Periodic shift-invariant multiresolution analysis

The Shannon multiresolution analysis and two methods of its periodization are considered. The class of all shift-invariant multiperiodic analyses with complex-valued scaling functions is described. All weakly shift-invariant periodic analogues of the Shannon scaling function are found. Among them, two shift-invariant periodic analogues of the sine function are revealed. A wavelet packet transform that generalizes the discrete Fourier transformation is found. An application of the shift-invariant periodic wavelet packet bases for time-frequency analysis of speech signals is discussed and illustrated