Discrete wavelet transform based on cyclic convolutions

A filterbank in which cyclic convolutions are used in place of linear convolutions will be referred to as a cyclic convolution filterbank (CCFB). A dyadic tree-structured CCFB can be used to perform a discrete wavelet transform suitable for coding based on symmetric extension methods. This paper derives two types of efficient implementation techniques for the tree-structured CCFB: one using complex arithmetic and one using only real arithmetic. In addition, the present paper analyzes in detail the perfect reconstruction (PR) condition for the two-channel CCFB and shows that this condition is much less restrictive than that of usual two-channel filterbanks. When each of the two-channel CCFBs constituting a tree-structured CCFB is designed to be PR, the whole system is PR. A quadrature mirror filter (QMF) CCFB having the PR property is demonstrated to be easily designed using a standard filter design subroutine. In contrast, designing a linear-phase PR FIR QMF bank that has good frequency response is not possible when filters are realized by linear convolutions.