Halfband filters and Hilbert transformers

Abstract

This paper presents in summarizing form a description of halfband filters and the related symmetrical Hilbert transformers. It starts with the two complemetary relations by which halfband filters are defined and the consequences for their impulse responses. The idealized versions of the frequency responses of halfband lowpasses and Hilbert transformers are introduced, and the related tolerance schemes that realized systems must satisfy are described. Using their frequency responses, the transformation of one filter type into the other is presented in general form. The design of finite impulse response (FIR)-halfband filters and their relation to corresponding Hilbert transformers are recalled, using maximally flat and Chebyshev approximations as examples. It is shown that the relation between both types of systems can be used for the infinite impulse response (IIR) case as well. The design of IIR-halfband filters is presented for systems with approximately linear phase and for those with minimum phase again for maximally flat and Chebyshev approximations. The design methods are partly new. The general procedure for the transformation into Hilbert transformers yields noncausal 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 published around 1950. Another system with approximately linear phase corresponds to a paper presented in 1987. Finally, the coupled form of these phase splitting allpasses is found to be a Hilbert transformer with precise phase difference, but with deviations of the magnitudes of the frequency responses.