Chapter 4 Karnaugh Map
Introduction • Another method to simplify the Boolean expression is by using Karnaugh map (K-map). • K-map is a graphical representation of the output for a given Boolean expression. • It contains the same information as the truth table. • It also contains a cell for each input combination. • A Boolean expression or a truth table with n input variables has 2n cells on the K-Map.
10/9/2012
2
2-variables K-Map • The 2- variables K-Map has 22 = 4 cells. • It has 4 combinations of input. Truth table
A B
10/9/2012
0
0
0
1
0
1
2
1
0
3
1
1
K-Map
Y
A
A
B 0
2
1
3
B
3
3-variables K-Map • The 3- variables K-Map has 23 = 8 cells. • It has 8 combinations of input. Truth table
10/9/2012
K-Map
A
B
C
0
0
0
0
1
0
0
1
2
0
1
0
3
0
1
1
4
1
0
0
5
1
0
1
6
1
1
0
7
1
1
1
Y A B AB
AB
AB
C 0
2
6
4
1
3
7
5
C
4
4-variables K-Map A B C D Y Truth table
10/9/2012
0
0 0 0 0
1
0 0 0 1
2
0 0 1 0
3
0 0 1
4
0 1 0 0
5
0 1 0 1
6
0 1
1 0
7
0 1
1
8
1 0 0 0
9
1 0 0 1
10
1 0 1 0
1
1
11
1 0 1
12
1
1 0 0
13
1
1 0 1
14
1
1
1 0
15
1
1
1
1
1
• The 4-variables K-Map has 24 = 16 cells. • It has 16 combinations of input. K-Map
AB
AB
AB
AB
CD CD CD CD
0
4
12
8
1
5
13
9
3
7
15
11
2
6
14
10
5
Simplification of 2-variables K-Map 1
Steps:
Identify the SOP expression from the truth table.
Example:
0
3
2 A B
Y
0
0
0
1
0
1
1
2
1
0
1
3
1
1
1
Plot a 1 on the K-map for each output Y=1 A
A
B 0
B
A
1 1
2
1 1
3
B
Draw loops around adjacent cells. The largest loop is a group of 4 cells followed by 2 and 1 cell. The loops may overlap.
4
Repeat step 2 and 3 until each bit 1 has been looped. Each loop produces a simplified product.
5
Then, logically OR the simplified product term. Y= A + B
10/9/2012
6
Simplification of 3-variables K-Map Steps:
1
Example:
A
B
C
Y
0
0
0
0
0
1
0
0
1
1
2
0
1
0
1
3
0
1
1
1
4
1
0
0
0
5
1
0
1
1
6
1
1
0
1
7
1
1
1
1
10/9/2012
Identify the SOP expression from the truth table.
2
2
Plot a 1 on the K-map for each output Y=1
A B AB C
1 0
C
1
3
AB
A B AB
1
C 6
2
1 1
AB
1 3
0
4
1 7
1
C 5
1
AB B
1 6
2
1 1
AB
1 3
4
1 7
C
5
Draw loops around adjacent cells. The largest loop is a group of 8 cells, followed by 4, 2 and 1 cell. The loops may overlap.
4
Repeat step 2 and 3 until each bit 1 has been looped. Each loop produces a simplified product.
5
Then, logically OR the simplified product term.
Y= B + C
7
Simplification of 4-variables K-Map A B C D Y 0
0 0 0 0
1
1
0 0 0 1
1
2
0 0 1 0
1
3
0 0 1
1
4
0 1 0 0 0
5
0 1 0 1
6
0 1
1 0 0
CD
1
7
0 1
1
CD
1
8
1
1
Example:
2A B
CD
1 0 0 0
1
9
1 0 0 1
1
10
1 0 1 0
1
11
1 0 1
1
3
12
1
1 0 0 0
13
1
1 0 1
4
14
1
1
15
1
0
1 0 0
10/9/2012
1
1
1
1
0
CD
Identify the SOP expression from the truth table. Plot a 1 on the K-map for each output Y=1
AB
AB
AB
1
AB
1 0
1 1
1 2
Steps:
4
12
5
13
1 1
8
1
1
9
1
3
7
15
2
6
14
1
11
1 10
CD
AB
AB
1
1 0
CD
1
CD
1
CD
1
4
12
1 1
3
8
1 5
13
1
9
1 7
15
11
1 2
B
5
AB
6
14
10
AD
Draw loops around adjacent cells. The largest loop is a group of 16 cells, followed by 8,4, 2 and 1 cell. The loops may overlap. Repeat step 2 and 3 until each bit 1 has been looped. Each loop produces a simplified product. Then, logically OR the simplified product term.
RA/Sept2012-Jan2013
Y = B + AD 8
Designing Combinational Logic Circuits Example
• The diagram shown below is a water filtering system. A water quality sensing detector will generate a quality scale from 0 – 7. From this scale, selected filter will function as follows to produce clean water. Filter
Water flow
A B
Water quality sensing detector
C
Clean water
Water Condition The cleanest
Logic circuit
Scale 0 - 7
The dirtiest
10/9/2012
Scale 0 1 2 3 4 5 6 7
Filters needed No filters are activated A only A, B and C B only A only A and B only A and C only A, B and C
Design the logic circuit. Your solution should include: a) Truth table. b) Simplified expression using Karnaugh map. c) Based on the simplified expression in (b), draw the logic circuit. 9
Designing Combinational Logic Circuits Solution:
1
Derive the truth table
A B C
FA
FB
FC
0 0 0
0
0
0
0 0
1
1
0
0
0
0
1
1
1
1
0
1
1
0
1
0
1
0 0
1
0
0
1
0
1
1
1
0
1
1
0
1
0
1
1
1
1
1
1
1
2
A B AB
BC
C
1 0
C
AB
AB
1
1 6
2
1
1 1
BC
C
A 4
1 7
3
A B AB
AB
FA A BC BC
5
AB
1 0
C
6
2
1 1
AB 10/9/2012
Simplify the Boolean expression using K-map
1 3
4
FB AB AC
1 7
5
AC 10
Designing Combinational Logic Circuits Solution:
1
10/9/2012
Derive the truth table
A B C
FA
FB
FC
0 0 0
0
0
0
0 0
1
1
0
0
0
1
0
1
1
1
0
1
1
0
1
0
1
0 0
1
0
0
1
0
1
1
1
0
1
1
0
1
0
1
1
1
1
1
1
1
2
Simplify the Boolean expression using K-map
A B AB
BC C
1 0
AB
AB
AB
1 2
C
6
4
7
5
FC BC AB
1 1
3
11