ECE 331 – Digital System Design Karnaugh Maps, Determining a Minimal Cover, and Circuit Design and Analysis (Lecture #8) 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.

ECE 331 - Digital System Design Four-variable K-map row # A B C D minterm m0 1 m1 2 m2 3 m3 4 m4 5 m5 … 11 m11 12 m12 13 m13 14 m14 15 m15 Spring 2011 ECE 331 - Digital System Design

ECE 331 - Digital System Design Four-variable K-map A B C D 0 0 0 1 1 1 1 0 Gray code m0 m4 m12 m8 m1 m5 m13 m9 m3 m7 m14 m11 m2 m6 m15 m10 Spring 2011 ECE 331 - Digital System Design

Minimization: Example #7 Use a Karnaugh map to determine the 1. minimum SOP expression 2. minimum POS expression For the following logic function: F(A,B,C,D) = S m(0,1,3,4,5,7,8,11,14) Specify the equivalent maxterm expansion. Spring 2011 ECE 331 - Digital System Design

Minimization: Example #8 Use a Karnaugh map to determine the 1. minimum SOP expression 2. minimum POS expression For the following logic function: F(A,B,C,D) = P M(0,2,5,7,8,11,13,15) Specify the equivalent minterm expansion. Spring 2011 ECE 331 - Digital System Design

Minimization: Example #9 Use a Karnaugh map to determine the 1. minimum SOP expression 2. minimum POS expression For the following logic function: F(A,B,C,D) = P M(0,1,2,3,6,11,14) What is the cost of each logic circuit? Spring 2011 ECE 331 - Digital System Design

ECE 331 - Digital System Design Karnaugh Maps Karnaugh maps can also be used to minimize incompletely specified functions. Spring 2011 ECE 331 - Digital System Design

Minimization: Example #10 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(0,2,4) + S d(1,6,7) Spring 2011 ECE 331 - Digital System Design

Minimization: Example #11 Use a Karnaugh map to determine the 1. minimum SOP expression 2. minimum POS expression For the following logic function: F(A,B,C,D) = P M(2,8,11,13,14,15) . P D(5,7,9,12) Spring 2011 ECE 331 - Digital System Design

Determining a Minimal Cover Spring 2011 ECE 331 - Digital System Design

Literals and Implicants Each occurrence of a variable or its complement in an expression Implicant (SOP) ← represents a product term A single 1 in the K-map A group of adjacent 1's in the K-map Implicant (POS) ← represents a sum term A single 0 in the K-map A group of adjacent 0's in the K-map Spring 2011 ECE 331 - Digital System Design

ECE 331 - Digital System Design Prime Implicants Prime Implicant (SOP) A product term implicant that cannot be combined with another product term implicant to eliminate a literal. Prime Implicant (POS) A sum term implicant that cannot be combined with another sum term implicant to eliminate a literal. Spring 2011 ECE 331 - Digital System Design

Implicants and Prime Implicants Spring 2011 ECE 331 - Digital System Design

Identifying Prime Implicants Spring 2011 ECE 331 - Digital System Design

Identifying Required Terms Is this term required? Spring 2011 ECE 331 - Digital System Design

Essential Prime Implicants 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 Spring 2011 ECE 331 - Digital System Design

Identifying Essential Prime Implicants 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. Spring 2011 ECE 331 - Digital System Design

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 Spring 2011 ECE 331 - Digital System Design

Determining a Minimal Cover 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. Spring 2011 ECE 331 - Digital System Design

ECE 331 - Digital System Design 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. Spring 2011 ECE 331 - Digital System Design

ECE 331 - Digital System Design Questions? Spring 2011 ECE 331 - Digital System Design