Low and High Shelving Filters

The analog transfer function for a low shelf is given by [103]

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.

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Subsections

• • Exercise

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.