Frequency Transformations in Filter Design

The problem of determining the possible functionsomega = varphi(y), that will transform the amplitude characteristicA (omega)of a given network to the amplitude characteristicA[varphi(y)]of another network, is considered. It is shown that the general form of these transformations is given byomega = sqrt{F(y^2)}, whereF(y^2)is a positive rational function ofy^2. The network function whose amplitude characteristic isA[varphi(y)]is determined from the original network and the functionF(y^2). The concept of frequency transformations is used in filter design and network analysis; the Butterworth and Tchebycheff filters result as special forms of such transformations.