Paraunitary Filter Banks Over Finite Fields - Signal Processing, IEEE Transactions on

In real and complex fields, unitary and paraunitary (PU) matrices have found many applications in signal processing. There has recently been interest in extending these ideas to the case of finite fields. In this paper, we will study the theory of PU filter banks (FB’s) in GF(y) with y prime. Various properties of unitary and PU matrices in finite fields will be studied. In particular, a number of factorization theorems will be given. We will show that i) all unitary matrices in GF(y) are factorizable in terms of Householder-like matrices and permutation matrices, and ii) the class of first-order PU matrices (the lapped orthogonal transform in finite fields) can always be expressed as a product of degree-one or degree-two building blocks. If Q > 2, we do not need degree-two building blocks. While many properties of PU matrices in finite fields are similar to those of PU matrices in complex field, there are a number of differences. For example, unlike the conventional PU systems, in finite fields, there are PU systems that are unfuctorizable in terms of smaller building blocks. In fact, in the special case of 2 x 2 systems, all PU matrices that are factorizable in terms of degree-one building blocks are diagonal matrices. We will derive results for both the cases of G F ( 2 ) and G F ( y ) with q > 2 . Even though they share some similarities, there are many differences between these two cases.