Control Systems II · ETH Zürich
z-transform · discrete-time
Presets
Mode
Poles & Zeros
Click on the z-plane to place poles or zeros.
Drag existing markers to move them.
Right-click a marker to delete it.

Pole × — unstable if |z| > 1
Zero ○ — affects gain at ω
Re: 0.000 | Im: 0.000 | |z|: 0.000 | ∠z: 0.0°
Stability
Stable — all poles inside unit circle
Transfer Function H(z)
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Frequency Response
|H(e^jω)| vs ω
Phase Response
∠H(e^jω) vs ω
Selected Point
Click a pole or zero
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Z-Transform Definition
X(z) = Σ x[n] · z⁻ⁿ Stability (BIBO)
All poles must lie strictly inside the unit circle |z| < 1.

Frequency Response
Evaluate H(z) on the unit circle: z = e^(jω), ω ∈ [0, 2π]

Region of Convergence
For causal systems: |z| > max|pₖ|