1. ECE 331 – Digital System Design
Karnaugh Maps
(Lecture #6)
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. ECE 331 - Digital System Design
Karnaugh Map
Switching functions can generally be simplified using Boolean algebra. However, two problems arise when algebraic procedures are used:
1. The procedures are difficult to apply in a systematic way.
2. It is difficult to tell when you have arrived at a minimum solution.
The Karnaugh map method is generally faster and easier to
apply than other simplification methods.
Fall 2010
ECE Digital System Design

3. Minimum Forms of Switching Functions
Given a minterm expansion, the minimum sum-of-products format can often be obtained by the following procedure:
Combine terms by using XY′ + XY = X. Do this repeatedly to eliminate as many literals as possible. A given term may be used more than once because X + X = X.
Eliminate redundant terms by using the consensus theorem or other theorems.
Unfortunately, the result of this procedure may depend on the order in which the terms are combined or eliminated so that the final expression obtained is not necessarily minimum.
Fall 2010
ECE Digital System Design

4. Minimum Forms of Switching Functions
Given a maxterm expansion, the minimum product-of-sums format can often be obtained by the following procedure:
Combine terms by using (X+Y′).(X+Y) = X. Do this repeatedly to eliminate as many literals as possible. A given term may be used more than once because X.X = X.
Eliminate redundant terms by using the consensus theorem or other theorems.
This procedure, too, may result in a final expression that is not necessarily minimum.
Fall 2010
ECE Digital System Design

5. Algebraic Simplification: Example
Find a minimum sum-of-products expression for
None of the terms in the above expression can be eliminated by consensus. However, combining terms in a different way leads directly to a minimum sum of products:
Fall 2010
ECE Digital System Design

6. ECE 331 - Digital System Design
Two-variable K-map
Like a Truth table, a K-map specifies the value of the function for all combinations of the inputs. 1 m 2 3 B A
Row # A B
minterm m0 1 m1 2 m2 3 m3
Fall 2010
ECE Digital System Design

7. Two-variable K-map: Example
Fall 2010
ECE Digital System Design

8. 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
Fall 2010
ECE Digital System Design

9. Three-variable K-map: Example
Fall 2010
ECE Digital System Design

10. Minimization using K-maps
K-maps can be used to derive the
Minimum Sum-of-Products (SOP)
Minimum Product-of-Sums (POS)
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
Fall 2010
ECE Digital System Design

11. 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, the more the function can be reduced.
Fall 2010
ECE Digital System Design

12. K-maps – Logical Adjacency
Gray code
Fall 2010
ECE Digital System Design

13. Minimization using K-maps
Example:
Minimize the following function using a K-map:
F(a, b, c) = m(1, 3, 5) =  M(0, 2, 4, 6, 7)
Fall 2010
ECE Digital System Design

14. Minimization using K-maps
Example:
Fall 2010
ECE Digital System Design

15. Minimization using K-maps
Exercise:
Using a K-map derive the minimum sum-of-products (SOP) for the following Boolean expression:
F(A,B,C) = Sm(1, 3, 4, 6)
What is the minterm expansion in terms of A, B, and C?
Fall 2010
ECE Digital System Design

16. ECE 331 - Digital System Design
Can we derive another simplified expression for F, namely the minimized product-of-sums?
Fall 2010
ECE Digital System Design

17. Minimization using K-maps
Exercise:
Using a K-map derive the minimum sum-of-products (SOP) for the following Boolean expression:
F(A,B,C) = P M(1, 3, 4, 6)
What is the minterm expansion in terms of A, B, and C?
Fall 2010
ECE Digital System Design

18. Minimization using K-maps
Exercise:
Using a K-map derive the minimum product-of-sums (POS) for the following Boolean expression:
F(A,B,C) = P M(0, 2, 6)
What is the maxterm expansion in terms of A, B, and C?
Fall 2010
ECE Digital System Design

19. Minimization using K-maps
Exercise:
Using a K-map derive the minimum product-of-sums (POS) for the following Boolean expression:
F(A,B,C) = Sm(0, 2, 6)
What is the maxterm expansion in terms of A, B, and C?
Fall 2010
ECE Digital System Design

20. Minimization using K-Maps
Exercise:
Given the following Truth table, determine the following:
1. Minterm expansion
2. Maxterm expansion
3. Minimized SOP expression
4. Minimized POS expression
Fall 2010
ECE Digital System Design

21. Minimization using K-Maps# A B C F 1 2 3 4 5 6 7
Fall 2010
ECE Digital System Design

22. Minimization using K-Maps
Exercise:
Given the following Truth table, determine the following:
1. Minterm expansion
2. Maxterm expansion
3. Minimized SOP expression
4. Minimized POS expression
Fall 2010
ECE Digital System Design

23. Minimization using K-Maps# A B C F 1 2 3 4 5 6 7
Fall 2010
ECE Digital System Design

24. Minimization using K-maps
Exercise:
Using a K-map derive the minimum sum-of-products (SOP) for the following Boolean expression:
F(a,b,c) = Sm(0, 1, 2, 5, 6, 7)
Is there more than one minimum SOP expression?
Fall 2010
ECE Digital System Design

25. K-maps – Two minimal forms
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ECE Digital System Design

26. Using a K-map to illustrate the Consensus Theorem
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ECE Digital System Design

27. ECE 331 - Digital System Design
Questions?
Fall 2010
ECE Digital System Design