All-Pass Filter
What it does
An all-pass filter lets all frequencies pass through unchanged in amplitude but alters their phase. It does not remove or amplify any frequency component.
Key Property
|H(jω)| = 1 for all ω
This means: Unity gain is ensured at all frequencies.
Continuous-Time Example
H(s) = (s - a) / (s + a)
Substituting s = jω:
H(jω) = (jω - a) / (jω + a)
Magnitude |H(jω)| = 1
Reason for Unity Gain
- Zeros are mirror images of poles across the imaginary axis
- Magnitude stays constant, phase changes with frequency
Digital (Discrete-Time) Version
H(z) = (z⁻¹ - a) / (1 - a z⁻¹)
|H(e^{jω})| = 1, phase depends on ω
Effect
- Amplitude remains constant
- Phase varies → causes time delay differences
Applications
- Phase correction
- Signal alignment
- Audio processing (phase equalization)
- Communication systems
Comparison with Other Filters
| Filter Type | Amplitude | Phase |
|---|---|---|
| Low-pass | Changes | Changes |
| Notch | Removes freq | Changes |
| All-pass | Constant (1) | Changes |
Summary
All-pass filter = “magnitude stays same, phase changes”