1. ECE 331 – Digital System Design Karnaugh Maps and Determining a Minimal Cover (Lecture #7) 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.

2. Fall 2010ECE 331 - Digital System Design2 Four-variable K-map Each minterm is located adjacent to the four terms with which it can combine. The 16 cells in the K-map correspond to the 16 rows in a 4-variable truth table.

3. Fall 2010ECE 331 - Digital System Design3 Minimization using K-maps Example: Minimize the following function using a K-map: F =  m(1, 3, 4, 5, 10, 12, 13)

4. Fall 2010ECE 331 - Digital System Design4 Minimization using K-maps Example:

5. Fall 2010ECE 331 - Digital System Design5 Minimization using K-maps Example: Minimize the following function using a K-map: F =  m(0, 2, 3, 5, 6, 7, 8, 10, 11, 14, 15)

6. Fall 2010ECE 331 - Digital System Design6 Minimization using K-maps Example:

7. Fall 2010ECE 331 - Digital System Design7 Minimization using K-maps Example: Minimize the following function using a K-map: F(A,B,C,D) =  M(1, 3, 9, 12)

8. Fall 2010ECE 331 - Digital System Design8 Minimization using K-maps Example:

9. Fall 2010ECE 331 - Digital System Design9 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,D) =  m(1, 5, 6, 8, 9, 12, 13, 14)

10. Fall 2010ECE 331 - Digital System Design10 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,D) =  m(1, 5, 6, 8, 9, 12, 13, 14)

11. Fall 2010ECE 331 - Digital System Design11 Minimization using K-maps Exercise: Using a K-map derive the minimum Boolean expression for the following function: F(A,B,C) =  M(0, 2, 3, 7, 9, 10, 11, 14) Note: the minimum Boolean expression may be in either SOP or POS form.

12. Fall 2010ECE 331 - Digital System Design12 Minimization using K-maps Example: Using a K-map, minimize the following incompletely specified function: F =  m(1, 3, 5, 7, 9) +  d(6, 12, 13)

13. Fall 2010ECE 331 - Digital System Design13 Minimization using K-maps Example:

14. Fall 2010ECE 331 - Digital System Design14 Minimization using K-maps Exercise: Using a K-map derive the minimum sum-of-products (SOP) expression for the following incompletely specified function: F(A,B,C,D) =  m(1, 5, 9, 13, 14) +  d(4, 7, 8, 15)

15. Fall 2010ECE 331 - Digital System Design15 Minimization using K-maps Exercise: Using a K-map derive the minimum product-of-sums (POS) expression for the following incompletely specified function: F(A,B,C,D) =  M(1, 3, 4, 9, 10, 12).  D(2, 6, 11, 14)

16. Fall 2010ECE 331 - Digital System Design16 Determining a Minimal Cover

17. Fall 2010ECE 331 - Digital System Design17 Implicants and Prime Implicants Literal  Each appearance of a variable or its complement in an expression. Implicant (SOP)  Any single 1 or any group of 1’s which can be combined together on a K-map of the function F  Represents a product term Prime Implicant (SOP)  A product term implicant that cannot be combined with another term to eliminate a literal

18. Fall 2010ECE 331 - Digital System Design18 Implicant Prime Implicant Implicant Prime Implicant Implicants and Prime Implicants

19. Fall 2010ECE 331 - Digital System Design19 Identifying Prime Implicants

20. Fall 2010ECE 331 - Digital System Design20 Identifying Required Terms Is this term required?

21. Fall 2010ECE 331 - Digital System Design21 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 Essential Prime Implicants

22. Fall 2010ECE 331 - Digital System Design22 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. Identifying Essential Prime Implicants

23. Fall 2010ECE 331 - Digital System Design23 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

24. Fall 2010ECE 331 - Digital System Design24 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. Determining a Minimal Cover

25. Fall 2010ECE 331 - Digital System Design25 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.

26. Fall 2010ECE 331 - Digital System Design26 Introduction to the 7-Segment Decoder

27. Fall 2010ECE 331 - Digital System Design27 Binary Coded Decimal Assign a 4-bit code to each decimal digit.  A 4-bit code can represent 16 values.  There are only 10 digits in the decimal number system. Unassigned codes are not used.  How do we interpret these unused codes? Hint: think about K-maps.

28. Fall 2010ECE 331 - Digital System Design28 BCD Digits Decimal DigitBCD Code 00000 10001 20010 30011 40100 50101 60110 70111 81000 91001

29. Fall 2010ECE 331 - Digital System Design29 7-Segment Display

30. Fall 2010ECE 331 - Digital System Design30 7-Segment Display

31. Fall 2010ECE 331 - Digital System Design31 7-Segment Decoder

32. Fall 2010ECE 331 - Digital System Design32 Exercise: Design a 7-Segment Decoder. 7-Segment Decoder

33. Fall 2010ECE 331 - Digital System Design33 Describing a Function

34. Fall 2010ECE 331 - Digital System Design34 Describing a Function (SOP) #ABCF 00000 10011 20101 30110 41000 51011 61100 71111 F = A'B'C + A'BC' + AB'C + ABC Minterm Expansion F =  (m 1, m 2, m 5, m 7 ) Shorthand Notation F =  m(1, 2, 5, 7) Shorter-hand Notation corresponds to the row #s

35. Fall 2010ECE 331 - Digital System Design35 Describing a Function (POS) #ABCF 00000 10011 20101 30110 41000 51011 61100 71111 F = (A+B+C)(A+B'+C')(A'+B+C)(A'+B'+C) Maxterm Expansion F =  (M 0, M 3, M 4, M 6 ) Shorthand Notation F =  M(0, 3, 4, 6) Shorter-hand Notation corresponds to the row #s

36. Fall 2010ECE 331 - Digital System Design36 Questions?