Extraction of Walsh Harmonics by Linear Combinations of Dyadic Shifts

We solve two problems of extracting any term from a signal in the form of a finite sum of the Fourier series with respect to the discrete Walsh functions (in the first case) and the Walsh functions (in the second case) by a linear combination of group shifts of the original signal. We propose a vector version of the discrete time- and frequencythinning wavelet Haar bases, which widely used in encoding and decoding algorithms for the discrete Haar transform. Bibliography: 10 titles.