# Digital Signal Processing/Spectral Transforms

When designing a particular filter, it is common practice to first design a Low-pass filter, and then using a spectral transform to convert that lowpass filter equation into the equation of a different type of filter. This is done because many common values for butterworth, cheybyshev and elliptical low-pass filters are already extensively tabulated, and designing a filter in this manner rarely requires more effort then looking values up in a table, and applying the correct transformations. This page will enumerate some of the common transformations for filter design.

It is important to note that spectral transformations are not exclusively used for analog filters. There are digital variants of these transformations that can be applied directly to digital filters to transform them into a different type. This page will only talk about the analog filter transforms, and the digital filter transforms will be discussed elsewhere.

## Low-Pass to Low-Pass

This spectral transform is used to convert between two lowpass filters with different cutoff frequencies.

${\displaystyle z^{-1}={\frac {1}{F({\hat {z}})}}=}$