Generic functions and Hartley transform in polybasic notations

An ingenious method for synthesizing new discrete parametric basis functions is proposed. It aims at solving problems of spectral processing of digital signals in real time information management systems, having different applications. The method uses the Hartley generalization of polybasic notations with free radix of a number system. The analytical description of the obtained basic functions is given. Their main characteristics are analyzed. Methods of forming various orthogonal systems and transforms based on them are studied. The authors propose some techniques for additional extension of the collection of new bases, which apply various approaches to reorder the basis functions, including both the classical methods of the inverted and Grey codes and the new ones with the help of some versions of the Chinese remainder theorem for indexes. The possibility of formulating the main spectral analysis theorems in terms of these functions is shown. The theorems could be applied in digital processing, in both theory and practice. Their validity is proved. The obtained results represent the basis of both the representation theory and the transform theory in new Hartley polybasic notations.

References

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