Hilbert transformers with a piecewise-polynomial-sinusoidal impulse response

Hilbert transformers are one of the very important special classes of finite impulse response (FIR) filters used in signal processing applications. A method is presented to synthesize Hilbert transformers by using wideband linear-phase FIR filters with a piecewise-polynomial-sinusoidal impulse response. The proposed method is based on merging the earlier synthesis scheme proposed by the authors with the method proposed by Chu and Burrus and by modifying it by using an arbitrary number of separately generated center coefficients instead of none 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. The arithmetic complexity of these filters is proportional to the number of branches, the common polynomial order for each branch and the number of separate center coefficients. The filter coefficients are optimized by using linear programming. An example shows the benefits of these filters compared to the frequency-response masking approach (FRM) with regard to reducing the number of coefficients as well as arithmetic complexity.