1. ECE 301 – Digital Electronics Minimizing Boolean Expressions using K-maps, The Minimal Cover, and Incompletely Specified Boolean Functions (Lecture #6)

2. ECE 301 - Digital Electronics2 Minimizing SOP Expressions

3. ECE 301 - Digital Electronics3 Remember … 1. A Canonical SOP expression can be derived from a Truth table. 2. A shorthand notation can be used to describe a SOP expression.

4. ECE 301 - Digital Electronics4 Minimizing SOP Expressions #ABCF 00000 10011 20101 30110 41000 51011 61100 71111 F = A'B'C + A'BC' + AB'C + ABC Canonical Sum-of-Products 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

5. ECE 301 - Digital Electronics5 Minimizing SOP Expressions Exercise: Given the following Canonical SOP expression: F(A,B,C) =  m(0, 2, 3, 6) 1. Write out the expression in terms of the minterms. 2. Minimize the SOP expression using a K-Map

6. ECE 301 - Digital Electronics6 Minimizing SOP Expressions Exercise: Given the following Canonical SOP expression: F(A,B,C) =  m(1, 4, 5, 6, 7) 1. Write out the expression in terms of the minterms. 2. Minimize the SOP expression using a K-Map

7. ECE 301 - Digital Electronics7 Minimizing SOP Expressions Exercise: Given the following Canonical SOP expression: F(A,B,C,D) =  m(0, 4, 8, 10, 11, 12, 13, 15) 1. Write out the expression in terms of the minterms. 2. Minimize the SOP expression using a K-Map

8. ECE 301 - Digital Electronics8 Minimizing POS Expressions

9. ECE 301 - Digital Electronics9 Minimizing POS Expressions #ABCF 00000 10011 20101 30110 41000 51011 61100 71111 F = (A+B+C)(A+B'+C')(A'+B+C)(A'+B'+C) Canonical Product-of-Sums 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

10. ECE 301 - Digital Electronics10 Minimizing POS Expressions Exercise: Given the following Canonical POS expression: F(A,B,C) =  M(4, 5, 6) 1. Write out the expression in terms of the minterms. 2. Minimize the POS expression using a K-Map

11. ECE 301 - Digital Electronics11 Minimizing POS Expressions Exercise: Given the following Canonical POS expression: F(A,B,C) =  M(1, 2, 3, 5) 1. Write out the expression in terms of the minterms. 2. Minimize the POS expression using a K-Map

12. ECE 301 - Digital Electronics12 Minimizing POS Expressions Exercise: Given the following Canonical POS expression: F(A,B,C,D) =  M(0, 1, 4, 8, 9, 12, 15) 1. Write out the expression in terms of the minterms. 2. Minimize the POS expression using a K-Map

13. ECE 301 - Digital Electronics13 Selecting a Minimal Cover

14. ECE 301 - Digital Electronics14 Definitions Literal: each appearance of a variable, either uncomplemented or complemented Implicant: a product term that implies F=1 Prime Implicant: an implicant that cannot be combined into another implicant that has fewer literals  Cannot be further minimized Essential Prime Implicant: a prime implicant that covers a minterm uniquely.

15. ECE 301 - Digital Electronics15 Definitions F A B C 0 0 11 1 0 0 1 Which are the implicants, prime implicants, and essential prime implicants? 1100 1110

16. ECE 301 - Digital Electronics16 Definition: Implicants F A B C 0 0 11 1 0 0 1 Implicants: A'B'C', A'B'C, A'BC', A'BC, ABC (minterms) 1100 1110

17. ECE 301 - Digital Electronics17 Definition: Implicants F A B C 0 0 11 1 0 0 1 Implicants: A'B'C', A'B'C, A'BC', A'BC, ABC (minterms) A'C', A'C, A'B', A'B, BC(pairs of minterms) 1100 1110

18. ECE 301 - Digital Electronics18 Definition: Implicants F A B C 0 0 11 1 0 0 1 Implicants: A'B'C', A'B'C, A'BC', A'BC, ABC (minterms) A'C', A'C, A'B', A'B, BC(pairs of minterms) A'(quartet of minterms) 1100 1110

19. ECE 301 - Digital Electronics19 Definition: Prime Implicants F A B C 0 0 11 1 0 0 1 Prime Implicants: BC,A' 1100 1110

20. ECE 301 - Digital Electronics20 Definition: Essential Prime Implicants F A B C 0 0 11 1 0 0 1 Essential Prime Implicants: BC,A' 1100 1110

21. ECE 301 - Digital Electronics21 Definitions Minimal Cover (SOP): the sum (ORing) of prime implicants  Solution may not be unique  For SOP, must cover each “1” at least once  A minimal solution is one with the fewest product terms in the SOP expression, excluding input inversions

22. ECE 301 - Digital Electronics22 Selecting a Minimal Cover (SOP) 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” Write minimum Boolean expression  May not be unique

23. ECE 301 - Digital Electronics23 Example: Determine the minimal cover for the following K-Map: Selecting a Minimal Cover a b c d 0 00011110 110 1110 1011 0011 00 01 11 10 F 1. Identify Prime Implicants 2. Identify Essential Prime Implicants 3. Determine Minimal Cover

24. ECE 301 - Digital Electronics24 Example #1 Prime Implicants: a'b'd, bc', ac, a'c'd, ab, b'cd Essential Prime Implicants: bc', ac Minimal Cover (SOP): F = a'b'd + bc' + ac a b c d 0 00011110 110 1110 1011 0011 00 01 11 10 F

25. ECE 301 - Digital Electronics25 Example: Determine the minimal cover for the following K-Map: Selecting a Minimal Cover 1. Identify Prime Implicants 2. Identify Essential Prime Implicants 3. Determine Minimal Cover y z w x 0 00011110 000 1110 1011 0000 00 01 11 10 F

26. ECE 301 - Digital Electronics26 Example #2 Prime Implicants: xy'z', w'xy', w'xz, xyz, wxy, wxz' Essential Prime Implicants: none Minimal Cover: F = xy'z' + w'xz + wxy F = w'xy' + xyz + wxz' y z w x 0 00011110 000 1110 1011 0000 00 01 11 10 F

27. ECE 301 - Digital Electronics27 Incompletely Specified Functions

28. ECE 301 - Digital Electronics28 Incompletely Specified Functions Some minterms should (or will) never occur.  For example, Binary Coded Decimal (BCD) does not use all 16 combinations of 4 variables. These are considered “don't care” outputs. When minimizing a SOP expression using a K- Map, treat a “don't care” as a “1” whenever it is beneficial. When minimizing a POS expression using a K- Map, treat a “don't care” as a “0” whenever it is beneficial.

29. ECE 301 - Digital Electronics29 Incompletely Specified Functions Exercise: Derive the minimum SOP expression for the following incompletely specified logic function: F(A,B,C,D) =  m(2, 4, 5, 6, 10) + D(12, 13, 14, 15)

30. ECE 301 - Digital Electronics30 Incompletely Specified Functions Exercise: Derive the minimum POS expression for the following incompletely specified logic function: F(A,B,C,D) =  m(2, 4, 5, 6, 10) + D(12, 13, 14, 15)