notchfiltering

Notch filtering is a signal processing technique that attenuates or suppresses a narrow band of frequencies while leaving others largely unaffected. It is a specific form of band-stop filtering aimed at removing an unwanted tone or interference without broadly distorting the rest of the spectrum.

In digital implementations, notch filters are typically realized as second-order IIR filters. A common discrete-time form places zeros on the unit circle at the target notch frequency and places poles near the unit circle to control the notch width. A standard transfer function is H(z) = (1 - 2 cos(ω0) z^-1 + z^-2) / (1 - 2 r cos(ω0) z^-1 + r^2 z^-2), where ω0 is the notch frequency (in radians per sample) and r (0 < r < 1) determines the bandwidth; tighter notches have r close to 1. The quality factor Q is defined as f0 / BW, with higher Q yielding a narrower notch. Analog notch filters use a similar pole-zero arrangement in continuous time, typically expressed as H(s) = (s^2 + ω0^2) / (s^2 + 2 ζ ω0 s + ω0^2).

Notch filters can be implemented with IIR or FIR structures. FIR implementations, while typically less efficient, can offer linear phase, but IIR notch filters are common for their steep attenuation with relatively low order. Adaptive notch filtering, using algorithms like LMS, tracks drifting interference such as power-line hum.

Applications span audio and music processing, biomedical signal cleaning (for example, removing 50/60 Hz interference in EEG or ECG data), communications, and vibration or seismic analysis. Limitations include potential distortion of nearby frequencies, phase nonlinearity in IIR designs, and the need for accurate notch frequency estimation or adaptation.