Pole-Zero Frequency Manipulation (with Example)

Pole-Zero Frequency Manipulation: 40 Hz, 50 Hz, 60 Hz

This example demonstrates how to cancel, amplify, and attenuate specific frequencies using pole-zero placement in analog and digital filters. We target sinusoids at 40 Hz, 50 Hz, and 60 Hz.

1. Analog Example (s-domain)

We have a signal:

x(t) = sin(2π·40 t) + sin(2π·50 t) + sin(2π·60 t)

Step 1: Convert frequencies to angular frequency

• 40 Hz → ω₁ = 2π·40 ≈ 251.33 rad/s
• 50 Hz → ω₂ = 2π·50 ≈ 314.16 rad/s
• 60 Hz → ω₃ = 2π·60 ≈ 376.99 rad/s

Step 2: Construct filters

Notch filter at 50 Hz (cancel 50 Hz):

H_notch(s) = (s² + ω₂²) / (s² + 2ζω₂s + ω₂²)

Zero at ±ω₂ cancels 50 Hz; poles define notch width via damping ζ.

Resonator at 40 Hz (amplify 40 Hz):

H_res(s) = 1 / (s² + 2ζ₁ω₁ s + ω₁²)

Poles near ±ω₁ amplify 40 Hz; damping ζ₁ controls resonance sharpness.

Low-pass effect for 60 Hz (attenuate 60 Hz):

H_LP(s) = ω_c / (s + ω_c), ω_c << ω₃

Step 3: Combine filters

H(s) = H_notch(s) · H_res(s) · H_LP(s)

Result: 50 Hz is cancelled, 40 Hz is amplified, 60 Hz is attenuated.

2. Digital Example (z-domain)

Sampling frequency: fs = 500 Hz

Step 1: Normalize frequencies (0–π rad/sample)

• 40 Hz → ω₁ = 2π·40 / 500 ≈ 0.502 rad/sample
• 50 Hz → ω₂ ≈ 0.628 rad/sample
• 60 Hz → ω₃ ≈ 0.754 rad/sample

Step 2: Notch filter at 50 Hz

H_notch(z) = (1 - 2 cos(ω₂) z⁻¹ + z⁻²) / (1 - 2 r cos(ω₂) z⁻¹ + r² z⁻²), r ≈ 0.95

Zero at z = e^{±jω₂} cancels 50 Hz; r controls notch width.

Step 3: Resonator at 40 Hz (all-pole)

H_res(z) = 1 / (1 - 2 r₁ cos(ω₁) z⁻¹ + r₁² z⁻²), r₁ ≈ 0.99

Sharp resonance at 40 Hz; amplifies amplitude.

Step 4: Low-pass effect for 60 Hz

H_LP(z) = 1 - α z⁻¹, 0 < α < 1

Attenuates high-frequency 60 Hz component.

Step 5: Combine filters

H(z) = H_notch(z) · H_res(z) · H_LP(z)

Result: Frequency response shows 50 Hz cancelled, 40 Hz amplified, 60 Hz degraded.

3. Teaching Insights

• Zero placement → cancels specific frequencies (notch filter)
• Pole placement → amplifies frequencies (resonator)
• Damping or r values → control bandwidth of effect
• Combined effect → shapes amplitude of multiple sinusoids in a signal
• Exact cancellation requires precise matching of pole and zero locations

4. Optional Visualization Tips

• Plot magnitude response |H(jω)| for analog filter
• Plot |H(e^{jω})| for digital filter
• Show time-domain input and output signals to illustrate amplification, cancellation, and attenuation