Closed-form design of maximally flat IIR half-band filters

Half-band (HB) filters are of great importance and are often used in multirate digital signal processing systems, filter banks and wavelets. In this paper, a new closed-form expression for the transfer function of the maximally flat (MF) infinite-impulse response (IIR) HB filters is presented. The filter coefficients are directly obtained by solving a linear system of Vandermonde equations that are derived from the maximal flatness conditions. The proposed IIR half-band filters are more general than the existing half-band filters, because they include the conventional finite-impulse response (FIR) half-band filters with exactly linear phase, the generalized FIR half-band filters with approximately linear phase and the all-pass-based IIR half-band filters, as special cases. Furthermore, the causal stable IIR HB filters and the IIR HB filters with exactly linear phase can be realized also. Finally, some design examples are presented to demonstrate the effectiveness of the proposed IIR HB filters.