Signals and Systems/Filter Transforms

Why Transform Filters?

When designing a filter, it is common practice to first design a general low-pass filter, and then use a spectral transform to transform that low-pass filter into a different type of filter (high-pass, band-pass, band-stop). This page will talk about some of the common transformations, for turning a low-pass filter model into another type of filter.

The reason why we design filters like this is because the necessary values for designing lowpass filters are extensively tabulated. Filter design then can be reduced to the task of looking up the appropriate values in a table, and then transforming the filter to meet our specific needs.

Lowpass to Highpass

Converting a lowpass filter to a highpass filter is one of the easiest transformations available. To transform to a highpass, we will replace all S in our equation with the following:

$S=\frac{{\mathrm{\Omega }}_p\widehat{{\mathrm{\Omega }}_p}}{\hat{S}}$

Lowpass to Bandpass

To Convert from a low-pass filter to a bandpass filter requires that we replace S with the following:

$S=$

Lowpass to Bandstop

To convert a lowpass filter to a bandstop filter, we replace every reference to S with:

$S=$