Pole–Zero & Frequency Response

A digital filter's whole character lives in a handful of poles and zeros. Move them around the z-plane and watch the frequency response morph.

Pole–Zero & Frequency ResponseLive

poles peak, zeros notch

Controls

Presets

Pole radius

Pole angle (°)

Zero angle (°)

A digital filter is defined by its poles and zeros in the z-plane. A pole near the unit circle creates a resonant peak at its angle; a zero on the circle carves a notch. Move them and watch the frequency response reshape. Educational tool.

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Data Inspector

Pole radius0.85

Resonance at45° (ω)

Stabilitystable ✓

Governing equation

H (z) = \frac{( z - e ^{j 12 0^{\circ}} ) ( z - e ^{- j 12 0^{\circ}} )}{( z - 0.85 e ^{j 4 5^{\circ}} ) ( z - 0.85 e ^{- j 4 5^{\circ}} )}, stable ⟺ ∣ p ∣ = 0.85 < 1

Reading this result: A pole at r=0.85, 45° gives a resonant peak while the zero at 120° carves a separate notch — a classic two-feature filter.

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How it works

In the z-plane, a pole near the unit circle produces a resonant peak in the frequency response at its angle, while a zero on the circle carves a notch. Poles must stay inside the unit circle for stability. This geometric view is central to digital filter design. Educational tool.

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Frequently asked questions

Is this pole zero plot filter tool really free?▾

Yes. Pole–Zero & Frequency Response runs entirely in your browser using your device's own compute, so local use is free forever. You only pay Compute Tokens if you scale a job to the cloud.

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No. Everything runs client-side in a modern browser — no downloads, no license, no account required to start.

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How accurate are the results?▾

The solver uses established numerical methods, but results are for research and educational purposes and should be validated against experiment or professional review before you rely on them.