What is the difference between a notch filter and a band-pass filter, and provide a simple transfer function form for the notch?

• A. Band-pass attenuates a narrow band; notch passes all frequencies.
• B. Notch (band-stop) attenuates a narrow band while passing others; H(s) for notch is H(s) = (s^2 + ω0^2)/(s^2 + (ω0/Q)s + ω0^2). Band-pass passes a band around ω0.
• C. Notch passes a wide band; band-pass attenuates a central band.
• D. Both have identical transfer functions but different naming.

Answer: (B)

Notch filters are a type of band-stop filter designed to remove a very specific frequency range while leaving the rest of the spectrum mostly unaffected. This is achieved by placing zeros on the imaginary axis at ±jω0 in the transfer function, so the response goes to zero exactly at that center frequency. A band-pass filter, on the other hand, is meant to pass frequencies within a certain band around ω0 and attenuate frequencies outside that band.

The simple notch form H(s) = (s^2 + ω0^2)/(s^2 + (ω0/Q)s + ω0^2) captures this behavior neatly. The zeros at s = ±jω0 create the notch, and the poles set by ω0 and Q shape how sharp that notch is. A higher Q gives a narrower notch, while a lower Q broadens it. At frequencies far from ω0, the gain typically returns toward unity, meaning those frequencies pass with little attenuation.