IIR discrete-time Hilbert transformers

A procedure for obtaining IIR Hilbert transformers from generalized halfband filters [5] is described. The solution is shown to be suited for specifications where the "care"-band for the Hilbert transformers is centered at 0.5π. Even when this condition is not satisfied, the proposed Hilbert transformers obtained by overdesign may often be preferable for implementation to those obtained from a general design procedure [6]. Analytic solutions for noncausal Hilbert transformers and 90° phase shifters can be derived from special classes of classical low-pass filters. The design procedure described by Gold et al. [6] is shown to be equivalent to applying a transformation on Hilbert transformers derived from elliptic low-pass filters. Numerical approximation can be used for designing IIR Hilbert transformers with a right-sided impulse response. This gives us a single causal IIR filter (instead of a pair of 90° phase shifters) whose output is the approximate Hilbert transform of the delayed version of the input.