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ETE 204 – Digital Electronics

Karnaugh Maps [Lecture: 6] Instructor: Sajib Roy Lecturer, ETE, ULAB. ETE 204 – Digital Electronics. Simplification of Logic Functions. Logic functions can generally be simplified using Boolean algebra . However, two problems arise:

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ETE 204 – Digital Electronics

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  1. Karnaugh Maps [Lecture: 6] Instructor: Sajib Roy Lecturer, ETE, ULAB ETE 204 – Digital Electronics

  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. Summer 2012 ETE 204 - Digital Electronics

  3. 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 Summer 2012 ETE 204 - Digital Electronics

  4. Like a truth table, a Karnaugh map specifies the value of a function for all combinations of the input variables. Karnaugh Maps Summer 2012 ETE 204 - Digital Electronics

  5. A B 0 1 m m 0 0 2 m m 1 1 3 Two-variable K-map Summer 2012 ETE 204 - Digital Electronics

  6. 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 Summer 2012 ETE 204 - Digital Electronics

  7. 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 Summer 2012 ETE 204 - Digital Electronics

  8. 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) Summer 2012 ETE 204 - Digital Electronics

  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 Summer 2012 ETE 204 - Digital Electronics

  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. Summer 2012 ETE 204 - Digital Electronics

  11. K-maps – Logical Adjacency Gray code Summer 2012 ETE 204 - Digital Electronics

  12. 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. Summer 2012 ETE 204 - Digital Electronics

  13. 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. Summer 2012 ETE 204 - Digital Electronics

  14. 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. Summer 2012 ETE 204 - Digital Electronics

  15. 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. Summer 2012 ETE 204 - Digital Electronics

  16. For the following truth table: Minimization: Example #5 Summer 2012 ETE 204 - Digital Electronics

  17. 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 Summer 2012 ETE 204 - Digital Electronics

  18. For the following truth table: Minimization: Example #6 Summer 2012 ETE 204 - Digital Electronics

  19. 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 Summer 2012 ETE 204 - Digital Electronics

  20. Can a logic function have more than one minimum SOP expression? Can a logic function have more than one minimum POS expression? Minimal Forms Summer 2012 ETE 204 - Digital Electronics

  21. K-maps – Two minimal forms F(A,B,C) = S m(0,1,2,5,6,7) = P M(3,4) Summer 2012 ETE 204 - Digital Electronics

  22. Questions? Summer 2012 ETE 204 - Digital Electronics

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