Frequency Transformations in Filter Design

The problem of determining the possible functions
\omega = \varphi(y)
, that will transform the amplitude characteristic
A (\omega)
of a given network to the amplitude characteristic
A[\varphi(y)]
of another network, is considered. It is shown that the general form of these transformations is given by
\omega = \sqrt{F(y^2)}
, where
F(y^2)
is a positive rational function of
y^2
. The network function whose amplitude characteristic is
A[\varphi(y)]
is determined from the original network and the function
F(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.