Constraint Based Design of Two-Channel Paraunitary Filter Banks of a Given Length Over ${\rm GF}(2^{r})$

Abstract

Over the real field all degree-J paraunitary (PU) multirate systems can be described by the multiplication of J degree-1 lattice blocks and a unitary matrix. Over the finite field GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sup> ) this degree-1 factorization is not complete, i.e., it only describes a subset of all possible PU systems. In the two-channel case degree-2tau blocks are also required to completely describe all PU systems over GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sup> ). Therefore, different factorizations can be considered. Each factorization generates a subset of PU systems. It is interesting to consider if these different factorizations have distinct properties. In this correspondence, we specifically consider constraining the length of the filter bank to be equal to N+1. This is required in certain error control coding applications. We contrast this factorization based method with an existing trial and error approach employing the Berlekamp factoring algorithm. A key advantage of the proposed method is the elimination of redundant polyphase factorizations. Further simplifications over GF(2) identified by this method are also discussed