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ECE2030 Introduction to Computer Engineering Lecture 6: Canonical (Standard) Forms

ECE2030 Introduction to Computer Engineering Lecture 6: Canonical (Standard) Forms. Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering Georgia Tech. Boolean Variables. A multi-dimensional space spanned by a set of n Boolean variables is denoted by B n

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ECE2030 Introduction to Computer Engineering Lecture 6: Canonical (Standard) Forms

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  1. ECE2030 Introduction to Computer EngineeringLecture 6: Canonical (Standard) Forms Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering Georgia Tech

  2. Boolean Variables • A multi-dimensional space spanned by a set of n Boolean variables is denoted by Bn • A literal is an instance (e.g. A) of a variable or its complement (Ā)

  3. SOP Form • A product of literals is called a product term or a cube (e.g. Ā·B·C in B3, or B·C in B3) • Sum-Of-Product (SOP) Form: OR of product terms, e.g. ĀB+AC • A minterm is a product term in which every literal (or variable) appears in Bn • ĀBC is a minterm in B3 but not in B4. ABCD is a minterm in B4. • A canonical (or standard) SOP function: • a sum of minterms corresponding to the input combination of the truth table for which the function produces a “1” output.

  4. Minterms in B3

  5. Canonical (Standard) SOP Function

  6. POS form (dual of SOP form) • A sum of literals is called a sum term (e.g. Ā+B+C in B3, or (B+C) in B3) • Product-Of-Sum (POS) Form: AND of sum terms, e.g. (Ā+B)(A+C) • A maxterm is a sum term in which every literal (or variable) appears in Bn • (Ā+B+C) is a maxterm in B3 but not in B4. A+B+C+D is a maxterm in B4. • A canonical (or standard) POS function: • a product of maxterms corresponding to the input combination of the truth table for which the function produces a “0” output.

  7. Maxterms in B3

  8. Canonical (Standard) POS Function

  9. Convert a Boolean to Canonical SOP • Expand the Boolean eqn into a SOP • Take each product term w/ a missing literal A, “AND” () it with (A+Ā)

  10. Convert a Boolean to Canonical SOP 0 1 Minterms listed as 1’s 3 7

  11. Convert a Boolean to Canonical SOP

  12. Convert a Boolean to Canonical POS • Expand Boolean eqn into a POS • Use distributive property • Take each sum term w/ a missing variable A and OR it with A·Ā

  13. Convert a Boolean to Canonical POS

  14. Convert a Boolean to Canonical POS Maxterms listed as 0’s 2 4 5 6

  15. Convert a Boolean to Canonical SOP 0 1 Minterms listed as 1’s 3 7

  16. Convert a Boolean to Canonical POS

  17. Convert a Boolean to Canonical SOP

  18. Interchange Canonical SOP and POS • For the same Boolean eqn • Canonical SOP form is complementary to its canonical POS form • Use missing terms to interchange  and  • Examples • F(A,B,C) =  m(0,1,4,6,7) Can be re-expressed by • F(A,B,C) = M(2,3,5) Where 2, 3, 5 are the missing minterms in the canonical SOP form

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