DSP 101
M4 · L2
Module 4 — The Z-Transform
Region of
Convergence

The Z-transform is only defined where its infinite sum converges. The Region of Convergence encodes the signal type, makes the inverse transform unique, and determines whether a system is stable.

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DSP 101
M4 · L2
Motivation
Why ROC Matters

Two completely different sequences can share the same X(z) expression. Without the ROC, the inverse Z-transform is ambiguous.

|z|
ROC depends on magnitude
∞
Annulus centered at origin
No
poles inside ROC
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DSP 101
M4 · L2
Definition
Absolute Convergence

The ROC is the set of z values where the sum is finite. Since |z⁻ⁿ| = r⁻ⁿ with r = |z|, convergence depends only on the magnitude of z — making the ROC an annulus.

Convergence Condition
\sum_{n=-\infty}^{\infty} |x[n]|\,|z|^{-n} < \infty
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DSP 101
M4 · L2
Causal Signals
Right-Sided → Exterior

For causal sequences, convergence requires |z| to be large enough. ROC is the exterior of a circle.

Example
aⁿu[n] ↔ z/(z−a), ROC: |z| > |a|
  • Pole at z = a sits on boundary, never inside ROC
  • Causal FIR: ROC includes z = ∞
  • ROC never contains any pole of X(z)
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DSP 101
M4 · L2
Anti-Causal Signals
Left-Sided → Interior

For anti-causal sequences, convergence requires |z| to be small enough. The same rational X(z), a completely different sequence.

The Key Ambiguity
aⁿu[n] and −aⁿu[−n−1] both give z/(z−a) — only the ROC distinguishes them.
Anti-Causal ROC
−aⁿu[−n−1] ↔ z/(z−a), ROC: |z| < |a|
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DSP 101
M4 · L2
Two-Sided Signals
Annular ROC

Two-sided sequences must converge in both directions simultaneously — requiring |z| to be neither too large nor too small.

Annular ROC
r_1 < |z| < r_2
Empty ROC
If r₁ ≥ r₂, the intersection is empty and the Z-transform does not exist.
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DSP 101
M4 · L2
System Stability
ROC & Stability

A system is BIBO stable if and only if the unit circle lies inside the ROC of H(z). For causal systems, this means all poles inside |z| = 1.

|z|=1
in ROC → stable
<1
all causal poles
LHP
↔ inside circle
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DSP 101
Knowledge Check

Check what stuck

Four questions from this lesson. Answer to see why — the explanation appears whether you were right or wrong. Nothing is scored or saved.

Question 1 of 0
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8 / 9
DSP 101
M4 · L2
Key Takeaways
What You Learned
  • ROC = set of z where X(z) sum converges — always an annulus
  • ROC never contains poles of X(z)
  • Right-sided (causal): ROC = |z| > r₁
  • Left-sided (anti-causal): ROC = |z| < r₂
  • Two-sided: ROC = r₁ < |z| < r₂ (annular)
  • Same X(z) expression → different signals depending on ROC
  • BIBO stable ⟺ unit circle inside ROC ⟺ all causal poles inside unit circle
Next Lesson
Poles, Zeros & Frequency Response: reading filter behavior directly from the z-plane.
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