Properties and methods of calculating generalised arithmetic and adding transforms

Abstract

The methods of generation of forward and inverse transformation kernels for generalised arithmetic and adding transforms are presented. Different methods of generation of transformation matrices or spectra in arbitrary polarities from a known transformation matrix or spectrum in some polarity have been developed. Using an optimised dyadic convolution process on the original data vector, a new method with a reduced number of operations is shown to calculate the generalised arithmetic and adding spectra. The properties and efficient ways of generation of permutation matrices used in the generation of arithmetic and adding transforms have been investigated. Based on the representation of transform matrices in the form of layered Kronecker matrices, a unified approach to the fast algorithms in terms of strand matrices has been developed. The computational complexities in the evaluation of arithmetic and adding spectral coefficients of an arbitrary order z and orders up to some z have been given.