Lapped unimodular transforms: lifting factorization and structural regularity

In this paper, the lifting factorization and structural regularity of the lapped unimodular transforms (LUTs) are studied. The proposed M-channel lifting factorization is complete, is minimal in the McMillan sense, and has diagonal entries of unity. In addition to allowing for integer-to-integer mapping and guaranteeing perfect reconstruction even under finite precision, the proposed lifting factorization structurally ensures unimodularity. For regular LUT design, structural conditions that impose (1,1)-, (1,2)- and (2,1)-regularity onto the filter banks (FBs) are presented. Consequently, the optimal filter coefficients can be obtained through unconstrained optimizations. A special lifting-based lattice structure is used for parameterizing nonsingular matrices, which not only helps impose regularity but also has rational-coefficient unimodular FBs as a by-product. The regular LUTs can be transformed to the lifting domain with the proposed factorization for faster and multiplierless implementations. The lifting factorization and the regularity conditions are derived for two different (Type-I and Type-II) factorizations of the first-order unimodular FBs. Design examples are presented to confirm the proposed theory.