Digital Design: Karnaugh Map and Minimization Procedures Part II

Chapter 5

Karnaugh Map and Minimization Procedures

Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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Lesson 2

Minimization of Boolean Expressions


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Outline

• Adjacency• Wrapping adjacency• Don't care condition• Octets• Quads• Pairs• Offset and diagonal adjacencies


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Each cell has the adjacent cells such that when moving from one cell to another left, right up or down, only one variable complements

Property


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CAB

C0

C1

AB 00

AB 01

AB 11 AB 10

Only one variable complements between two neighboring cells


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Map for F = A.B.C + A.B.C +A.B.C + A.B.C + A.B.C

CAB

C0

C1

AB 00 1

AB 01 1

AB 11 1

AB 10 1 1

ΣΣΣΣΣΣΣΣ m(0, 3, m(0, 3, 4, 5, 7 )4, 5, 7 )

Adjacent Adjacent CellsCells——variable A variable A complementcomplementinging


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Only one variable complements between two neighboring cellsC

ABCD00

CD01

AB 00

AB 01

AB 11 AB 10

CD11

CD10


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Map for F = A.B.C.D + A.B.C.DD complementing

CAB

CD00

CD01

AB 00

1AB 01 1

AB 11 AB 10

ΣΣΣΣΣΣΣΣ m(4, 5)m(4, 5)CD11

CD10


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Outline

• Adjacency•• Wrapping adjacencyWrapping adjacency• Don't care condition• Octets• Quads• Pairs• Offset and diagonal adjacencies


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Each cell has the adjacent cells such that only one variable complements when moving from one cell to another when• Considering the first row and last row also as adjacent rows • Consider the first column and last Column also as adjacent columns

Wrapping Adjacency Property


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Wrap the map in horizontal axis cylindrical form

• Two cells, one each at the upper most row and lower most row can also be considered as adjacent if we wrap the map in horizontal axis cylindrical form and there is only a single-variable, which changes into its complement when we consider two cells of same column in upper most and lower most rows.


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Wrap the map in vertical axis cylindrical form

• Two cells, one each at the left most column and right most column can also be considered as adjacent if we wrap the map in vertical axis cylindrical form and there is only a single-variable, which changes into its complement when we consider two cells of same row in left most and rightmost columns


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CAB

C0

C1

AB 00 1

AB 01 1

AB 11 1

AB 10 1 1

ΣΣΣΣΣΣΣΣ m(0, 3, m(0, 3, 4, 5, 7 )4, 5, 7 )

Adjacent Adjacent CellsCells——variable A variable A complementicomplementingng


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Map for F = A.B.C.D + A.B.C.DC complementing

CAB

CD00

CD01

AB 00

AB 01 1

AB 11 AB 10

ΣΣΣΣΣΣΣΣ m(4, 6)m(4, 6)CD11

CD10

1


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Outline

• Adjacency• Wrapping adjacency•• Don't care conditionDon't care condition• Octets• Quads• Pairs• Offset and diagonal adjacencies


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CAB

C0

C1

AB 00 X

AB 01 1

AB 11 1

AB 10 1 1

==ΣΣΣΣΣΣΣΣ m(1, 3, m(1, 3, 4, 5, 7 )4, 5, 7 )==ΣΣΣΣΣΣΣΣ m(3, 4, m(3, 4, 5, 7 )5, 7 )

DonDon’’t care t care condition if F condition if F same whether same whether first term first term present or notpresent or not


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Same for F = D and A complementing

CAB

CD00

CD01

AB 00

1X

AB 01 1

AB 11 1

AB 10

ΣΣΣΣΣΣΣΣ m(4, 5, 12, 13) and m(4, 5, 12, 13) and ΣΣΣΣΣΣΣΣ m(4, 5, 12)m(4, 5, 12)

CD11

CD10

DonDon’’t care conditiont care condition


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Don’t care condition

Do not care condition means that a combination of input states do occur and whether the outputs for those states taken as 1s or 0s, it does not matter.


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Don’t care condition in SOP form Map

• Place 1s at the corresponding mini-term places in SOP form of the Karnaugh map. The 1s are placed only at the cells where it leads to make or improve adjacencies.


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Same for F = D and A complementing

CAB

CD00

CD01

AB 00

1X

AB 01 1

AB 11 1

AB 10

ΣΣΣΣΣΣΣΣ m(4, 5, 12, 13) and m(4, 5, 12, 13) and ΣΣΣΣΣΣΣΣ m(4, 5, 12)m(4, 5, 12)

CD11

CD10

Put 1 to makePut 1 to makeit adjacent it adjacent


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Don’t care condition in POS form Map

• Place 0s at the corresponding Max--term places in POS form of the Karnaugh map. The 0s are placed only at the cells where it leads to make or improve adjacencies.


