A Fast Discrete Periodic Wavelet Transform

The discrete wavelet transform (DWT) is extended to functions on the discrete circle to create a fast and complete discrete periodic wavelet transform (DPWT) for bounded periodic sequences. This extension also solves the problem of non-invertibility that arises in the application of the DWT to finite-dimensional sequences and provides the proper theoretical setting for 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∞ may be incorporated through the DPWT to create an orthonormal basis of discrete periodic wavelets. By exploiting transform symmetry and periodicity, easily implementable, fast, and recursive synthesis and analysis algorithms are derived. Matlab functions for DPWT experimentation are included. ∗This work was supported in part by NASA under grant NAG 2-243 and by NSF under grant IRI 90-14490.