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) = 0

Effect 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).

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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) = 0

Effect 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.

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3. Summary Table

Concept	Location	Effect on Frequency
Pole	Denominator root	Boosts/amplifies frequency, creates resonance
Zero	Numerator root	Attenuates/cancels frequency, creates notch

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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

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Further Reading

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