Digital Design: Karnaugh Map and Minimization Procedures Part II
-
Upload
atush-jain -
Category
Education
-
view
125 -
download
1
Transcript of Digital Design: Karnaugh Map and Minimization Procedures Part II
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
2
Lesson 2
Minimization of Boolean Expressions
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
3
Outline
• Adjacency• Wrapping adjacency• Don't care condition• Octets• Quads• Pairs• Offset and diagonal adjacencies
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
4
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
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
5
CAB
C0
C1
AB 00
AB 01
AB 11 AB 10
Only one variable complements between two neighboring cells
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
6
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
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
7
Only one variable complements between two neighboring cellsC
ABCD00
CD01
AB 00
AB 01
AB 11 AB 10
CD11
CD10
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
8
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
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
9
Outline
• Adjacency•• Wrapping adjacencyWrapping adjacency• Don't care condition• Octets• Quads• Pairs• Offset and diagonal adjacencies
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
10
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
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
11
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.
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
12
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
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
13
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 complementicomplementingng
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
14
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
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
15
Outline
• Adjacency• Wrapping adjacency•• Don't care conditionDon't care condition• Octets• Quads• Pairs• Offset and diagonal adjacencies
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
16
Map for F = A.B.C + A.B.C +A.B.C + A.B.C + A.B.C
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
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
17
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
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
18
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.
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
19
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.
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
20
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
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
21
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.
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
22
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
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
23
Outline
• Adjacency• Wrapping adjacency• Don't care condition•• OctetsOctets• Quads• Pairs• Offset and diagonal adjacencies
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
24
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
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
25
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
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
26
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
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
27
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
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
28
Outline
• Adjacency• Wrapping adjacency• Don't care condition• Octets•• QuadsQuads• Pairs• Offset and diagonal adjacencies
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
29
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
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
30
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
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
31
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
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
32
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
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
33
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
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
34
Outline
• Adjacency• Wrapping adjacency• Don't care condition• Octets• Quads•• PairsPairs• Offset and diagonal adjacencies
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
35
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
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
36
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
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
37
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
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
38
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
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
39
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
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
40
Outline
• Adjacency• Wrapping adjacency• Don't care condition• Octets• Quads• Pairs•• Offset and diagonal adjacenciesOffset and diagonal adjacencies
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
41
CAB
CD00
CD01
AB 00 1
AB 01 1
11
AB 11 AB 10
CD11
CD10
Offset adjacency
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
42
CAB
CD00
CD01
AB 00 1
AB 01
AB 11 1
AB 10 1
CD11
CD10
Diagonal adjacency
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
43
XOR and XNOR gates are used offset and diagonal adjacencies
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
45
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
Ch05L2--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
46
End of Lesson 2
Minimization of Boolean Expressions