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ECE 301 – Digital Electronics

Karnaugh Maps (Lecture #7). ECE 301 – Digital Electronics. The slides included herein were taken from the materials accompanying Fundamentals of Logic Design, 6 th Edition , by Roth and Kinney, and were used with permission from Cengage Learning. Simplification of Logic Functions.

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ECE 301 – Digital Electronics

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  1. Karnaugh Maps (Lecture #7) ECE 301 – Digital Electronics 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 301 - Digital Electronics 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.

  3. ECE 301 - Digital Electronics 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: Simplification using Boolean Algebra

  4. ECE 301 - Digital Electronics Like a truth table, a Karnaugh map specifies the value of a function for all combinations of the input variables. Karnaugh Maps

  5. ECE 301 - Digital Electronics A B 0 1 m m 0 0 2 m m 1 1 3 Two-variable K-map

  6. ECE 301 - Digital Electronics 0 2 1 3 Two-variable K-map: Example 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

  7. ECE 301 - Digital Electronics 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 Gray Code

  8. ECE 301 - Digital Electronics 0 4 1 5 3 7 2 6 Three-variable K-map: Example Minterm expansion: F(A,B,C) = S m(2, 3, 4, 6) Maxterm expansion: F(A,B,C) = P M(0, 1, 5, 7)

  9. ECE 301 - Digital Electronics 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

  10. ECE 301 - Digital Electronics 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.

  11. ECE 301 - Digital Electronics K-maps – Logical Adjacency Gray code

  12. ECE 301 - Digital Electronics Minimize the following logic function using a Karnaugh map: F(A,B,C) = S m(2, 6, 7) Minimization: Example #1 Specify the equivalent maxterm expansion.

  13. ECE 301 - Digital Electronics Minimize the following logic function using a Karnaugh map: F(A,B,C) = P M(1, 3, 5, 6, 7) Minimization: Example #2 Specify the equivalent minterm expansion.

  14. ECE 301 - Digital Electronics 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) Minimization: Example #3 Specify the equivalent maxterm expansion.

  15. ECE 301 - Digital Electronics 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) Minimization: Example #4 Specify the equivalent minterm expansion.

  16. ECE 301 - Digital Electronics For the following truth table: Minimization: Example #5

  17. ECE 301 - Digital Electronics Specify the: 1. minterm expansion 2. maxterm expansion Use a K-map to determine the: 1. minimum SOP expression 2. minimum POS expression Example #5

  18. ECE 301 - Digital Electronics For the following truth table: Minimization: Example #6

  19. ECE 301 - Digital Electronics Specify the: 1. minterm expansion 2. maxterm expansion Use a K-map to determine the: 1. minimum SOP expression 2. minimum POS expression Example #6

  20. ECE 301 - Digital Electronics Can a logic function have more than one minimum SOP expression? Can a logic function have more than one minimum POS expression? Minimal Forms

  21. ECE 301 - Digital Electronics K-maps – Two minimal forms F(A,B,C) = S m(0,1,2,5,6,7) = P M(3,4)

  22. ECE 301 - Digital Electronics Questions?

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