Karnaugh Graph or K-Map
32
KARNAUGH MAP
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Transcript of Karnaugh Graph or K-Map
KARNAUGH MAP
CONTENTS Introduction. Advantages of Karnaugh Maps. SOP & POS. Properties. Simplification Process Different Types of K-maps Simplyfing logic expression by different types of K-Map Don’t care conditions Prime Implicants References.
Also known as Veitch diagram or K-Map.
Invented in 1953 by Maurice Karnaugh.
A graphical way of minimizing Boolean expressions.
It consists tables of rows and columns with entries represent 1`s or 0`s.
Introduction
Advantages of Karnaugh Maps
Data representation’s simplicity. Changes in neighboring variables are easily displayed Changes Easy and Convenient to implement. Reduces the cost and quantity of logical gates.
SOP & POS The SOP (Sum of Product) expression represents
1’s .
SOP form such as (A.B)+(B.C).
The POS (Product of Sum) expression represents the low (0) values in the K-Map.
POS form like (A+B).(C+D)
PropertiesAn n-variable K-map has 2n cells with n-variable truth
table value.
Adjacent cells differ in only one bit .
Each cell refers to a minterm or maxterm.
For minterm mi , maxterm Mi and don’t care of f we place 1 , 0 , x .
Simplification Process
No diagonals.
Only 2^n cells in each group.
Groups should be as large as possible.
A group can be combined if all cells of the group have same set of variable.
Overlapping allowed.
Fewest number of groups possible.
Different Types of
K-maps
Two Variable K-map(continued)
The K-Map is just a different form of the truth table. V
W X FWX
Minterm – 0 0 0 1Minterm – 1 0 1 0Minterm – 2 1 0 1Minterm – 3 1 1 0
V
0 1
2 3
X
W
W
X
1 0
1 0
Two Variable K-map Grouping
V
0 0
0 0
B
A
A
Groups of One – 4
1
A B
B
Groups of Two – 2
Two Variable K-Map Groupings
Group of Four
V
0 0
0 0
B
A
A
B
1
B
1
V
1 1
1 1
B
A
A
1
B
Three Variable K-map (continued) K-map from truth table.
W X Y FWXY
Minterm – 0 0 0 0 1Minterm – 1 0 0 1 0Minterm – 2 0 1 0 0Minterm – 3 0 1 1 0Minterm – 4 1 0 0 0Minterm – 5 1 0 1 1Minterm – 6 1 1 0 1Minterm – 7 1 1 1 0
V
0 1
2 3
6 7
4 5
Y
X W
Y
1
X W
X W
X W
0
0 0
0 1
1 0
Only onevariable changes
for every row cnge
12
Three Variable K-Map Groupings
V
0 0
0 0
0 0
0 0
C C
B A
B A
BA
BA
B A
1 1
B A
1 1
B A
1 1
B A
1 1
1
C A
1
1
C A
1
1
C A
1
1
C B
1
1
C B
1
1
C A
11
C B
1
1
C B
1
Groups of One – 8 (not shown)Groups of Two – 12
Three Variable K-Map GroupingsGroups of Four – 6 Group of Eight - 1
V
1 1
1 1
1 1
1 1
C C
B A
B A
BA
BA
1
V
0 0
0 0
0 0
0 0
C C
B A
B A
BA
BA
1
C
1
1
1
1
C
1
1
1
A
1 1
1 1
B
1 1
1 1
A
1 1
1 1
B
1 1
1 1
Truth Table to K-Map MappingFour Variable K-Map
W X Y Z FWXYZ
Minterm – 0 0 0 0 0 0Minterm – 1 0 0 0 1 1Minterm – 2 0 0 1 0 1Minterm – 3 0 0 1 1 0Minterm – 4 0 1 0 0 1Minterm – 5 0 1 0 1 1Minterm – 6 0 1 1 0 0Minterm – 7 0 1 1 1 1Minterm – 8 1 0 0 0 0Minterm – 9 1 0 0 1 0Minterm –
10 1 0 1 0 1
Minterm – 11 1 0 1 1 0
Minterm – 12 1 1 0 0 1
Minterm – 13 1 1 0 1 0
Minterm – 14 1 1 1 0 1
Minterm – 15 1 1 1 1 1
V
0 1 3 2
4 5 7 6
12 13 15 14
8 9 11 10
X W
X W
X W
X W
Z Y Z Y ZY Z Y
1 01 1
1 10 1
0 10 0
0 11 0
FOUR VARIABLE K-MAP GROUPINGS
V
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
B A
B A
BA
BA
D C D C D C D CC B
1 1
1 1
D B
1 1
1 1D A
1
1
1
1
C B
1 1
1 1
D B1
1
1
1D A
1
1
1
1 D B11
11
FOUR VARIABLE K-MAP GROUPINGS
Groups of Eight – 8 (two shown)
V
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
B A
B A
BA
BA
D C D C D C D C
B
1 1 1 1
1 1 1 1
D1
1
1
1
1
1
1
1
Group of Sixteen – 1
V
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
B A
B A
BA
BA
D C D C D C D C
1
Simplyfing Logic Expression
by Different types of K-Map
TWO VARIABLE K-MAP Differ in the value of y in m0 and m1.
