Low and High Shelving Filters

The analog transfer function for a low shelf is given by [102,6]

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

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.,

$$H(s) \;=\; 1 + \frac{B_0\omega_1}{\frac{1}{s}+\omega_1} \;=\; (1... ...{\omega_z}{\omega_1} \cdot \frac{s + \frac{1}{\omega_z}}{s+\frac{1}{\omega_1}}$$

In this case, the dc gain is 1 and the high-frequency gain approaches $1+B_0 = \omega_z/\omega_1$.

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

$$s = \frac{2}{T}\frac{1-z^{-1}}{1+z^{-1}}$$

where $T$ denotes the sampling interval in seconds.