ECE2030 Introduction to Computer Engineering Lecture 6: Canonical (Standard) Forms Prof. Hsien-Hsin Sean Lee Prof. Hsien-Hsin Sean Lee School of Electrical and Computer School of Electrical and Computer Engineering Engineering Georgia Tech Georgia Tech
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Lec6 Intro to Computer Engineering by Hsien-Hsin Sean Lee Georgia Tech -- Canonical Forms
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ECE2030 Introduction to Computer Engineering
Lecture 6: Canonical (Standard) Forms
Prof. Hsien-Hsin Sean LeeProf. Hsien-Hsin Sean LeeSchool of Electrical and Computer EngineeringSchool of Electrical and Computer EngineeringGeorgia TechGeorgia Tech
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Boolean Variables• A multi-dimensional space spanned
by a set of n Boolean variables is denoted by BBnn
• A literalliteral is an instance (e.g. A) of a variable or its complement (Ā)
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SOP Form • A product of literals is called a product term product term
or a cube cube (e.g. Ā·B·C in BB33, or B·C in BB33)• Sum-Of-Product (SOP)Sum-Of-Product (SOP) Form: OROR of product
terms, e.g. ĀB+AC• A minterm minterm is a product term in which every
literal (or variable) appears in BBnn
– ĀBC is a minterm in ĀBC is a minterm in BB3 3 but not in but not in BB44. ABCD is a . ABCD is a minterm in minterm in BB44. .
• A canonicalcanonical (or standardstandard) SOP function:SOP function: – a sum of minterms corresponding to the input
combination of the truth table for which the function produces a “1”1” output.
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Minterms in BB33
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Canonical (Standard) SOP Function
m5 m4 m1 m0 CBACBACBACBAC)B,F(A,
5) 4, 1, set(0,one5) 4, 1, m(0,C)B,F(A,
m14 m9 m4 DABCDCBADCBAD)C,B,F(A,
14) 9, set(4,one14) 9, m(4,D)C,B,F(A,
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POS form (dual of SOP form)• A sum of literals is called a sum term sum term (e.g.
Ā+B+C in BB33, or (B+C) in BB33)• Product-Of-Sum (POS)Product-Of-Sum (POS) Form: ANDAND of sum
terms, e.g. (Ā+B)(A+C)• A maxterm maxterm is a sum term in which every
literal (or variable) appears in BBnn
– (Ā+B+C) is a maxterm in (Ā+B+C) is a maxterm in BB3 3 but not in but not in BB44. . A+B+C+D is a maxterm in A+B+C+D is a maxterm in BB44. .
• A canonicalcanonical (or standardstandard) POS function:POS function: – a product of maxterms corresponding to the input
combination of the truth table for which the function produces a “0”” output.
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Maxterms in BB33
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Canonical (Standard) POS Function
M2 M3 M6 M7 C)B)(ACBC)(ABA)(CBA(C)B,F(A,
7) 6, 3, set(2,zeroM(2,3,6,7)C)B,F(A,
M1M6M11 )DCBD)(ACB)(ADCBA(D)C,B,F(A,
11) 6, set(1,zero11) 6, M(1,D)C,B,F(A,
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Convert a Boolean to Canonical SOP• Expand the Boolean eqn into a SOP• Take each product term w/ a missing