要让 DTFT 存在，所有极点必须在单位圆内。极点在圆上 = 临界稳定（u[n]、cosine 都是这种情况）；极点在圆外 = 发散。

Q3: Why does the same z keep "appearing" when I slide Ω past 2π?为什么 Ω 每过 2π，z 平面上的点就重合？

Because is periodic in with period . Algebraically:

Geometrically: infinitely many horizontal strips of height in the s-plane all wrap onto the same z-plane circle. Continuous-time spectra repeat after sampling — this is the spectral aliasing of §4.2 viewed from the z side.

采样在频域引入周期，所以 s 平面里宽的水平条带都会被映到同一圈 z。

Q4: Where do the poles of sit, and why?cos(Ω₀n)·u[n] 的极点为何落在单位圆上？

By Euler: . Each complex exponential is a special case of with — so its pole is at , exactly on the unit circle. The cosine is a sum of these two, giving conjugate poles on the unit circle at angles . Switching to retracts the poles inward to radius — a decaying oscillation.

复指数的极点在，余弦 = 两个共轭复指数之和，所以共轭极点对落在单位圆上、辐角 ±Ω₀ 处。 a<1 时极点向原点缩，对应衰减振荡。

Q5: A finite-length pulse has poles only at the origin and zeros all around the unit circle. Why?有限长矩形脉冲的极点全在原点，零点为何均匀分布在单位圆上？

For on :

The numerator has roots at the -th roots of unity, but the root at cancels with the denominator's . The survivors are zeros evenly spaced on the unit circle at , and poles all at . ROC is "everywhere except " — finite-length sequences have essentially the entire z-plane as ROC.

长度 L 的矩形脉冲是几何级数和：分子 z^L−1 的 L 个根中去掉与分母约掉的 z=1，剩下 L−1 个零点均匀分布在单位圆上；极点全在原点。ROC = 整个 z 平面（除 0 外）。