Notch Filter (Band-Stop Filter)
What it does
A notch filter removes or strongly attenuates a very narrow range of frequencies while allowing all other frequencies to pass.
Example: Removing 50 Hz or 60 Hz power-line noise from a signal
Applications
- Audio noise removal (hum elimination)
- Biomedical signals (ECG/EEG)
- Communication systems
Mathematical Model (Continuous-Time)
H(s) = (s² + ω₀²) / (s² + 2ζω₀s + ω₀²)
Where:
- ω₀ = notch frequency
- ζ = damping factor (controls width)
Key Idea
At ω = ω₀ → H(jω₀) = 0
This means the output becomes zero at that frequency, effectively removing it.
Frequency Response
- At ω₀ → signal is blocked
- Near ω₀ → partially attenuated
- Far from ω₀ → passes normally
Digital Notch Filter
H(z) = (1 - 2cos(ω₀)z⁻¹ + z⁻²) / (1 - 2r cos(ω₀)z⁻¹ + r² z⁻²)
Where:
- ω₀ = notch frequency
- r ≈ 1 → sharper notch
- Zeros cancel the unwanted frequency
- Poles control how sharp the notch is
Summary
- Removes one specific frequency
- Output becomes zero at that frequency
- Based on transfer function design