Time-frequency localisation of Shannon wavelet packets

It is well known that the 2pi minimally supported frequency scaling function phi(alpha)(x) satisfying (φ) over cap (alpha)(w) = X-[-alpha,X-2pi-alpha) (w), 0 < alpha < 2pi is not time localised. The Shannon wavelet packets phi(n)(alpha) (x) are generated from phi(alpha) (x) via an arbitrary pair of low-pass and high-pass filters (h, g), which is associated with an orthogonal multiresolution analysis. The authors prove that phi(n)(alpha) (x) is time localised if alpha and n satisfy certain conditions. They also show that the decay properties of phi(n)(alpha)(x) n depend on the multiplicity of the zero w = pi of the symbol m(0)(w)) of the low-pass filter h.Engineering, Electrical & ElectronicSCI(E)EI0ARTICLE6365-36915