From Fourier Expansions to Arithmetic-Haar Expressions on Quaternion Groups |
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Authors: | Radomir S Stankovi? Claudio Moraga Jaakko Astola |
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Affiliation: | (1) Department of Computer Science, Faculty of Electronics, 18000 Nis✓, Yugoslavia, YU;(2) Department of Computer Science, University of Dortmund, 44221 Dortmund, Germany, DE;(3) Int. Center for Signal Processing, Tampere University of Technology, Tampere, Finland, FI |
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Abstract: | Arithmetic expressions for switching functions are introduced through the replacement of Boolean operations with arithmetic
equivalents. In this setting, they can be regarded as the integer counterpart of Reed-Muller expressions for switching functions.
However, arithmetic expressions can be interpreted as series expansions in the space of complex valued functions on finite
dyadic groups in terms of a particular set of basic functions. In this case, arithmetic expressions can be derived from the
Walsh series expansions, which are the Fourier expansions on finite dyadic groups.
In this paper, we extend the arithmetic expressions to non-Abelian groups by the example of quaternion groups. Similar to
the case of finite dyadic groups, the arithmetic expressions on quaternion groups are derived from the Fourier expansions.
Attempts are done to get the related transform matrices with a structure similar to that of the Haar transform matrices, which
ensures efficiency of computation of arithmetic coefficients.
Received: October 5, 1999; revised version: June 14, 2000 |
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Keywords: | : Fourier transform Quaternion groups Arithmetic expressions Haar expressions |
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