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Low and High Shelf Filters

The analog transfer function for a low shelf is given by [103]B.5

$\displaystyle H(s) \;=\; 1 + \frac{B_0}{(s/\omega_1)+1} \;=\; \frac{s+\omega_1(B_0+1)}{s+\omega_1} \;\isdef \; \frac{s+\omega_z}{s+\omega_1} \protect$ (B.11)

where $ B_0$ is the dc boost amount (at $ s=0$ ), and the high-frequency gain ($ s=\infty$ ) is constrained to be $ 1$ . The transition frequency dividing low and high frequency regions is $ \omega_1$ . See Appendix E for a development of $ s$ -plane analysis of analog (continuous-time) filters.

A high shelf is obtained from a low shelf by the conformal mapping $ s \leftarrow 1/s$ , which interchanges high and low frequencies, i.e.,

$\displaystyle H(s) \;=\; 1 + \frac{B_\pi}{\frac{1}{s/\omega_1}+1} \;=\; 1 + \frac{B_\pi s}{s+\omega_1} \;=\; (1 + B_\pi)\frac{s + \frac{\omega_1}{1+B_\pi}}{s+\omega_1}\,. \protect$ (B.12)

In this case, the dc gain is 1 and the high-frequency gain approaches $ 1+B_\pi$ .

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

$\displaystyle s = c\frac{1-z^{-1}}{1+z^{-1}}

where typically $ c=2/T$ and $ T$ 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.

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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-05-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University