Low and High Shelf Filters

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

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

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

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

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.