Synthesis of narrowband differentiators with a piecewise-polynomial impulse response with parallel-branch structures

A synthesis method is proposed for linear-phase narrowband FIR differentiators. This method is based on two computationally efficient recursive structures to synthesize linear-phase narrowband FIR filters with a piecewise-polynomial impulse response proposed by Saramaki and Mitra. The efficiency of these structures is based on implementing the overall transfer function as a parallel connection of a few branches of the form G_l(z^L)F_l(z), where each F_l(z) requires no real multipliers. Such F_l(z)s by thorough derivation for both structures were proposed by Lehto, Saramaki and Vainio. This paper proposes the synthesis method of linear-phase narrowband FIR filters with an antisymmetrical piecewise-polynomial impulse response for differentiators. The arithmetic complexity of the proposed method is based on the number of branches and the polynomial degree. An example is included, illustrating the benefits of the proposed filters, in terms of a reduced number of unknowns in the optimization and implementation.