Matrix decomposition and butterfly diagrams for mutual relations between Hadamard-Haar and arithmetic spectra

The mutual relationships between Hadamard-Haar and Arithmetic transforms and their corresponding spectra in the form of matrix decomposition as layered vertical and horizontal Kronecker matrices are discussed here together with their proofs, fast algorithms, and computational costs. The new relations apply to an arbitrary dimension of the transform matrices and allow performing direct conversions between Arithmetic and Hadamard-Haar functions and their corresponding spectra. In addition, analysis of butterfly diagrams for these new relations is also introduced and it is shown that they are more efficient than the matrix decomposition method.