1. ECE 331 – Digital System Design
Karnaugh Maps,
Determining a Minimal Cover,
and
Circuit Design and Analysis
(Lecture #8)
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
Four-variable K-map
row # A B C D
minterm m0 1 m1 2 m2 3 m3 4 m4 5 m5 … 11
m11 12
m12 13
m13 14
m14 15
m15
Spring 2011
ECE Digital System Design

3. ECE 331 - Digital System Design
Four-variable K-map
A B
C D
0 0
0 1
1 1
1 0
Gray code m0 m4
m12 m8 m1 m5
m13 m9 m3 m7
m14
m11 m2 m6
m15
m10
Spring 2011
ECE Digital System Design

4. Minimization: Example #7
Use a Karnaugh map to determine the
1. minimum SOP expression
2. minimum POS expression
For the following logic function:
F(A,B,C,D) = S m(0,1,3,4,5,7,8,11,14)
Specify the equivalent maxterm expansion.
Spring 2011
ECE Digital System Design

5. Minimization: Example #8
Use a Karnaugh map to determine the
1. minimum SOP expression
2. minimum POS expression
For the following logic function:
F(A,B,C,D) = P M(0,2,5,7,8,11,13,15)
Specify the equivalent minterm expansion.
Spring 2011
ECE Digital System Design

6. Minimization: Example #9
Use a Karnaugh map to determine the
1. minimum SOP expression
2. minimum POS expression
For the following logic function:
F(A,B,C,D) = P M(0,1,2,3,6,11,14)
What is the cost of each logic circuit?
Spring 2011
ECE Digital System Design

7. ECE 331 - Digital System Design
Karnaugh Maps
Karnaugh maps can also be used to minimize incompletely specified functions.
Spring 2011
ECE Digital System Design

8. Minimization: Example #10
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(0,2,4) + S d(1,6,7)
Spring 2011
ECE Digital System Design

9. Minimization: Example #11
Use a Karnaugh map to determine the
1. minimum SOP expression
2. minimum POS expression
For the following logic function:
F(A,B,C,D) = P M(2,8,11,13,14,15) . P D(5,7,9,12)
Spring 2011
ECE Digital System Design

10. Determining a Minimal Cover
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ECE Digital System Design

11. Literals and Implicants
Each occurrence of a variable or its complement in an expression
Implicant (SOP) ← represents a product term
A single 1 in the K-map
A group of adjacent 1's in the K-map
Implicant (POS) ← represents a sum term
A single 0 in the K-map
A group of adjacent 0's in the K-map
Spring 2011
ECE Digital System Design

12. ECE 331 - Digital System Design
Prime Implicants
Prime Implicant (SOP)
A product term implicant that cannot be combined with another product term implicant to eliminate a literal.
Prime Implicant (POS)
A sum term implicant that cannot be combined with another sum term implicant to eliminate a literal.
Spring 2011
ECE Digital System Design

13. Implicants and Prime Implicants
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ECE Digital System Design

14. Identifying Prime Implicants
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ECE Digital System Design

15. Identifying Required Terms
Is this term
required?
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ECE Digital System Design

16. Essential Prime Implicants
If a minterm is covered by only one prime implicant, that prime implicant is said to be essential, and must be included in the minimum sum of products (SOP).
Essential Prime Implicants
Prime Implicants
Implicants
Spring 2011
ECE Digital System Design

17. Identifying Essential Prime Implicants
Note: 1’s shaded in blue are covered by only one prime implicant. All other 1’s are covered by at least two prime implicants.
Spring 2011
ECE Digital System Design

18. Determining a Minimal Cover
Identify all prime implicants
Select all essential prime implicants
Select prime implicant(s) to cover remaining terms by considering all possibilities
Sometimes selection is obvious
Sometimes “guess” next prime implicant
Continue, perhaps recursively
Try all possible “guesses”
Determine the Boolean expression
May not be unique
Spring 2011
ECE Digital System Design

19. Determining a Minimal Cover
Shaded 1’s are covered by only one prime implicant.
Essential prime implicants:
A′B, AB′D′
Then AC′D covers the remaining 1’s.
Spring 2011
ECE Digital System Design

20. ECE 331 - Digital System Design
A Minimal Cover
Thus …
A minimal cover is an expression that consists of the fewest product terms (for a SOP expression) or sum terms (for a POS expression) and the fewest literals in each term.
Spring 2011
ECE Digital System Design

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