Filter Transformations
See slide.
Other types of filters can be designed by transforming them to low-pass filters, determining their order, and then transforming them back.
Creating a high-pass filter
We can mirror the filter over the cutoff frequency by dividing the nominated filter by the cutoff frequency itself ($\omega_{s}$).
$$H_{hp}(s) = H_{lp}(\bar{s})|_{\bar{s} = \frac{1}{s}}$$
Denomering $$s \rightarrow \frac{s}{\omega_{a}}$$
Creating a band-pass filter
Lektion 1 - Filterfunktioner.pdf>page=66
A band pass filer is a combination of a high-pass and a low-pass filter.
$$H_{bp} = H_{lp} \cdot H_{hp}$$
$\omega_{a}$: Pass-band width
$\omega_{s}$: Stop-band width
$\omega$: Center-frequency: The middle of the stop-band-frequencies, on a *logarithmic scale*.
$A_{s}$: The filter stop-dampening.
NOTE: We use the center-frequency to nominate the filter.
Formfaktor $$W_{a} = \frac{\Delta f_{a}}{f_{c}} \s W_{s} = \frac{\Delta f_{s}}{f_{c}}, \s F = \frac{W_{s}}{W_{a}}$$
The transformation back to band-pass: $$H_{bp}(s) = H_{lp}(\bar{s})|_{\bar{s} = \frac{1}{\omega_{a}}\left(\frac{s+1}{s}\right)}$$
%% TODO: List low-pass TODO: List high-pass TODO: List band-stop %%