Closed-Form Design of Generalized Maxflat$R$-Regular FIR$M$th-Band Filters Using Waveform Moments

$M$th-band filters have found numerous applications in multirate signal processing systems, filter banks, and wavelets. In this paper, the design problem of generalized maxflat$R$-regular finite impulse response (FIR)$M$th-band filters with a specified integer group delay at$ omega =0 $is considered, and the closed-form expression for its impulse response is presented. The filter coefficients are directly derived by solving a linear system of Vandermonde equations that are obtained from the regularity condition of the maxflat$R$-regular FIR$M$th-band filters via the blockwise waveform moments. Differing from the conventional FIR$M$th-band filters with exactly linear phase responses, the generalized FIR$M$th-band filters proposed in this paper have an arbitrarily specified integer group delay at$ omega =0 $. Moreover, a new efficient implementation of the generalized maxflat$R$-regular FIR$M$th-band filters is proposed by making use of the relationship between the filter coefficients in the closed-form solution. Finally, several design examples are presented to demonstrate the effectiveness of the proposed FIR$M$th-band filters.