On biorthogonal nonuniform filter banks and tree structures

This paper concerns biorthogonal nonuniform filter banks. It is shown that a tree structured filter bank is biorthogonal if it is equivalent to a tree structured filter bank whose matching constituent levels on the analysis and synthesis sides are themselves biorthogonal pairs. We then show that a stronger statement can be made about dyadic filter banks in general: That a dyadic filter bank is biorthogonal if both the analysis and synthesis banks can be decomposed into dyadic trees. We further show that these decompositions are stability and FIR preserving. These results, derived for filter banks having filters with rational transfer functions, thus extend some of the earlier comparable results for orthonormal filter banks.