ECE 301 – Digital Electronics Karnaugh Maps (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.

Spring 2011ECE Digital Electronics2 Simplification of Logic Functions Logic functions can generally be simplified using Boolean algebra. However, two problems arise: – It is difficult to apply to Boolean algebra laws and theorems in a systematic way. – It is difficult to determine when a minimum solution has been achieved. Using a Karnaugh map is generally faster and easier than using Boolean algebra.

Spring 2011ECE Digital Electronics3 Simplification using Boolean Algebra Given: F(A,B,C) =  m(0, 1, 2, 5, 6, 7) Find: minimum SOP expression Combining terms in one way: Combining terms in a different way:

Spring 2011ECE Digital Electronics4 Karnaugh Maps Like a truth table, a Karnaugh map specifies the value of a function for all combinations of the input variables.

Spring 2011ECE Digital Electronics5 Two-variable K-map m 0 m 2 m 3 m 1 B A row #ABminterm 000m0m0 101m1m1 210m2m2 311m3m3

Spring 2011ECE Digital Electronics6 Two-variable K-map: Example Minterm expansion: F(A,B) =  m(0, 1) = A'B' + A'B Maxterm expansion: F(A,B) =  (2, 3) = (A'+B).(A'+B') numericalgebraic row #ABF

Spring 2011ECE Digital Electronics7 Three-variable K-map row #AB C minterm m0m m1m m2m2 3011m3m3 4100m4m4 5101m5m5 6110m6m m7m7 m 0 m 4 m 5 m 1 BC A m 3 m 7 m 6 m Gray Code

Spring 2011ECE Digital Electronics8 Three-variable K-map: Example Minterm expansion: F(A,B,C) =  m(2, 3, 4, 6) Maxterm expansion: F(A,B,C) =  (0, 1, 5, 7) row #ABCF

Spring 2011ECE Digital Electronics9 Minimization using K-maps K-maps can be used to derive the  Minimum Sum of Products (SOP) expression  Minimum Product of Sums (POS) expression Procedure:  Enter functional values in the K-map  Identify adjacent cells with same logical value Adjacent cells differ in only one bit  Use adjacency to minimize logic function Horizontal and Vertical adjacency K-map wraps from top to bottom and left to right

Spring 2011ECE Digital Electronics10 Minimization using K-maps Logical Adjacency is used to  Reduce the number number of literals in a term  Reduce the number of terms in a Boolean expression. The adjacent cells  Form a rectangle  Must be a power of 2 (e.g. 1, 2, 4, 8, …) The greater the number of adjacent cells that can be grouped together (i.e. the larger the rectangle), the more the function can be reduced.

Spring 2011ECE Digital Electronics11 K-maps – Logical Adjacency Gray code

Spring 2011ECE Digital Electronics12 Minimization: Example #1 Minimize the following logic function using a Karnaugh map: F(A,B,C) =  m(2, 6, 7) Specify the equivalent maxterm expansion.

Spring 2011ECE Digital Electronics13 Minimization: Example #2 Minimize the following logic function using a Karnaugh map: F(A,B,C) =  M(1, 3, 5, 6, 7) Specify the equivalent minterm expansion.

Spring 2011ECE Digital Electronics14 Minimization: Example #3 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(0, 1, 5, 7) Specify the equivalent maxterm expansion.

Spring 2011ECE Digital Electronics15 Minimization: Example #4 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(0, 1, 5, 7) Specify the equivalent minterm expansion.

Spring 2011ECE Digital Electronics16 Minimization: Example #5 For the following truth table: #ABC F

Spring 2011ECE Digital Electronics17 Example #5 Specify the: 1. minterm expansion 2. maxterm expansion Use a K-map to determine the: 1. minimum SOP expression 2. minimum POS expression

Spring 2011ECE Digital Electronics18 Minimization: Example #6 For the following truth table: #ABC F

Spring 2011ECE Digital Electronics19 Example #6 Specify the: 1. minterm expansion 2. maxterm expansion Use a K-map to determine the: 1. minimum SOP expression 2. minimum POS expression

Spring 2011ECE Digital Electronics20 Minimal Forms Can a logic function have more than one minimum SOP expression? Can a logic function have more than one minimum POS expression?

Spring 2011ECE Digital Electronics21 K-maps – Two minimal forms F(A,B,C) =  m(0,1,2,5,6,7) =  M(3,4)

Spring 2011ECE Digital Electronics22 Questions?