## Low and High Shelving Filters

The analog transfer function for a low shelf is given by 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 E 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: where denotes the sampling interval in seconds.B.5

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.

#### Exercise

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.

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Peaking Equalizers
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DC Blocker