Fractional nonuniform multiresolution analysis in L2(ℝ)

Abstract

To provide a significantly richer representation of non‐stationary signals appearing in various disciplines of science and engineering, we introduce a novel fractional nonuniform multiresolution analysis (FrNUMRA) on the spectrum Λ given by normalΛ={}0,rN+2ℤ, where N≧1 is an integer and r is an odd integer with 1≦r≦2N−1, such that r and N are relatively prime. The necessary and sufficient condition for the existence of nonuniform wavelets of fractional order is derived and an algorithm is also presented for the construction of fractional NUMRA starting from a fractional low‐pass filter h0α with appropriate conditions. Besides, we provide a complete characterization for the biorthogonality of the translates of the scaling functions of two fractional nonuniform multiresolution analyses and the associated fractional biorthogonal wavelet families.

Filtering of Audio Signals Using Discrete Wavelet Transforms

“…Readers may consult the works of [10][11][12][13][14] for further details on wavelet decomposition and related topics. Readers may also consult the most recent works of [15][16][17][18] for further details on the direction of present work.…”