Butterfly filter banks

Considers the properties of a class of delay tree-structured and sub-tree-structured filter banks, which can be formulated as Hadamard-Haar transforms operating on all-pass functions. The structures of such filter banks may be configured to provide an arbitrary number of weighted channels with desired combination of bandwidths. Analysis and synthesis filter banks which are related through a transposition and scaling operations, such that the cascade of analysis and synthesis filter banks achieves an all-pass function, are obtained. One attractive feature offered is that all-pass complementary and power complementary filter banks may be configured changing only scaled butterfly operations. The tree-structured approach to sub-band filtering results in sub-trees of the binary tree structure of analysis and synthesis filter banks. The main advantages of butterfly filter banks (BFB) are structural simplicity and a number of butterfly operations proportional to Nlog₂N-N/2 or less up to 2(N-1)