Bandstop filter

A bandstop filter (also called a notch filter when narrow) passes signal frequencies everywhere except a band that it attenuates. The textbook gives the second-order bandstop as

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

which corresponds to a HPF and LPF connected in parallel with cutoffs ${\omega }_{ca}$ (highpass) and ${\omega }_{cb}$ (lowpass). It has the same poles as the bandpass (so passband behavior outside the stop region is similar).

[The textbook calls this the standard bandstop and says it “kills the response in a narrow band.” This appears to be slightly inaccurate: the zeros of the given numerator are at $s=-{\omega }_{cb}\pm \sqrt{{\omega }_{cb}^2-{\omega }_{ca}{\omega }_{cb}}$, which are on the negative real axis (not the imaginary axis) when ${\omega }_{cb}>{\omega }_{ca}$. Real zeros produce a deep dip but never zero out the response — a true notch needs zeros at $\pm j{\omega }_0$. The “HPF + LPF in parallel” form gives a band-rejection behavior with a frequency-dependent dip whose minimum occurs around the geometric mean $\sqrt{{\omega }_{ca}{\omega }_{cb}}$, but it does not completely block any specific frequency. Verify against an authoritative 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 $s=\pm j{\omega }_0$ on the imaginary axis. This completely blocks $\omega ={\omega }_0$ while passing nearby frequencies (with sharpness controlled 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 textbook 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, so the zeros are real and negative on the negative real axis.
• If ${\omega }_{cb}<{\omega }_{ca}$ (the actual bandstop topology, with LPF cutoff below HPF cutoff so the band $({\omega }_{cb},{\omega }_{ca})$ gets rejected): discriminant is negative, so the zeros are complex-conjugate with real part $-{\omega }_{cb}$ — off the negative real axis but still off the imaginary axis.

In either case the zeros are not on the imaginary axis, so the filter does not completely block any specific frequency. The 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 — these completely block ${\omega }_0$.
• Two complex-conjugate poles near $-{\omega }_0/(2Q)\pm j{\omega }_0\sqrt{1-1/(4Q^2)}$, with the pole locations setting 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.

The narrower the notch, the cleaner the surrounding band — but the harder to design and the more careful you have to be that the line frequency hasn’t drifted. Power-line frequency varies by ±0.1% under normal grid conditions, which can be a significant 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 “notch” and “bandstop” 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.