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

Spring 2015 Week 2 Module 8 Digital Circuits and Systems Karnaugh Maps Shankar Balachandran* Associate Professor, CSE Department Indian Institute of Technology Madras *Currently a Visiting Professor at IIT Bombay

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 2

 1 variable K-map: f ( a ) a m 0 0 1 0 OR m 0 m 1 a m 1 1  2 variable K-map: f(a,b) b a f(a,b) f(a,b) a b b a 0 1 0 1 m 00 m 01 0 m 00 m 10 0 OR m 10 m 11 a m 01 m 11 1 b 1 Karnaugh Maps 3

 3-variable K-map: f(a,b,c) f(a,b,c) b bc a 00 01 11 10 m 000 m 001 m 011 m 010 0 OR a m 100 m 101 m 111 m 110 1 f(a,b,c) a ab c c 00 01 11 10 m 000 m 110 m 100 m 010 0 m 101 c m 001 m 011 m 111 1 b Karnaugh Maps 4

 4-variable K-map: f(a,b,c,d) f(a,b,c,d) c f(a,b,c,d) a cd ab ab 00 01 11 10 cd 00 01 11 10 m 0000 m 0001 m 0011 m 0010 m 0100 m 1100 m 1000 m 0000 00 00 OR m 0100 m 0101 m 0111 m 0110 01 m 1001 m 0001 m 0101 m 1101 01 b d m 1101 m 1111 m 1110 m 1100 m 1111 m 1011 m 0011 m 0111 11 11 a c m 1000 m 1001 m 1011 m 1010 10 m 0010 m 0110 m 1110 m 1010 10 d b Karnaugh Maps 5

K-map Example 1    f a bc d f ( a , b , c , d ) cd ab 00 01 11 10 00 d d bc bc 01 d d a bc a a a bc 11 d d a a a a 10 d d Karnaugh Maps 6

   f a bc d f ( a , b , c , d ) cd ab 00 01 11 10 1 1 00 d d bc bc 1 01 1 1 d d a bc a a a bc 11 1 1 1 1 d d a a a a 10 1 1 1 1 d d Karnaugh Maps 7

K- map Example 1’    f a bc d f ( a , b , c , d ) ab cd 00 01 11 10 a a 00 d d d d a a 01 bc a bc a 11 bc a bc a 10 d d d d Note: This K-map is drawn by swapping the placement of variable pairs ab and cd Karnaugh Maps 8

   f a bc d f ( a , b , c , d ) ab cd 01 11 10 00 a a 1 1 1 1 00 d d d d a a 01 0 0 1 1 bc a bc a 11 0 1 1 1 bc a bc a 10 1 1 1 1 d d d d Karnaugh Maps 9

K-map Example 2    f ( w , x , y , z ) w z x y x f ( w , x , y , z ) yz wx 00 01 11 10 00 01 11 10 Karnaugh Maps 10

K-map Example 2    f ( w , x , y , z ) w z x y x f ( w , x , y , z ) yz wx 00 01 11 10 1 1 1 1 00 x x x x xy xy 01 1 1 wz xy xy wz 11 1 1 1 wz wz 10 1 1 1 1 x x x x Karnaugh Maps 11

K-Map Properties  Minterms mapped to any two adjacent cells differ in exactly one bit position   f ( w , x , y , z ) ( 0 , 2 , 3 , 4 , 6 , 8 , 10 , 11 ) Example f ( w , x , y , z ) wx yz 00 01 11 10 1 1 1 00 1 1 01 11 1 1 1 10 Karnaugh Maps 12

 The sum of two minterms in adjacent cells can be simplified to a single product (AND) term with one less variable . Example f ( a , b , c ) bc a 00 01 11 10 1 1 1 0 1 1 1 If we combine adjacent minterms in the first column we get,       a b c a b c b c a a b c That is, variable a is eliminated. Karnaugh Maps 13

Example f ( a , b , c ) bc a 00 01 11 10 1 1 1 0 1 1 1 If we combine adjacent minterms like what is shown above we get,       abc a bc ab c c ab That is, variable c is eliminated. Karnaugh Maps 14

End of Week 2: Module 8 Thank You Karnaugh Maps 15