Algebraic theory of optimal filter banks

We approach the problem of characterizing an optimal FIR filter bank from an algebraic point of view. We introduce the concept of majorization ordering to compare the performance of various filter banks in an admissible set L. Using the properties of this ordering, we show that a principal component filter bank is associated with the greatest element in L. A greatest element does not necessarily exist in L hence one has to deal with the closely related notion of a maximal element. We show by construction that a maximal element always exist in L. An interesting result of the presented algebraic theory is that the connection between principal component filter banks and filter banks with maximum coding gain is clearly revealed. In fact, we show that coding gain is a Schur (1973) convex function preserving the order of majorization