Low and High Shelving Filters
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:
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.
Perform the bilinear transform defined above and calculate the coefficients of a first-order digital low shelving filter. Find the pole and zero as a function of , , and . Set and verify that you get a gain of . Set and verify that you get a gain of 1 there.