Complex frequency transformations for DSP applications

A simple idea of the frequency transformation where each delay of the original real lowpass filter transfer function is replaced by the same allpass filter allows a lot of flexibility in manipulating the original filter to fit the required specification. The simple cases of real first-order transformations and a very few second-order ones have been suggested by Constantinides in the early 1970s. However, the existing real mappings do not give enough flexibility for the transformations to be generally accepted as an add-on to other design tools. This paper offers the complex extension to the frequency transformation problem, attempting to formulate general equations describing most possible design situations. Example filter mappings presented herein illustrate the concepts for the cases of first and second order complex mappings, most suitable for industrial custom applications.

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