1. ECE 331 – Digital System Design
Karnaugh Maps
(Lecture #7)
The slides included herein were taken from the materials accompanying
Fundamentals of Logic Design, 6th Edition, by Roth and Kinney,
and were used with permission from Cengage Learning.

2. Simplification of Logic Functions
Logic functions can generally be simplified using Boolean algebra.
However, two problems arise:
It is difficult to apply to Boolean algebra laws and theorems in a systematic way.
It is difficult to determine when a minimum solution has been achieved.
Using a Karnaugh map is generally faster and easier than using Boolean algebra.
Spring 2011
ECE Digital System Design

3. Simplification using Boolean Algebra
Given: F(A,B,C) = Sm(0, 1, 2, 5, 6, 7)
Find: minimum SOP expression
Combining terms in one way:
Combining terms in a different way:
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ECE Digital System Design

4. ECE 331 - Digital System Design
Karnaugh Maps
Like a truth table, a Karnaugh map specifies the value of a function for all combinations of the input variables.
Spring 2011
ECE Digital System Design

5. ECE 331 - Digital System Design
Two-variable K-map 1 m 2 3 B A
row # A B
minterm m0 1 m1 2 m2 3 m3
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ECE Digital System Design

6. Two-variable K-map: Example2 1 3
row # A B F 1 2 3
Minterm expansion: F(A,B) = S m(0, 1) = A'B' + A'B
Maxterm expansion: F(A,B) = P M(2, 3) = (A'+B).(A'+B')
numeric
algebraic
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ECE Digital System Design

7. ECE 331 - Digital System Design
Three-variable K-map
row # A B C
minterm m0 1 m1 2 m2 3 m3 4 m4 5 m5 6 m6 7 m7 m 4 5 1 BC A 3 7 6 2
0 0
0 1
1 1
1 0
Gray Code
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ECE Digital System Design

8. Three-variable K-map: Example3 7 2 6 4 1 5
row # A B C F 1 2 3 4 5 6 7
Minterm expansion: F(A,B,C) = S m(2, 3, 4, 6)
Maxterm expansion: F(A,B,C) = P M(0, 1, 5, 7)
Spring 2011
ECE Digital System Design

9. Minimization using K-maps
K-maps can be used to derive the
Minimum Sum of Products (SOP) expression
Minimum Product of Sums (POS) expression
Procedure:
Enter functional values in the K-map
Identify adjacent cells with same logical value
Adjacent cells differ in only one bit
Use adjacency to minimize logic function
Horizontal and Vertical adjacency
K-map wraps from top to bottom and left to right
Spring 2011
ECE Digital System Design

10. Minimization using K-maps
Logical Adjacency is used to
Reduce the number number of literals in a term
Reduce the number of terms in a Boolean expression.
The adjacent cells
Form a rectangle
Must be a power of 2 (e.g. 1, 2, 4, 8, …)
The greater the number of adjacent cells that can be grouped together (i.e. the larger the rectangle), the more the function can be reduced.
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ECE Digital System Design

11. K-maps – Logical Adjacency
Gray code
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ECE Digital System Design

12. Minimization: Example #1
Minimize the following logic function using a Karnaugh map:
F(A,B,C) = S m(2, 6, 7)
Specify the equivalent maxterm expansion.
Spring 2011
ECE Digital System Design

13. Minimization: Example #2
Minimize the following logic function using a Karnaugh map:
F(A,B,C) = P M(1, 3, 5, 6, 7)
Specify the equivalent minterm expansion.
Spring 2011
ECE Digital System Design

14. Minimization: Example #3
Use a Karnaugh map to determine the
1. minimum SOP expression
2. minimum POS expression
For the following logic function:
F(A,B,C) = S m(0, 1, 5, 7)
Specify the equivalent maxterm expansion.
Spring 2011
ECE Digital System Design

15. Minimization: Example #4
Use a Karnaugh map to determine the
1. minimum SOP expression
2. minimum POS expression
For the following logic function:
F(A,B,C) = P M(0, 1, 5, 7)
Specify the equivalent minterm expansion.
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ECE Digital System Design

16. Minimization: Example #5
For the following truth table: # A B C F 1 2 3 4 5 6 7
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ECE Digital System Design

17. Example #5 Specify the: Use a K-map to determine the:
1. minterm expansion
2. maxterm expansion
Use a K-map to determine the:
1. minimum SOP expression
2. minimum POS expression
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ECE Digital System Design

18. Minimization: Example #6
For the following truth table: # A B C F 1 2 3 4 5 6 7
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ECE Digital System Design

19. Example #6 Specify the: Use a K-map to determine the:
1. minterm expansion
2. maxterm expansion
Use a K-map to determine the:
1. minimum SOP expression
2. minimum POS expression
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ECE Digital System Design

20. Minimal Forms
Can a logic function have more than one minimum SOP expression?
Can a logic function have more than one minimum POS expression?
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ECE Digital System Design

21. K-maps – Two minimal forms
F(A,B,C) = S m(0,1,2,5,6,7) = P M(3,4)
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ECE Digital System Design

22. ECE 331 - Digital System Design
Questions?
Spring 2011
ECE Digital System Design