A generic ideal band-stop filter, showing both positive and negative angular frequencies

In signal processing, a band-stop filter or band-rejection filter is a filter that passes most frequencies unaltered, but attenuates those in a specific range to very low levels.^[1] It is the opposite of a band-pass filter. A notch filter is a band-stop filter with a narrow stopband (high Q factor).

Narrow notch filters (optical) are used in Raman spectroscopy, live sound reproduction (public address systems, or PA systems) and in instrument amplifiers (especially amplifiers or preamplifiers for acoustic instruments such as acoustic guitar, mandolin, bass instrument amplifier, etc.) to reduce or prevent audio feedback, while having little noticeable effect on the rest of the frequency spectrum (electronic or software filters). Other names include "band limit filter", "T-notch filter", "band-elimination filter", and "band-reject filter".

Typically, the width of the stopband is 1 to 2 decades (that is, the highest frequency attenuated is 10 to 100 times the lowest frequency attenuated). However, in the audio band, a notch filter has high and low frequencies that may be only semitones apart. From the figure of the frequency response of an ideal band-stop filter, it's obvious that the band-stop filter is simply an inverted band-pass filter where they share same definition of bandwidth, pass band, stop band and center frequency. The attenuation should be infinite in the stop band and be zero in the two pass bands for an ideal band-stop filter. Band-stop filters are designed by the combination of a low-pass filter and a high-pass filter in a parallel configuration. Overlapping does not occur in the summation of high-pass filter and low-pass filter during the design of band-stop filter. The difference in the starting and ending frequency points causes the two filters to connect effectively without any overlapping.

Generic electrical schematic of a simple band-stop filter

Mathematical description

Band-stop filter can be represented as a combination of low-pass and high-pass filters if the bandwidth is wide enough that the two filters do not interact too much. A more general approach is to design as a low-pass prototype filter which can then be transformed into a bandstop. The simple notch filter shown can be directly analysed. The transfer function is,

${\displaystyle H(s)={\frac {s^{2}+\omega _{z}^{2}}{s^{2}+{\frac {\omega _{p}}{Q}}s+\omega _{p}^{2}}}}$

Here ${\displaystyle \omega _{z}}$ is zero circular frequency and ${\displaystyle \omega _{p}}$ is the pole circular frequency. Zero frequency is the cutoff frequency and ${\displaystyle \omega _{p}}$ sets the type of the notch filter: standard notch when ${\displaystyle \omega _{z}=\omega _{p}}$, low-pass notch (${\displaystyle \omega _{z}>\omega _{p}}$) and high-pass notch (${\displaystyle \omega _{z}<\omega _{p}}$) filters. ${\displaystyle Q}$ denotes the Q-factor.^[2]

For standard notch filter the formulation can be rewritten as

${\displaystyle H(s)={\frac {s^{2}+\omega _{0}^{2}}{s^{2}+\omega _{c}s+\omega _{0}^{2}}},}$

where ${\displaystyle \omega _{0}}$ is the central rejected frequency and ${\displaystyle \omega _{c}}$ is the width of the rejected band.

Examples

See also

• Parametric equalizer

References

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