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All-Pass Filter

Unity magnitude at all frequencies - only phase changes

r = 0.80
θ = 1.57 rad (π/2)

Magnitude Response |H(e^jω)|

Phase Response ∠H(e^jω)

Group Delay τ(ω)

Pole-Zero Plot

H(z) = (z⁻¹ - r·e^(jθ)) / (1 - r·e^(-jθ)·z⁻¹) × (z⁻¹ - r·e^(-jθ)) / (1 - r·e^(jθ)·z⁻¹)

All-Pass Filter Properties

Unity Magnitude: The magnitude response is exactly 1 at all frequencies. The filter only changes the phase of the signal.

Mirror Poles and Zeros: For each pole at z = r·e^(jθ), there's a zero at z = (1/r)·e^(jθ). This creates the unity magnitude property.

Group Delay: The rate of phase change varies with frequency. Higher pole radius = more concentrated delay near the pole frequency.

Applications: Phase equalization, audio effects (phaser), group delay compensation, building blocks for other filters.