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ECE 331 – Digital System Design

ECE 331 – Digital System Design. Minimizing Boolean Expressions using K-maps, The Minimal Cover, and Incompletely Specified Boolean Functions (Lecture #6). Minimizing SOP Expressions. Remember … 1. A Canonical SOP expression can be derived from a Truth table.

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ECE 331 – Digital System Design

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  1. ECE 331 – Digital System Design Minimizing Boolean Expressions using K-maps, The Minimal Cover, and Incompletely Specified Boolean Functions (Lecture #6)

  2. ECE 331 - Digital System Design Minimizing SOP Expressions

  3. ECE 331 - Digital System Design 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 331 - Digital System Design Minimizing SOP Expressions corresponds to the row #s F = S m(1, 2, 5, 7) Shorter-hand Notation F = S (m1, m2, m5, m7) F = A'B'C + A'BC' + AB'C + ABC Canonical Sum-of-Products Shorthand Notation

  5. ECE 331 - Digital System Design Minimizing SOP Expressions Exercise: Given the following Canonical SOP expression: F(A,B,C) = S m(1, 2, 5, 7) 1. Write out the expression in terms of the minterms. 2. Minimize the SOP expression using a K-Map

  6. ECE 331 - Digital System Design Minimizing SOP Expressions Exercise: Given the following Canonical SOP expression: F(A,B,C) = S m(0, 2, 3, 6) 1. Write out the expression in terms of the minterms. 2. Minimize the SOP expression using a K-Map

  7. ECE 331 - Digital System Design Minimizing SOP Expressions Exercise: Given the following Canonical SOP expression: F(A,B,C,D) = S 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 331 - Digital System Design Minimizing POS Expressions

  9. ECE 331 - Digital System Design Minimizing POS Expressions corresponds to the row #s F = P M(0, 3, 4, 6) Shorter-hand Notation F = P (M0, M3, M4, M6) F = (A+B+C)(A+B'+C')(A'+B+C)(A'+B'+C) Canonical Product-of-Sums Shorthand Notation

  10. ECE 331 - Digital System Design Minimizing POS Expressions Exercise: Given the following Canonical POS expression: F(A,B,C) = P 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 331 - Digital System Design Minimizing POS Expressions Exercise: Given the following Canonical POS expression: F(A,B,C) = P 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 331 - Digital System Design Minimizing POS Expressions Exercise: Given the following Canonical POS expression: F(A,B,C,D) = P 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 331 - Digital System Design Selecting a Minimal Cover

  14. ECE 331 - Digital System Design 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 331 - Digital System Design Definitions F A B 0 0 0 1 1 1 1 0 C 0 1 Which are the implicants, prime implicants, and essential prime implicants?

  16. ECE 331 - Digital System Design Definition: Implicants F A B 0 0 0 1 1 1 1 0 C 0 1 Implicants: A'B'C', A'B'C, A'BC', A'BC, ABC(minterms)

  17. ECE 331 - Digital System Design Definition: Implicants F A B 0 0 0 1 1 1 1 0 C 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)

  18. ECE 331 - Digital System Design Definition: Implicants F A B 0 0 0 1 1 1 1 0 C 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)

  19. ECE 331 - Digital System Design Definition: Prime Implicants F A B 0 0 0 1 1 1 1 0 C 0 1 Prime Implicants: BC,A'

  20. ECE 331 - Digital System Design Definition: Essential Prime Implicants F A B 0 0 0 1 1 1 1 0 C 0 1 Essential Prime Implicants: BC,A'

  21. ECE 331 - Digital System Design 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, and the fewest literals in each product term.

  22. ECE 331 - Digital System Design 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 331 - Digital System Design Example: Determine the minimal cover for the following K-Map: Selecting a Minimal Cover F a b c d 00 01 11 10 00 0 1 1 0 01 1 1 1 0 11 1 0 1 1 10 0 0 1 1 1. Identify Prime Implicants 2. Identify Essential Prime Implicants 3. Determine Minimal Cover

  24. ECE 331 - Digital System Design 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 F a b c d 00 01 11 10 00 0 1 1 0 01 1 1 1 0 11 1 0 1 1 10 0 0 1 1

  25. ECE 331 - Digital System Design Example: Determine the minimal cover for the following K-Map: Selecting a Minimal Cover F y z w x 00 01 11 10 00 0 0 0 0 01 1 1 1 0 11 1 0 1 1 10 0 0 0 0 1. Identify Prime Implicants 2. Identify Essential Prime Implicants 3. Determine Minimal Cover

  26. ECE 331 - Digital System Design 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' F y z w x 00 01 11 10 00 0 0 0 0 01 1 1 1 0 11 1 0 1 1 10 0 0 0 0

  27. ECE 331 - Digital System Design Incompletely Specified Functions

  28. ECE 331 - Digital System Design Incompletely Specified Functions Some minterms should (or will) never occur. For example, Binary Coded Decimal (BCD) 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 331 - Digital System Design Incompletely Specified Functions Exercise: Derive the minimum SOP expression for the following incompletely specified logic function: F(A,B,C,D) = S m(2, 4, 5, 6, 10) + D(12, 13, 14, 15)

  30. ECE 331 - Digital System Design Incompletely Specified Functions Exercise: Derive the minimum POS expression for the following incompletely specified logic function: F(A,B,C,D) = S m(2, 4, 5, 6, 10) + D(12, 13, 14, 15)

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