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F = (A+ B + C) . (A+ B+ C)

CAB

C0

C1

A+B 00

A+B 01 X

A+B 11 0

A+B 10

F= F= ΠΠΠΠΠΠΠΠ M(2, M(2, 6) and 6) and also F = also F = M(6). M(6). M(2) is M(2) is dondon’’t care t care conditioncondition

X = 0Put Put X = X = 00


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Outline

• Adjacency• Wrapping adjacency• Don't care condition•• OctetsOctets• Quads• Pairs• Offset and diagonal adjacencies


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F = Octet C, D and A complementing

CAB

CD00

CD01

AB 00 1

1AB 01 1

AB 11 1 11 AB 10 1

1

ΣΣΣΣΣΣΣΣ m(0, 1, 4, 5, 12, 13, 8, 9)m(0, 1, 4, 5, 12, 13, 8, 9)

CD11

CD10


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Octet Property

Minimises the Boolean expressionEight terms become one termRemaining term has three variables removedOnly one variable common in four variable map

Eight cells show adjacency or wrapping adjacency


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F = Octet— B, D and A complementing

CAB

CD00

CD01

AB 00 1

1AB 01 1

AB 11 1 11 AB 10 1

1

ΣΣΣΣΣΣΣΣ m(0, 1, 4, 5, 12, 13, 8, 9) = C m(0, 1, 4, 5, 12, 13, 8, 9) = C

CD11

CD10


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F = Octet C, B and A complementing

CAB

CD00

CD01

AB 00 1

AB 01 1

AB 11 1

AB 10 1

ΣΣΣΣΣΣΣΣ m(0, 2, 4, 6, 12, 14, 8, 10) = Dm(0, 2, 4, 6, 12, 14, 8, 10) = D

CD11

CD10

1

111


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Outline

• Adjacency• Wrapping adjacency• Don't care condition• Octets•• QuadsQuads• Pairs• Offset and diagonal adjacencies


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F = Quad D and A complementing

CAB

CD00

CD01

AB 00 1

AB 01

AB 11 1 AB 10 1

1

ΣΣΣΣΣΣΣΣ m(0, 1, 8, 9)= B.Cm(0, 1, 8, 9)= B.C

CD11

CD10


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Quad Property

Minimises the Boolean expression

Four terms become one termRemaining term has two variables removedTwo variables common in four or three variable map

Four cells show adjacency or wrapping adjacency


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F = Quad— B and C complementing

CAB

CD00

CD01

AB 00

1AB 01

1

AB 11 AB 10

1

ΣΣΣΣΣΣΣΣ m(1, 3, 5, 7) = A.D m(1, 3, 5, 7) = A.D

CD11

CD10

1


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F = Quad C, and B complementing

CAB

CD00

CD01

AB 00 1

AB 01 1

AB 11 AB 10 1

ΣΣΣΣΣΣΣΣ m(0, 2, 4, 6) =A. Dm(0, 2, 4, 6) =A. D

CD11

CD10

1

1


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Map for F = A.B.C + A.B.C +A.B.C + A.B.C = B

CAB

C0

C1

AB 00

AB 01 11

1

AB 11 1

AB 10

ΣΣΣΣΣΣΣΣ m(4, 5, m(4, 5, 12, 13)12, 13)

4 Adjacent 4 Adjacent CellsCells——variable B variable B commoncommon


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Outline

• Adjacency• Wrapping adjacency• Don't care condition• Octets• Quads•• PairsPairs• Offset and diagonal adjacencies


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F = Pair A complementing (wrapping adjacency)

CAB

CD00

CD01

AB 00 1

AB 01

AB 11 AB 10 1

ΣΣΣΣΣΣΣΣ m(0, 8)= B.C.Dm(0, 8)= B.C.D

CD11

CD10


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Pair Property

Minimises the Boolean expression

Two terms become one termRemaining term has one variables removedOne variable common in map

Two cells show adjacency or wrapping adjacency


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CAB

CD00

CD01

AB 00

AB 01 1

1

AB 11 AB 10

1

F= F= ΣΣΣΣΣΣΣΣ m(1, 3, 4, 6) = A.B.D + A.B.Dm(1, 3, 4, 6) = A.B.D + A.B.D

CD11

CD10

1

Pairs (one adjacency and one wrapping adjacency


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Pair B complementingC

ABCD00

CD01

AB 00

AB 01

AB 11 AB 10 1

1

F = F = ΣΣΣΣΣΣΣΣ m(1, 8, 2, 6) =A.B.C. D+A.B.C.D + A.C.Dm(1, 8, 2, 6) =A.B.C. D+A.B.C.D + A.C.D

CD11

CD10

1

1


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Map for F = A.B.C + A.B.C = B.C

CAB

C0

C1

AB 00

AB 01 11AB 11

AB 10

ΣΣΣΣΣΣΣΣ m(4, 12)m(4, 12)

2 Adjacent 2 Adjacent CellsCells——variables B variables B and C and C commoncommon


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Outline

• Adjacency• Wrapping adjacency• Don't care condition• Octets• Quads• Pairs•• Offset and diagonal adjacenciesOffset and diagonal adjacencies


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CAB

CD00

CD01

AB 00 1

AB 01 1

11

AB 11 AB 10

CD11

CD10

Offset adjacency


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CAB

CD00

CD01

AB 00 1

AB 01

AB 11 1

AB 10 1

CD11

CD10

Diagonal adjacency


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XOR and XNOR gates are used offset and diagonal adjacencies


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Summary


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We learnt• Octets, Quads and pairs reduces a set

of 8, 4 and 2, respectively miniterms or maxterms into 1.

• Adjacency of wrapping the map is also taken into account


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End of Lesson 2

Minimization of Boolean Expressions


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THANK YOU

Digital Design: Karnaugh Map and Minimization Procedures Part II

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Transcript of Digital Design: Karnaugh Map and Minimization Procedures Part II