ECE 301 – Digital Electronics Karnaugh Maps and Determining a Minimal Cover (Lecture #8) 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.

Spring 2011ECE Digital Electronics2 Four-variable K-map row #ABCDminterm 00000m0m m1m m2m m3m m4m m5m5 …… m m m m m 15

Spring 2011ECE Digital Electronics3 Four-variable K-map A B C D Gray code m0 m0 m4m4 m 12 m8m8 m1 m1 m5m5 m 13 m9m9 m3 m3 m7m7 m 14 m 11 m2 m2 m6m6 m 15 m 10

Spring 2011ECE Digital Electronics4 Minimization: Example #7 Use a Karnaugh map to determine the minimum POS expression For the following logic function: F(A,B,C,D) =  m(0,1,3,4,5,7,8,11,14) Specify the equivalent maxterm expansion.

Spring 2011ECE Digital Electronics5 Minimization: Example #8 Use a Karnaugh map to determine the minimum SOP expression For the following logic function: F(A,B,C,D) =  M(0,2,5,7,8,11,13,15) Specify the equivalent minterm expansion.

Spring 2011ECE Digital Electronics6 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) =  M(0,1,2,3,6,11,14) What is the cost of each logic circuit?

Spring 2011ECE Digital Electronics7 Karnaugh Maps Karnaugh maps can also be used to minimize incompletely specified functions.

Spring 2011ECE Digital Electronics8 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) =  m(4,7) +  d(1,3)

Spring 2011ECE Digital Electronics9 Minimization: Example #11 Use a Karnaugh map to determine the minimum SOP expression For the following logic function: F(A,B,C,D) =  M(0,2,5,6,8,13,15).  D(3,4,10)

Spring 2011ECE Digital Electronics10 Minimization: Example #12 Use a Karnaugh map to determine the minimum POS expression For the following logic function: F(A,B,C,D) =  m(0,1,2,4,6,8,9,10) +  d(3,7,11,13,14)

Spring 2011ECE Digital Electronics11 Determining a Minimal Cover

Spring 2011ECE Digital Electronics12 Literals and Implicants Literal  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 2011ECE Digital Electronics13 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 2011ECE Digital Electronics14 Implicant Prime Implicant Implicant Prime Implicant Implicants and Prime Implicants Additional Prime Implicants?

Spring 2011ECE Digital Electronics15 Identifying Prime Implicants

Spring 2011ECE Digital Electronics16 Identifying Required Terms Is this term required?

Spring 2011ECE Digital Electronics17 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

Spring 2011ECE Digital Electronics18 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

Spring 2011ECE Digital Electronics19 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 2011ECE Digital Electronics20 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

Spring 2011ECE Digital Electronics21 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 2011ECE Digital Electronics22 Questions?