G-LETS: a new signal processing algorithm

Abstract

This work proposes a new algorithm for signal processing in time and frequency domains using group theory called G-lets. In this algorithm, discrete signal is taken to be a member of a nite transformation group. The algorithm is illustrated using dihedral groups. The group may be represented through matrices or functions by following representation theory. The dimension of the signal space is given by the number of conjugacy classes of the group, which is also equal to the number of irreducible representations and the number of characters. G-let matrices are constructed as sparse block diagonal matrices with the two-dimensional irreducible representations forming the diagonal blocks. The signal is projected onto the G-let matrices to obtain the G-let coe cients. It is noticed that at the end of each irreducible representation block, abrupt changes in G-let coe cients arise, resulting in oscillations in the G-lets. The width of the oscillations is inversely proportional to the local amplitude gradient, and their height is directly proportional to the local amplitude gradient. Since the number of conjugacy classes is xed for a nite-dimensional group, there are no approximations involved in processing the discrete signal, and the signal can be reconstructed perfectly. This method does not explicitly use characters, nor does it choose a predetermined pattern and search for its variations in the signal, nor does it convolve the signal with a kernel comprising irreducible representations. By avoiding such approaches that might result in loss of signal information, this algorithm results in a set of error-free orthonormal bases of the signal space in time domain.