where is the dc boost amount (at ), and the high-frequency
gain () is constrained to be . The transition
frequency dividing low and high frequency regions is .
See
Appendix C for a development of -plane analysis of analog
(continuous-time) filters.

A high shelf is obtained from a low shelf by the conformal mapping
, which interchanges high and low frequencies, i.e.,

In this case, the dc gain is 1 and the high-frequency gain approaches
.

To convert these analog-filter transfer functions to digital form, we
apply the bilinear transform: