The first step in the filter design is to design a high-order analog lowpass
shelving filter that meets the specified gain and bandwidth. A high-order
Butterworth filter is used for this purpose. The analog Butterworth filter is then
transformed into a digital lowpass shelving filter, and finally into a peaking
equalizer that is centered at the specified peak frequency.

The design specifications for the digital equalizer are the order of the
equalizer, *N*, the gain of the equalizer, *G*,
the center frequency of the equalizer, *W*_{0},
and the bandwidth of the equalizer, *BW*.

The transfer function of the high-order parametric equalizer is given by:

where *b*_{00},
*b*_{01},
*b*_{02},
*a*_{01}, and
*a*_{02} are coefficients of the
second-order section of the equalizer.
*b*_{i0},
*b*_{i1},
*b*_{i2},
*b*_{i3},
*b*_{i4},
*a*_{i1},
*a*_{i2}, and
*a*_{i3} are coefficients of the
fourth-order sections of the equalizer. *L* = (*N*
–*r*) / `2`

, where *r* =
`1`

when *N* is odd, and *r*
= `0`

when *N* is even. The fourth-order sections
are factored into second-order sections so that you can implement them using biquad
filters.

For more information on how coefficients are computed in terms of the design
specifications, see the "Butterworth Designs" section in [1].