Properties and methods of calculating generalised arithmetic andadding transforms

摘要

The methods of generation of forward and inverse transformationnkernels for generalised arithmetic and adding transforms are presented.nDifferent methods of generation of transformation matrices or spectra innarbitrary polarities from a known transformation matrix or spectrum innsome polarity have been developed. Using an optimised dyadic convolutionnprocess on the original data vector, a new method with a reduced numbernof operations is shown to calculate the generalised arithmetic andnadding spectra. The properties and efficient ways of generation ofnpermutation matrices used in the generation of arithmetic and addingntransforms have been investigated. Based on the representation ofntransform matrices in the form of layered Kronecker matrices, a unifiednapproach to the fast algorithms in terms of strand matrices has beenndeveloped. The computational complexities in the evaluation ofnarithmetic and adding spectral coefficients of an arbitrary order z andnorders up to some z have been given