On orthonormal wavelets and paraunitary filter banks

Binary tree-structured filter banks have been employed in the past to generate wavelet bases. While the relation between paraunitary filter banks and orthonormal bases is known to some extent, there are some extensions which are either not known, or not published so far. In particular it is known that a binary tree-structured filter bank with the same paraunitary polyphase matrix on all levels generates an orthonormal basis. First, we generalize the result to binary trees having different paraunitary matrices on each level. Next, we prove a converse result: that every discrete-time orthonormal wavelet basis can be generated by a tree-structured filter bank having paraunitary polyphase matrices. We then extend the concept of orthonormal bases to generalized (i.e., nonbinary) tree structures, and see that a close relationship exists between orthonormality and paraunitariness in this case too. We prove that a generalized tree structure with paraunitary polyphase matrices produces an orthonormal basis. Since not all bases can be generated by tree-structured filter banks, we prove that if an orthonormal basis can be generated using a tree structure, it can be generated specifically by a paraunitary tree.