Combinational Circuits

• Christian Posthoff
• Bernd Steinbach
Chapter

Abstract

The analysis, synthesis, and test of combinational circuits is a major field of applications of logic functions and equations. We introduce models which can be used to describe either the behavior or the structure of several realizations of combinational circuits. Based on these models we provide methods for the fundamental analysis task and the calculation of the behavior of a given circuit structure. Additionally, several analysis tasks will be solved. There are two main approaches for the synthesis of combinational circuits: covering and decomposition methods. Covering methods are widely used for the synthesis of several types of two-level circuit structures. Due to restrictions of the technology covering methods are suitable for circuits of a small number of variables. We give an overview of these methods and demonstrate their application by means of synthesis examples. Decomposition methods facilitate the synthesis of multilevel circuits for larger numbers of variables, and their theory is more complicated. We give an overview of different decomposition methods and explain the newest results with regard to strong, weak, and vectorial bi-decompositions for both single logic functions and lattices of logic functions. We explain new possibilities of bi-decompositions utilizing the extensions of the Boolean Differential Calculus, provided in this book. The test of combinational circuits is needed to discard circuits which do not show the expected behavior. The basic method to calculate the needed test pattern uses the network model of the sensible path. Due to some drawbacks of this model we suggest the network model of the sensible point for internal signals and internal branches. Using this new model test patterns for all non-redundant gate connection in the circuit can be computed. The provided synthesis by mean of bi-decompositions leads to completely testable circuits and allows the generation of the test patterns in parallel to the synthesis of the circuit.

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Authors and Affiliations

• Christian Posthoff
• 1
• Bernd Steinbach
• 2
1. 1.Computing and Information TechnologyUniversity of the West Indies (retired)ChemnitzGermany
2. 2.Computer ScienceTU Bergakademie Freiberg (retired)ChemnitzGermany