The analog transfer function for a low shelf is given by B.5
A high shelf is obtained from a low shelf by the conformal mapping , which interchanges high and low frequencies, i.e.,
To convert these analog-filter transfer functions to digital form, we apply the bilinear transform:
where typically and denotes the sampling interval in seconds.
Low and high shelf filters are typically implemented in series, and are typically used to give a little boost or cut at the extreme low or high end (of the spectrum), respectively. To provide a boost or cut near other frequencies, it is necessary to go to (at least) a second-order section, often called a ``peaking equalizer,'' as described in §B.5 below.