Fourier Analysis of Sequences over a Composition Algebra of the Real Number Field

Takao MAEDA  Takafumi HAYASHI  

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E96-A   No.12   pp.2452-2456
Publication Date: 2013/12/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.E96.A.2452
Print ISSN: 0916-8508
Type of Manuscript: Special Section LETTER (Special Section on Information Theory and Its Applications)
Category: Sequence
Keyword: 
Fourier analysis,  convolution,  composition algebra,  correlation function,  perfect sequence,  

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Summary: 
To analyze the structure of a set of perfect sequences over a composition algebra of the real number field, transforms of a set of sequences similar to the discrete Fourier transform (DFT) are introduced. The discrete cosine transform, discrete sine transform, and generalized discrete Fourier transform (GDFT) of the sequences are defined and the fundamental properties of these transforms are proved. We show that GDFT is bijective and that there exists a relationship between these transforms and a convolution of sequences. Applying these properties to the set of perfect sequences, a parameterization theorem of such sequences is obtained.