Time–frequency localisation of Shannon wavelet packets

Abstract

It is well known that the 2π minimally supported frequency scaling function φα(x) satisfying φ̂α(ω)=χ[-α,2π-α)(ω), 0<α<2π, is not time localised. The Shannon wavelet packets φnα(x) are generated from φα(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 φnα(x) is time localised if α and n satisfy certain conditions. They also show that the decay properties of φnα(x) depend on the multiplicity of the zero ω=π of the symbol m0(ω) of the low-pass filter h.