DSP 101
M4 · L3
Module 4 — The Z-Transform
Poles, Zeros &
Frequency Response

Every rational filter is a constellation of poles and zeros in the complex plane. Their positions relative to the unit circle dictate the entire frequency response — magnitude, phase, and stability — at a glance.

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DSP 101
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Definition
The Pole-Zero Plot

Factor H(z) = g · ∏(z − z_k) / ∏(z − p_k). Zeros (circles ○) are where H = 0; poles (crosses ×) are where H → ∞.

○
Zeros — nulls
×
Poles — resonances
g
Gain constant
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DSP 101
M4 · L3
Frequency Response
Unit Circle Evaluation

Substitute z = ejω: the magnitude response is the product of zero distances divided by the product of pole distances to ejω.

Geometric Formula
|H(e^{j\omega})| = |g|\,\frac{\prod_k |e^{j\omega}-z_k|}{\prod_k |e^{j\omega}-p_k|}
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DSP 101
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Geometric Intuition
Distance Rules

Sweep ejω around the unit circle. The magnitude |H| rises and falls as distances to poles and zeros change.

Pole near circle
Small denominator distance → magnitude peak (resonance)
Zero on circle
Numerator distance = 0 → perfect null (|H| = 0)
Pole on circle (r = 1)
Neither decays nor grows → sustained oscillation. Marginally stable: an oscillator, never a filter.
Interactive: drag the poles and zeros, watch |H| peak and null — AI-generated
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DSP 101
M4 · L3
Phase Response
Sum of Angles

∠H(ejω) = sum of angles from zeros − sum of angles from poles. Group delay = −d∠H/dω.

Linear Phase
Symmetric FIR designs achieve linear phase — all frequencies delayed equally. IIR filters with poles generally do not.
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DSP 101
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Real Filters
Conjugate Pairs

If h[n] is real, poles and zeros off the real axis always appear as conjugate pairs (p, p*). This ensures a real-valued impulse response.

Building Block
Each conjugate pole pair → one second-order section with real coefficients. This is the standard IIR implementation unit.
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DSP 101
M4 · L3
Design Example
60 Hz Notch

Place zeros on the unit circle at ±ω₀ for a perfect null. Poles just inside at radius r control bandwidth.

Notch Transfer Function
H(z) = \frac{z^2 - 2\cos\omega_0\,z + 1}{z^2 - 2r\cos\omega_0\,z + r^2}
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DSP 101
Knowledge Check

Check what stuck

Four questions from this lesson. Answer to see why — the explanation appears whether you were right or wrong. Nothing is scored or saved.

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DSP 101
M4 · L3
Key Takeaways
What You Learned
  • Poles (×) → resonances; zeros (○) → nulls in frequency response
  • |H(ejω)| = product of zero distances / product of pole distances
  • Pole close to unit circle → magnitude peak at that angle
  • Zero on unit circle → |H| = 0 at that angle
  • Phase = sum of zero angles − sum of pole angles
  • Real filters: poles and zeros appear in conjugate pairs
  • Notch filter: zeros on circle at ω₀, poles just inside at same angle
Next Lesson
Inverse Z-Transform Methods: partial fractions, long division, and recovering h[n] from H(z).
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