Low and High Shelving Filters
The analog transfer function for a low shelf is given by [103]
![$\displaystyle 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}
$](http://www.dsprelated.com/josimages_new/filters/img1466.png)
![$ B_0$](http://www.dsprelated.com/josimages_new/filters/img1467.png)
![$ s=0$](http://www.dsprelated.com/josimages_new/filters/img1468.png)
![$ s=\infty$](http://www.dsprelated.com/josimages_new/filters/img1469.png)
![$ 1$](http://www.dsprelated.com/josimages_new/filters/img1440.png)
![$ \omega_1$](http://www.dsprelated.com/josimages_new/filters/img1470.png)
![$ s$](http://www.dsprelated.com/josimages_new/filters/img1471.png)
A high shelf is obtained from a low shelf by the conformal mapping
, which interchanges high and low frequencies, i.e.,
![$\displaystyle H(s) \;=\; 1 + \frac{B_\pi\omega_1}{\frac{1}{s}+\omega_1}
\;=\; ...
...{\omega_z}{\omega_1} \cdot \frac{s + \frac{1}{\omega_z}}{s+\frac{1}{\omega_1}}
$](http://www.dsprelated.com/josimages_new/filters/img1473.png)
![$ 1+B_\pi = \omega_z/\omega_1$](http://www.dsprelated.com/josimages_new/filters/img1474.png)
To convert these analog-filter transfer functions to digital form, we apply the bilinear transform:
![$\displaystyle s = \frac{2}{T}\frac{1-z^{-1}}{1+z^{-1}}
$](http://www.dsprelated.com/josimages_new/filters/img1475.png)
![$ T$](http://www.dsprelated.com/josimages_new/filters/img96.png)
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.
Next Section:
Peaking Equalizers
Previous Section:
DC Blocker