M-channel recursive DFT perfect and near perfect reconstruction QMF banks

Perfect and Near Perfect Reconstruction (NPR) DFT QMF banks with general stable, causal IIR analysis and synthesis prototype filters are presented. The PR property is obtained by constraining type I polyphase components E_k(z) of the analysis filter to correspond to the exact inverses of type II polyphase components R_k(z) of the synthesis prototype filter. Stable and causal solutions corresponding to a stable prototype filter with minimum-phase polyphase elements are obtained through a search procedure involving an optimisation procedure with respect to filter parameters such as cut-off frequency or attenuation rates. The design algorithm is easy to implement and convergence was observed in numerous cases when the filter bank order and number of channels is moderate. Although resulting adjacent filters present a large frequency overlap, the resulting M-channel FB delay is optimal equal to M-1. Allowing some zeros of the polyphase filters outside the unit-circle could decrease the frequency overlap, and NPR DFT QMF banks are obtained by approximating each of the non-causal parts of R_k(z) with FIR filters. The approximation of the FIR coefficients is performed using a simple time-domain least square pseudo-inverse technique also described in this paper. The resulting filter bank will allow some aliasing error and its system delay will increase significantly. An analysis of quantisation effects as well as simulation examples are also included