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Department of Computer Science

DCS

COMSATS Institute of Information Technology

Karnaugh Maps Simplification

Rab Nawaz Khan Jadoon

Lecturer COMSATS Lahore Pakistan

Digital Logic and Computer Design

K-Map Simplification

 After an SOP expression has been mapped, a minimum SOP expression is obtained by grouping the Is and determining the minimum SOP expression from the map.

 You can group Is on the Karnaugh map according to the following rules by enclosing those adjacent cells containing 1s.

 The goal is to maximize the size of the groups and to minimize the number of groups.

K-Maps

 Steps for Grouping

 A group must contain either 1, 2, 4, 8, or 16 cells, which are all powers of two. In the case of a 3- variable map, 2^3 = 8 cells is the maximum group.  Each cell in a group must be adjacent to one or more cells in that same group. but all cells in the group do not have to be adjacent to each other.  Always include the largest possible number of 1’s in a group in accordance with rule 1.  Each 1 on the map must be included in at least one group. The Is already in a group can be included in another group as long as the overlapping groups include noncommon 1’s.

For Example

K-Maps

K-Maps

Wrap around adjacency

The 4-Variable Karnaugh Map

 Determining the minimum term for each group

 For a 3-veriable map. (1) A 1-cell group yields a 3-variable product term (2) A 2-cell group yields a 2-variable product term (3) A 4-cell group yields a 1-variable term (4) An 8-cell group yields a value of 1 for the expression  For a 4-veriable map (1) A 1-cell group yields a 4-variable product term (2) A 2-cell group yields a 3-variable product term (3) A 4-cell group yields a 2-variable product term (4) An 8-cell group yields a 1-variable term (5) A 16-cell group yields a value of 1 for the expression

Note: When all the minimum product terms are derived from the Karnaugh map, they are summed to form the minimum SOP expression.

Related Example

•  F(x, y, z) = Σ(0,2,6,7)   Minimize the following SOP expression,
•  F(x, y, z) = Σ(0,2,3,4,6) 
•  F(x, y, z) = Σ(0,1,2,3,7) 
•  F(x, y, z) = Σ(3,5,6,7) 
•  F(x, y, z) = Σ(0,1,5,7) 
•  F(x, y, z) = Σ(0,1,6,7) 
•  F(x, y, z) = Σ(1,2,3,6,7) 

Solutions

 Solution

F(x, y, z) = Σ(0,2,3,4,6)  2

F = z' + x'y

Solution

 Solution

F(x, y, z) = Σ(0,1,2,3,7)  3

F = x' + y z

Solution of 5, 6, 7

5. F = x‘ y' + x z 7. F = y + x‘ z 6. F = x‘ y' + x y

K-Maps

 Example

 Determine the product terms for the Karnaugh map given and write the resulting minimum SOP expression?

Self Assessment

Problem: For the Karnaugh map on the previous slide, add a 1 in the lower right cell (1010) and determine the resulting SOP expression.

Example 1

a b