# Atlantis Essay

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Points Addressed in this Lecture • Standard form of Boolean Expressions Lecture 5: Logic Simplication & Karnaugh Map Professor Peter Cheung Department of EEE, Imperial College London (Floyd 4.5-4.11) (Tocci 4.1-4.5) – Sum-of-Products (SOP), Product-of-Sums (POS) – Canonical form • Boolean simplification with Boolean Algebra • Boolean simplification using Karnaugh Maps • “Don’t cares” E1.2 Digital Electronics I 5.1 Cot 2007 E1.2 Digital Electronics I Cot 2007 Forms of Boolean Expressions Canonical Form • Sum-of-products form (SOP) – first the product (AND) terms are formed then these are summed (OR) – eg: ABC + DEF + GHI • Product-of-sum form (POS) – first the sum (OR) terms are formed then the products are taken (AND) – eg: (A+B+C) (D+E+F) (G+H+I) • Canonical form is not efficient but sometimes useful in analysis and design • In an expression in canonical form, every variable appears in every term f(A, B,C, D) = ABCD + ABCD + ABCD – note that the dot (meaning AND) is often omitted • It is possible to convert between these two forms using Boolean algebra (DeMorgan’s) E1.2 Digital Electronics I Cot 2007 E1.2 Digital Electronics I Cot 2007 – An SOP expression can be forced into canonical form by ANDing the incomplete terms with terms of the form (X + X ) where X is the name of the missing variable – eg: A Notation using Canonical Form • Previous example: • Construct the truth table for this function – use a 0 when the variable is complemented, 1 otherwise f ( A, B, C) = AB + BC f ( A, B , C ) = ABC + ABC + ABC R o w N u m ber 0 1 2 3 4 5 6 7 A 0 0 0 0 1 1 1 1 B 0 0 1 1 0 0 1 1 C 0 1 0 1 0 1 0 1 f 0 0 0 1 0 0 1 1 = AB(C + C) + ( A + A) BC = ABC + ABC + ABC + ABC = ABC + ABC + ABC – The product term in a canonical SOP expression is called a 'minterm' E1.2 Digital Electronics I Cot 2007 – f can