Synthesis of Wideband Linear-Phase FIR Filters with a Piecewise-Polynomial-Sinusoidal Impulse Response

A method is presented to synthesize wideband linear-phase finite-impulse-response (FIR) filters with a piecewise-polynomial-sinusoidal impulse response. The method is based on merging the earlier synthesis scheme proposed by the authors to design piecewise-polynomial filters with the method proposed by Chu and Burrus. The method uses an arbitrary number of separately generated center coefficients instead of only one or none as used in the method by Chu–Burrus. The desired impulse response is created by using a parallel connection of several filter branches and by adding an arbitrary number of center coefficients to form it. This method is especially effective for designing Hilbert transformers by using Type 4 linear-phase FIR filters, where only real-valued multipliers are needed in the implementation. The arithmetic complexity is proportional to the number of branches, the common polynomial order for each branch, and the number of separate center coefficients. For other linear-phase FIR filter types the arithmetic complexity depends additionally on the number of complex multipliers. Examples are given to illustrate the benefits of this method compared to the frequency-response masking (FRM) technique with regard to reducing the number of coefficients as well as arithmetic complexity.