In the papers below the discrete wavelet transform (DWT) is extended to functions on the discrete circle to create a fast and complete discrete periodic wavelet transform (DPWT). It is proven that the same filter coefficients used with the DWT may be used with the DPWT to create an orthonormal basis of discrete periodic wavelets. Unlike the DWT, the DPWT is perfectly invertible when applied to sequences of finite length.
The illustration shows examples of elements of a periodic wavelet basis for length 128 sequences. This basis set is based on Daubechies filters of length 4. The sequences have been normalized to have peak magnitudes of 1. The outer dotted circle corresponds to a sample value of 1 and the inner to a sample value of -1. The intermediate solid circle corresponds to 0.
This paper is a slightly abbreviated version of the Memorandum UCB/ERL M92-138 above.
This is a collection of Matlab routines designed to enable easy experimentation with the DPWT and its inverse. Some knowledge of the background contained in either of the above two publications is suggested.