Fractional Wavelet Transform Using an Unbalanced Lifting Structure

In this article, we introduce the concept of fractional wavelet transform. Using a two-channel unbalanced lifting structure it is possible to decompose a given discrete-time signal x[n] sampled with period T into two sub-signals x1[n] and x2[n] whose average sampling periods are pT and qT , respectively. Fractions p and q are rational numbers satisfying the condition: 1/p + 1/q = 1. The low-band sub-signal x1[n] comes from [0, π/p] band and the high-band wavelet signal x2[n] comes from (π/p, π] band of the original signal x[n]. Filters used in the lifting structure are designed using the Lagrange interpolation formula. It is straightforward to extend the proposed fractional wavelet transform to two or higher dimensions in a separable or non separable manner.