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Frequency Transformations in Filter Design

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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.

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Circuit Theory, IRE Transactions on  (Volume:3 ,  Issue: 2 )