A Bell or Peaking filter is a type of audio equalisation filter that boosts or attenuates the magnitude of a specified set of frequencies around a centre frequency in order to perform magnitude equalisation. As seen in the plot in the below, the filter gets its name from the shape of the its magnitude spectrum (blue line) which resembles a Bell curve.
Frequency response (magnitude shown in blue, phase shown in purple) of a 2nd order Bell filter peaking at 125Hz.
Central to the Bell filter is the so called All-pass filter. All-pass filters provide a simple way of altering/improving the phase response of an IIR without affecting its magnitude response. As such, they are commonly referred to as phase equalisers and have found particular use in digital audio applications.
Notice how the numerator and denominator coefficients are arranged as a mirror image (mirror-image pair) of one another. The mirror image property is what gives the all-pass filter its desirable property, namely allowing the designer to alter the phase response while keeping the magnitude response constant or flat over the complete frequency spectrum.
A Bell filter can be constructed from the \(A(z)\) filter by the following transfer function:
After some algebraic simplication, we obtain the transfer function for the Peaking or Bell filter as: