Perfectly invertible fast and complete wavelet transform for finite-length sequences: the discrete periodic wavelet transform

摘要

The discrete wavelet transform (DWT) is adapted to functions on the discrete circle to create a discrete periodic wavelet transform (DPWT) for bounded periodic sequences. This extension also offers a solution to the problem of non-invertibility that arises in the application of the DWT to finite length sequences and provides the proper theoretical setting for the completion of some previous incomplete solutions to the invertibility problem. It is proven that the same filter coefficients used with the DWT to create orthonormal wavelets on compact support in l$+$INF$/ (Z) may be incorporated through the DPWT to create an orthonormal basis of discrete periodic wavelets. By exploiting transform symmetry and periodicity we arrive at easily implementable and fast synthesis and analysis algorithms.