Poles and Zeros in Signal Processing
1. Poles
Poles are values of z (or s in analog) that make the filter’s denominator zero, i.e., locations where the filter’s transfer function theoretically goes to infinity.
Digital filter example:
H(z) = B(z) / A(z) // Poles are roots of A(z) = 0Effect of poles:
- Correspond to resonant frequencies
- Close to the unit circle → narrowband, high amplification
- Far from the unit circle → damped, broad frequency response
- Poles essentially generate/boost frequency components
Intuition: Poles create peaks in the frequency spectrum (resonances).
2. Zeros
Zeros are values of z (or s) that make the numerator zero, i.e., frequencies where the filter output is zero.
Zeros are roots of B(z) = 0Effect of zeros:
- Correspond to frequencies that are attenuated or cancelled
- Close to the unit circle → strong notch (deep null)
- Far from the unit circle → little effect
Intuition: Zeros suppress or cancel specific frequencies.
3. Summary Table
| Concept | Location | Effect on Frequency |
|---|---|---|
| Pole | Denominator root | Boosts/amplifies frequency, creates resonance |
| Zero | Numerator root | Attenuates/cancels frequency, creates notch |
4. Summary
- Poles → Peaks in the spectrum
- Zeros → Dips or notches in the spectrum
- a notch is a sharp cut or indentation. In signal processing, it refers to a narrow band of attenuation at a specific frequency.
- a dip typically refers to a form of attenuation in a signal.
Examples:
- All-pole filter (e.g., LPC) → only resonances, no zeros
- Notch filter → uses zeros to remove a narrow frequency