ECE 331 – Digital System Design

1. Karnaugh Maps (Lecture #6) ECE 331 – Digital System Design 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.

3. ECE 331 - Digital System Design 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.

4. ECE 331 - Digital System Design 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.

5. ECE 331 - Digital System Design 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:

6. ECE 331 - Digital System Design A B 0 1 m m 0 0 2 m m 1 1 3 Two-variable K-map • Like a Truth table, a K-map specifies the value of the function for all combinations of the inputs.

7. ECE 331 - Digital System Design Two-variable K-map: Example

8. ECE 331 - Digital System Design A 0 1 BC 0 0 m m 0 4 0 1 m m 1 5 1 1 m m 3 7 1 0 m m 2 6 Three-variable K-map

9. ECE 331 - Digital System Design Three-variable K-map: Example

10. ECE 331 - Digital System Design 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

11. ECE 331 - Digital System Design 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.

12. ECE 331 - Digital System Design K-maps – Logical Adjacency Gray code

13. ECE 331 - Digital System Design Example: Minimize the following function using a K-map: F(a, b, c) = m(1, 3, 5) =  M(0, 2, 4, 6, 7) Minimization using K-maps

14. ECE 331 - Digital System Design Example: Minimization using K-maps

15. ECE 331 - Digital System Design 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) Minimization using K-maps What is the minterm expansion in terms of A, B, and C?

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

17. ECE 331 - Digital System Design 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) Minimization using K-maps What is the minterm expansion in terms of A, B, and C?

18. ECE 331 - Digital System Design 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) Minimization using K-maps What is the maxterm expansion in terms of A, B, and C?

19. ECE 331 - Digital System Design 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) Minimization using K-maps What is the maxterm expansion in terms of A, B, and C?

20. ECE 331 - Digital System Design Exercise: Given the following Truth table, determine the following: 1. Minterm expansion 2. Maxterm expansion 3. Minimized SOP expression 4. Minimized POS expression Minimization using K-Maps

21. ECE 331 - Digital System Design Minimization using K-Maps

22. ECE 331 - Digital System Design Exercise: Given the following Truth table, determine the following: 1. Minterm expansion 2. Maxterm expansion 3. Minimized SOP expression 4. Minimized POS expression Minimization using K-Maps

23. ECE 331 - Digital System Design Minimization using K-Maps

24. ECE 331 - Digital System Design 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) Minimization using K-maps Is there more than one minimum SOP expression?

25. ECE 331 - Digital System Design K-maps – Two minimal forms

26. ECE 331 - Digital System Design Using a K-map to illustrate the Consensus Theorem

27. ECE 331 - Digital System Design Questions?