Authors: Stanković, Radomir 
Astola, Jaakko
Title: Interpretations of the sampling theorem in multiple-valued logic
Journal: Journal of Multiple-Valued Logic and Soft Computing
Volume: 13
Issue: 4-6
First page: 467
Last page: 486
Issue Date: 29-Nov-2007
Rank: M23
ISSN: 1542-3980
Signals described by functions of continuous and discrete variables can be uniformly studied in a group theoretic framework. This paper presents a consideration which shows that in the case of multiple-valued (MV) functions, the notion of bandwidth relates to the concept of essential variables. Sampling conditions convert into requirements for periodicity and regularity in the truth-vectors of MV functions. Due to that, by starting from the sampling theorem, we derive generalized Shannon decomposition rules for MV functions that include the classical Shannon decomposition rule in binary-valued logic as a particular case. The sampling theorem provides a regular way for the decomposition of a MV function into subfunctions of smaller numbers of variables. In circuit synthesis, this allows decomposition of a network to realize a function into subnetworks realizing subfunctions depending on subsets of variables, where the cardinality of the subsets is determined by the bandwidth selected. As there are no convergence problems, the sampling theorem for discrete functions can be formulated in terms of a class of Fourier-like transforms satisfying certain properties. Different decompositions of a given function can then be determined by selecting various Fourier-like transforms.
Keywords: Bandwidth | Galois field transform | Multiple-valued logic | Reed-muller transform | Sampling theorem | Shannon expansion rule
Publisher: Old City Publishing

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