Bandstop filter

A bandstop filter (a notch filter when narrow) passes everything except a band it attenuates. The second-order bandstop given in the course is

$H(s)=\frac{s^2+2{\omega }_{cb}s+{\omega }_{ca}{\omega }_{cb}}{s^2+({\omega }_{ca}+{\omega }_{cb})s+{\omega }_{ca}{\omega }_{cb}},$

which is a HPF and LPF in parallel, cutoffs ${\omega }_{ca}$ (highpass) and ${\omega }_{cb}$ (lowpass). Same poles as the bandpass, so passband behavior outside the stop region is similar.

One caveat on that formula: the zeros of the numerator sit at $s=-{\omega }_{cb}\pm \sqrt{{\omega }_{cb}^2-{\omega }_{ca}{\omega }_{cb}}$, which land on the negative real axis (not the imaginary axis) when ${\omega }_{cb}>{\omega }_{ca}$. Real zeros give a deep dip but never zero out the response; a true notch needs zeros at $\pm j{\omega }_0$. The HPF+LPF parallel form gives band rejection with a frequency-dependent dip whose minimum is around the geometric mean $\sqrt{{\omega }_{ca}{\omega }_{cb}}$, but it doesn’t completely block any specific frequency. So calling it a filter that “kills the response in a narrow band” is loose. Worth checking against a proper filter-design reference.

A true notch filter has the standard form

$H_{\text{notch}}(s)=\frac{s^2+{\omega }_0^2}{s^2+({\omega }_0/Q)s+{\omega }_0^2},$

with zeros exactly at $s=\pm j{\omega }_0$. This blocks $\omega ={\omega }_0$ completely while passing nearby frequencies, sharpness set by the quality factor $Q$. Most engineering literature uses this form for the 60 Hz notch in audio and similar applications.

Pole-zero structure

For the parallel-HPF/LPF formula, the zeros sit at $s=-{\omega }_{cb}\pm \sqrt{{\omega }_{cb}^2-{\omega }_{ca}{\omega }_{cb}}$:

• If ${\omega }_{cb}>{\omega }_{ca}$ (cascade-bandpass topology): discriminant is positive, zeros are real and negative on the negative real axis.
• If ${\omega }_{cb}<{\omega }_{ca}$ (the actual bandstop topology, LPF cutoff below HPF cutoff so the band $({\omega }_{cb},{\omega }_{ca})$ gets rejected): discriminant is negative, zeros are complex-conjugate with real part $-{\omega }_{cb}$, off the negative real axis but still off the imaginary axis.

Either way the zeros are not on the imaginary axis, so the filter doesn’t completely block any specific frequency. Poles sit in the left half-plane (real at $-{\omega }_{ca}$ and $-{\omega }_{cb}$, or complex-conjugate if their discriminant is negative).

For the true-notch formula:

• Two zeros at $\pm j{\omega }_0$ on the imaginary axis, blocking ${\omega }_0$ completely.
• Two complex-conjugate poles near $-{\omega }_0/(2Q)\pm j{\omega }_0\sqrt{1-1/(4Q^2)}$; the pole locations set the bandwidth.

Behavior

• At DC: passes through with some attenuation.
• At high frequency: passes through (since the highest-order terms in $H(s)$ have the same coefficient in numerator and denominator).
• At the notch frequency ${\omega }_0$: $|H(j{\omega }_0)|\to 0$. The notch kills this frequency completely.

So the response is “flat-flat-notch-flat-flat”: flat everywhere except a deep dip at ${\omega }_0$.

The classic application: 60 Hz hum

Audio recordings and measurement instruments pick up 60 Hz hum from nearby power lines (50 Hz in some countries). A bandstop filter centered at 60 Hz removes the line-frequency hum without affecting nearby audio frequencies.

Narrower notch means a cleaner surrounding band, but it’s harder to design and you have to watch that the line frequency hasn’t drifted. Power-line frequency varies by ±0.1% under normal grid conditions, a sizeable fraction of a few-Hz-wide notch.

Other applications

• EEG/ECG: remove 60 Hz interference from biological signal recordings.
• Radio receivers: notch out an adjacent strong interferer to prevent receiver overload.
• Audio equalizers: a deep notch can be used to cancel a specific resonance (e.g. a room mode).
• Power systems: notch filters in instrument transformers reduce specific harmonics.

Notch vs general bandstop

The terms overlap. Notch usually means a narrow, deep response at a single frequency (Q > 5, very sharp). Bandstop is the more general term, covering both narrow notches and broader rejection bands.