Design of FIR Hilbert transformers and differentiators in the complex domain

This paper presents a method for the design of FIR Hilbert transformers and differentiators in the complex domain. The method can be used to obtain conjugate-symmetric designs with smaller group delay compared to linear-phase designs. Non-conjugate symmetric Hilbert transformers are also designed. This paper is an extension of our previous work, which presented the algorithm for the design of standard frequency selective filters. The minimax criterion is used and the Chebychev approximation is posed as a linear optimization problem. The primal problem is converted to its dual and is solved using an efficient quadratically convergent algorithm developed by Tang (1988). When a constant group delay is specified, the filter designs have almost linear phase in the passbands. When the specified group delay is half the filter length, the algorithm results in exactly linear-phase designs