On the existence of complete order-one lattice for linear phase perfect reconstruction filter banks

In this letter, we revisit the completeness of the lattice factorization for M-channel linear phase perfect reconstruction filter banks (LPPRFBs) with equal length L =KM and further investigate a more fundamental problem, i.e., the existence of complete lattice factorization by using LPPR propagating blocks of order-one causal FIR with anticausal FIR inverse (CAFACAFI) matrices. Reviewing the previous works, we point out the limitation of the existing LPPR propagating blocks and then show its consequence for incompleteness of the existing lattice factorizations. Furthermore, we show the nonexistence of any order-one LPPR propagating block by using CAFACAFI matrices. In addition, the completeness of lattice factorizations has been re-examined for generalized lapped orthogonal transforms (GenLOTs) and lapped biorthogonal transforms (LBTs) with linear phase based on the analysis developed in this letter.