ece2030 introduction to computer engineering lecture 6: canonical (standard) forms prof. hsien-hsin...
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ECE2030 Introduction to Computer Engineering
Lecture 6: Canonical (Standard) Forms
Prof. Hsien-Hsin Sean LeeProf. Hsien-Hsin Sean Lee
School of Electrical and Computer EngineeringSchool of Electrical and Computer Engineering
Georgia 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
literal A, “AND” () it with (A+Ā)
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Convert a Boolean to Canonical SOP
3in BCBAF B
7) 3, 1, m(0,
ABCBCACBACBAC)B,F(A,
A B C F
ABC 0 0 0 1
ABC 0 0 1 1
ABC 0 1 0 0
ABC 0 1 1 1
ABC 1 0 0 0
ABC 1 0 1 0
ABC 1 1 0 0
ABC 1 1 1 1
0
1
3
7
Minterms listedas 1’s
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Convert a Boolean to Canonical SOP
4Bin BCBAF
15) 14, 7, 6, 3, 2, 1, m(0,
ABCDDABCBCDADBCA
CDBADCBADCBADCBAD)C,B,F(A,
3Bin )CA(BABF
7) 6, 4, 1, m(0,
ABCCABCBACBACBAC)B,F(A,
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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
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Convert a Boolean to Canonical POS
ive)(Distribut Z)Y)(X(X YZX Use
Bin BCBAF 3
6) 5, 4, M(2,
C)BC)(ABA)(CBAC)(BA(F
C)BAC)(BC)(ABAC)(BA)(CBAC)(BA(F
C)BAC)(ABBA)(CCBA(F
C)BC)(AB)(BB)(A(F
C)BAB)(BA(F
BCBAF
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Convert a Boolean to Canonical POS
in BCBAF 3B
M(2,4,5,6)
C)BA)(CBAC)(BAC)(B(AF
BCBAF
A B C F
ABC 0 0 0 1
ABC 0 0 1 1
ABC 0 1 0 0
ABC 0 1 1 1
ABC 1 0 0 0
ABC 1 0 1 0
ABC 1 1 0 0
ABC 1 1 1 1
4
6
2
5
Maxterms listedas 0’s
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Convert a Boolean to Canonical SOP
3in BCBAF B
7) 3, 1, m(0,
ABCBCACBACBAC)B,F(A,
A B C F
ABC 0 0 0 1
ABC 0 0 1 1
ABC 0 1 0 0
ABC 0 1 1 1
ABC 1 0 0 0
ABC 1 0 1 0
ABC 1 1 0 0
ABC 1 1 1 1
0
1
3
7
Minterms listedas 1’s
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Convert a Boolean to Canonical POS
ive)(Distribut Z)Y)(X(X YZX Use
in )CA(BABF 3
B
M(2,3,5)
)CBA)(CBC)(AB(AF
)CBA)(CCB(AF
)CBA)(B(AF
)CA)(BCA)(AB)(BB(AF
)CA(AB )B(AB F
)CA(BABF
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Convert a Boolean to Canonical SOP
3Bin )CA(BABF
7) 6, 4, 1, m(0,
ABCCABCBACBACBAC)B,F(A,
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Interchange Canonical SOP and POS
• For the same Boolean eqn– Canonical SOP form is complementarycomplementary 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