On orthonormal wavelets and paraunitary filter banks

Abstract

The known result that a binary-tree-structured filter bank with the same paraunitary polyphase matrix on all levels generates an orthonormal basis is generalized to binary trees having different paraunitary matrices on each level. A converse result that every orthonormal wavelet basis can be generated by a tree-structured filter bank having paraunitary polyphase matrices is then proved. The concept of orthonormal bases is extended to generalized (nonbinary) tree structures, and it is seen that a close relationship exists between orthonormality and paraunitariness. It is proved that a generalized tree structure with paraunitary polyphase matrices produces an orthonormal basis. Since not all phases can be generated by tree-structured filter banks, it is proved that if an orthonormal basis can be generated using a tree structure, it can be generated specifically by a paraunitary tree.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>