Differ in the value of x in m0 and m2.
y = 0 y = 1
x = 0 m 0 = m 1 =
x = 1 m2 =
m
3 =
yx yx
yx yx
Two Variable K-MapSimplified sum-of-products (SOP) logic expression for the logic function F1.
V
1 1
0 0
K
J
J
K
J
JF 1
J K F1
0 0 10 1 11 0 01 1 0
20
Three Variable Maps
A three variable K-map :
yz=00 yz=01 yz=11 yz=10
x=0 m0 m1 m3 m2
x=1 m4 m5 m7 m6
Where each minterm corresponds to the product terms:
yz=00 yz=01 yz=11 yz=10
x=0
x=1
zyx zyx zyx zyx
zyx zyx zyx zyx
Four Variable K-MapSimplified sum-of-products (SOP) logic expression for the logic function F3.
T SU RU T SU S RF 3
R S T U F3
0 0 0 0 00 0 0 1 10 0 1 0 00 0 1 1 10 1 0 0 00 1 0 1 10 1 1 0 10 1 1 1 11 0 0 0 01 0 0 1 11 0 1 0 01 0 1 1 01 1 0 0 11 1 0 1 01 1 1 0 11 1 1 1 1
V
0 1 1 0
0 1 1 1
1 0 1 1
0 1 0 0
S R
S R
S R
S R
U T U T U T U T
U R
T S
U S R
U T S
Five variable K-map is formed using two connected 4-variable maps:
Chapter 2 - Part 2 2323
0
1 5
4
VWX
YZ
V
Z
000 001
00
13
12
011
9
8
010
X
3
2 6
7
14
15
10
11
01
11
10
Y
16
17 21
20
29
28
25
24
19
18 22
23
30
31
26
27
100 101 111 110
W W
X
Five Variable K-Map
Don’t-care condition
Minterms that may produce either
0 or 1 for the function.
Marked with an ‘x’
in the K-map.
These don’t-care conditions can
be used to provide further simplification.
SOME YOU GROUP, SOME YOU DON’T
V
X 0
1 0
0 0
X 0
C C
B A
B A
BA
BA
C A
This don’t care condition was treated as a (1).
There was no advantage in treating this don’t care condition as a (1), thus it was treated as a (0) and not grouped.
Don’t Care ConditionsSimplified sum-of-products (SOP) logic expression for the logic function F4.
S RT RF 4
R S T U F4
0 0 0 0 X0 0 0 1 00 0 1 0 10 0 1 1 X0 1 0 0 00 1 0 1 X0 1 1 0 X0 1 1 1 11 0 0 0 11 0 0 1 11 0 1 0 11 0 1 1 X1 1 0 0 X1 1 0 1 01 1 1 0 01 1 1 1 0
V
X 0 X 1
0 X 1 X
X 0 0 0
1 1 X 1
S R
S R
S R
S R
U T U T U T U TT R
S R
ImplicantsThe group of 1s is called implicants.Two types of Implicants:Prime Implicants.Essential Prime Implicants.
Prime and Essential Prime Implicants
Chapter 2 - Part 2 28
DB
CB
1 1
1 1
1 1
B
D
A
1 1
1 1
1
ESSENTIAL Prime Implicants
C
BD
CD
BD
Minterms covered by single prime implicant
DB
1 1
1 1
1 1
B
C
D
A
1 1
1 1
1
AD
BA
Example with don’t Care
Chapter 2 - Part 2 29
x
x
1
1 1
1
1
B
D
A
C
1
1
1
x
x
1
1 1
1
1
B
D
A
C
1
1
EssentialSelected
Besides some disadvantages like usage of limited variables K-Map is very efficient to simplify logic expression.
Conclusion
References
Wikipedia.com. Digital Design by Morris Mano
Thank You
