Digital Circuits and Systems Karnaugh Maps Shankar Balachandran* - - PowerPoint PPT Presentation

Shankar Balachandran* Associate Professor, CSE Department Indian Institute of Technology Madras

*Currently a Visiting Professor at IIT Bombay

Digital Circuits and Systems

Spring 2015 Week 2 Module 8

Karnaugh Maps

Karnaugh Maps 2

Karnaugh Maps

 Truth tables are a convenient form to represent equations

but they don’t aid in the simplification of logic equations.

 Karnaugh maps (K-maps) are similar to TT’s and they

lead to graphical methods for boolean expression simplification.

 A K-map is a multi-dimensional tabulation of function

values.

 Each minterm is assigned an entry (a cell) in the table.

The cell contains the value of the function for the corresponding minterm.

Karnaugh Maps 3

 1 variable K-map: f(a)  2 variable K-map: f(a,b)

m0 m1

1

m0 m1

1

a a OR m00 m01

1

b m10 m11

1

a

f(a,b)

a b OR m00 m01

1

a m10 m11

1

b

f(a,b)

b a

Karnaugh Maps 4

 3-variable K-map: f(a,b,c) m000 m001 m010 m011 m100 m101 m110 m111 00 01

c

1

a b bc OR

11 10 f(a,b,c)

a

m000 m001 m010 m011 m100 m101 m110 m111 00 01

b

1

c a ab

11 10 f(a,b,c)

c

Karnaugh Maps 5

 4-variable K-map: f(a,b,c,d) m0000 m0001 m0010 m0011 m0100 m0101 m0110 m0111 00 01

d

00 01

ab c cd OR

11 10 f(a,b,c,d)

a

m1100 m1101 m1110 m1111 m1000 m1001 m1010 m1011 11 10

b

m0000 m0001 m0010 m0011 m0100 m0101 m0110 m0111 00 01

d

00 01

cd c ab

11 10 f(a,b,c,d)

a

m1100 m1101 m1110 m1111 m1000 m1001 m1010 m1011 11 10

b

Karnaugh Maps 6

K-map Example 1

d bc a f   

) d , c , b , a ( f

00

01 11 10

00 01 11 10

ab cd a a a a a a a a bc bc bc bc d d d d d d d d

Karnaugh Maps 7

d bc a f   

) d , c , b , a ( f

00

01 11 10

00 01 11 10

ab cd a a a a a a a a bc bc bc bc d d d d d d d d 1 1 1 1 1 1 1 1 1 1 1 1 1

Karnaugh Maps 8

K-map Example 1’

) d , c , b , a ( f

00

01 11 10

00 01 11 10

d bc a f   

ab cd

Note: This K-map is drawn by swapping the placement of variable pairs ab and cd

a a a a a a a a bc bc bc bc d d d d d d d d

Karnaugh Maps 9

) d , c , b , a ( f

00

01 11 10

00 01 11 10

ab cd a a a a a a a a bc bc bc bc d d d d d d d d 1 1 1 1 1 1 1 1 1 1 1 1 1 d bc a f   

Karnaugh Maps 10

K-map Example 2

x y x z w ) z , y , x , w ( f   

) z , y , x , w ( f

00

01 11 10

00 01 11 10

wx yz

Karnaugh Maps 11

K-map Example 2

x y x z w ) z , y , x , w ( f   

) z , y , x , w ( f

00

01 11 10

00 01 11 10

wx yz wz wz wz wz xy xy xy xy x x x x x x x x 1 1 1 1 1 1 1 1 1 1 1 1 1

Karnaugh Maps 12

K-Map Properties

 Minterms mapped to any two adjacent cells differ in

exactly one bit position

  ) , , , , , , , ( ) z , y , x , w ( f 11 10 8 6 4 3 2 Example

00

01 11 10

00 01 11 10

) z , y , x , w ( f wx yz 1 1 1 1 1 1 1 1

Karnaugh Maps 13

 The sum of two minterms in adjacent cells can be

simplified to a single product (AND) term with one less variable.

) c , b , a ( f Example a bc

00 01 11 10

1 1 1

1

1 1 If we combine adjacent minterms in the first column we get, That is, variable a is eliminated.

 

c b a a c b c b a c b a    

Karnaugh Maps 14

) c , b , a ( f Example a bc

00 01 11 10

1 1 1

1

1 1 If we combine adjacent minterms like what is shown above we get, That is, variable c is eliminated.

 

abc a bc ab c c ab    

End of Week 2: Module 8

Thank You

Karnaugh Maps 15