HALFBAND FILTERS AND HILBERT TRANSFORMERS

This paper presents in summarizing form a description of halfband filt ers and the related symmetrical Hilbert transformers. It starts with t he two complementary relations by which halfband filters are defined a nd the consequences for their impulse responses. The idealized version s of the frequency responses of halfband lowpasses and Hilbert transfo rmers are introduced, and the related tolerance schemes that realized systems must satisfy are described. Using their frequency responses, t he transformation of one filter type into the other is presented in ge neral form. The design of finite impulse response (FIR)-halfband filte rs and their relation to corresponding Hilbert transformers are recall ed, using maximally hat and Chebyshev approximations as examples. It i s shown that the relation between both types of systems can be used fo r the infinite impulse response (IIR) case as well. The design of IIR- halfband filters is presented for systems with approximately linear ph ase and for those with minimum phase again for maximally flat and Cheb yshev approximations. The design methods are partly new. The general p rocedure for the transformation into Hilbert transformers yields nonca usal solutions, one of which is already known from the literature. By modifying this operation, phase-splitting systems are obtained, one of them related to corresponding continuous ones, discussed in papers pu blished around 1950. Another system with approximately linear phase co rresponds to a paper presented in 1987. Finally, the coupled form of t hese phase splitting allpasses is found to be a Hilbert transformer wi th precise phase difference, but with deviations of the magnitudes of the frequency responses.