Radomir S. Stankovic and Jaakko T. Astola

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) func-tions, the notion of bandwidth relates to the concept of essential variables. Sampling conditions convert into requirements for periodicity and regular-ity in the truth-vectors of MV functions. Due to that, by starting from the sampling theorem, we derive generalized Shannon decomposition rules for MVfunctions 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 vari-ables, where the cardinality of the subsets is determined by the bandwidth selected.

As there are no convergence problems, the sampling theorem for dis-crete 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.