Transforming an analog filter to analog low pass prototype
Frequency transformation in s-domain is a common method to design an analog filter (low pass, high pass, band pas, band stop, narrow band...) from an analog low pass prototype and it can be used to transform from an analog filter back to its analog low pass prototpye as shown in the diagram below. With the supporting of the inverting matrix of [Tc], [Tt] and [Th], which are easy for hand-calculation and programming, involved in an equation called "Inverting Pascal matrix equations" and they will be discussed in this section.
The "Inverting Pascal matrix equations" can be found from "Analog low pass prototype Pascal matrix equation" and it can be written as below
Inverting low pass to low pass prototype Pascal matrix equation
Inverting high pass to low pass prototype Pascal matrix equation
Inverting band pass to low pass prototype Pascal matrix equation
Inverting band stop to low pass prototype Pascal matrix equation
Transforming an analog band stop filter to aa analog low pass prototype filter is the same way as inverting the band pass filter to low pass protototype filter. Only one different is the subscript of the coefficients in the matrix [Ai] and [Bi] are written in descending order.