Algebraic theory of optimal filter banks

We introduce an optimality theory for finite impulse response (FIR) filterbanks using a general algebraic point of view. We consider an admissible set of FIR filterbanks and use scalability as the main notion based on which performance of the elements in are compared. We show that quantification of scalability leads naturally to a partial ordering on the set . An optimal solution is, therefore, represented by the greatest element in . It turns out that a greatest element does not necessarily exist in . Hence, one has to settle with one of the maximal elements that exist in . We provide a systematic way of finding a maximal element by embedding the partial ordering at hand in a total ordering. This is done by using a special class of order-preserving functions known as Schur-convex. There is, however, a price to pay for achieving a total ordering: There are infinitely many possible choices for Schur-convex functions, and the optimal solution specified in depends on this (subjective) choice. An interesting aspect of the presented algebraic theory is that the connection between several concepts, namely, principal component filterbanks (PCFBs), filterbanks with maximum coding gain, and filterbanks with good scalability, is clearly revealed. We show that these are simply associated with different extremal elements of the partial ordering induced on by scalability.