Utilization of Eight-Variable Karnaugh Maps in the Digital Design of n-bit Comparators

Abstract

An -bit comparator is a celebrated combinational circuit that compares two -bit inputs  and  and produces three orthonormal outputs: G (indicating that  is strictly greater than ), E (indicating that  and  are equal or equivalent), and L (indicating that  is strictly less than ). The symbols ‘G’, ‘E’, and ‘L’ are deliberately chosen to convey the notions of ‘Greater than,’ ‘Equal to,’ and ‘Less than,’ respectively. This paper analyzes an -bit comparator in the general case of arbitrary  and visualizes the analysis for  on a regular and modular version of the 8-variable Karnaugh-map. The cases  3, 2 and 1 appear as special cases on 6-variable, 4-variable, and 2-variable submaps of the original map. The analysis is a tutorial exposition of many important concepts in switching theory including those of implicants, prime implicants, essential prime implicants, irredundant disjunctive forms, minimal sums, the complete sum and disjoint sums of products (or probability-ready expressions).