Butterfly orthogonal structure for fast transforms, filter banks and wavelets

Spectral analysis/synthesis ideas that are common for orthogonal transforms, multichannel and multirate filtering, and wavelet transforms are discussed and generalized. Some recently developed unconventional applications of the butterfly orthogonal decomposition technique are reviewed and its usefulness in developing efficient multiresolution digital signal processing systems is discussed. A generalized multirate filtering structure is developed that is based on fast algorithms of orthogonal transforms and their orthogonal subtransforms. In particular the structural subband decomposition of a discrete signal in sequency and frequency spectral domains is given. A generalized butterfly tree structure with all-pass branches and arbitrary weighting constants as well as its multilevel filter application is discussed. Wavelet filter bank realizations appear as a subset of presented structures