Theory of paraunitary filter banks over fields of characteristic two

Motivated by our wavelet framework for error-control coding, we proceed to develop an important family of wavelet transforms over finite fields. Paraunitary (PU) filter banks that are realizations of orthogonal wavelets by multirate filters are an important subclass of perfect reconstruction (PR) filter banks. A parameterization of PU filter banks that covers all possible PU systems is very desirable in error-control coding because it provides a framework for optimizing the free parameters to maximize coding performance. This paper undertakes the problem of classifying all PU matrices with entries from a polynomial ring, where the coefficients of the polynomials are taken from finite fields. It constructs Householder transformations that are used as elementary operations for the realization of all unitary matrices. Then, it introduces elementary PU building blocks and a factorization technique that is specialized to obtain a complete realization for all PU filter banks over fields of characteristic two. This is proved for the 2 × 2 case, and conjectured for the M × M case, where M ⩾ 3. Using these elementary building blocks, we can construct all PU filter banks over fields of characteristic two. These filter banks can be used to implement transforms which, in turn, provide a powerful new perspective on the problems of constructing and decoding arbitrary-rate error-correcting codes