Design digital filters by hand. Drag poles and zeros around the z-plane and watch the frequency response, phase, and impulse response react in real time.
Z-Transform StudioLive
pole–zero design of digital filters
Controls
Drag poles (×) and zeros (○) around the z-plane and watch the digital filter's frequency response, phase, and impulse response update live. Poles/zeros move as conjugate pairs. A filter is stable only while every pole stays inside the unit circle.
Reading this result: A zero lies essentially on the unit circle, so |H| drops to a deep null at that frequency — this is how notch and band-reject filters kill a specific tone. The response peaks near ω = 0.00π.
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How it works
The z-transform maps a digital filter to its poles and zeros in the complex z-plane, and the transfer function H(z)=∏(z−z_i)/∏(z−p_i) follows directly from their positions. Poles pull the magnitude response up into resonant peaks; zeros push it down into nulls. Evaluating H on the unit circle z=e^{jω} gives the frequency response you hear. The filter is stable only while every pole lies strictly inside the unit circle — drag one outside and the impulse response blows up. Try the low-pass, high-pass, notch, resonator, and Butterworth-ish presets, then move a pole toward the circle to sharpen its resonant peak.
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The math, the assumptions, real-world uses, or a code translation — explained for this exact simulation.
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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.