the karnaugh map
TRANSCRIPT
THE THE KARNAUGH KARNAUGH
MAPMAP
GROUP MEMBERS
K MAP
• MAURICE KARNAUGH INTRODUCED K-MAP IN 1953• THE KARNAUGH MAP, ALSO KNOWN AS THE K-MAP, IS A METHOD TO SIMPLIFY
BOOLEAN ALGEBRA EXPRESSIONS.• THE KARNAUGH MAP REDUCES THE NEED FOR EXTENSIVE CALCULATIONS BY
TAKING ADVANTAGE OF HUMANS' PATTERN-RECOGNITION CAPABILITY.• THE REQUIRED BOOLEAN RESULTS ARE TRANSFERRED FROM A TRUTH TABLE
ONTO A TWO-DIMENSIONAL GRID WHERE THE CELLS ARE ORDERED IN GRAY CODE, AND EACH CELL POSITION REPRESENTS ONE COMBINATION OF INPUT CONDITIONS, WHILE EACH CELL VALUE REPRESENTS THE CORRESPONDING OUTPUT VALUE. OPTIMAL GROUPS OF 1S OR 0S ARE IDENTIFIED.
• THESE TERMS CAN BE USED TO WRITE A MINIMAL BOOLEAN EXPRESSION REPRESENTING THE REQUIRED LOGIC.
• ADVANTAGES• REDUCES EXTENSIVE CALC.• REDUCES EXPRESSION WITHOUT BOOLEAN THEOREMS• USED FOR MINIMIZING CIRCUITS• LESS TIME CONSUMING• LESS SPACE CONSUMING
• DISADVANTAGES• TEDIOUS FOR MORE THAN 5 VARIABLES• SOME EXAMPLES ARE SOLVED IN FEW SECONDS BY BOOLEAN THEOREMS EASILY
THE KARNAUGH MAPTHE KARNAUGH MAP
• NUMBER OF SQUARES = NUMBER OF COMBINATIONS
• EACH SQUARE REPRESENTS A MINTERM
• 2 VARIABLES 4 SQUARES
• 3 VARIABLES 8 SQUARES
• 4 VARIABLES 16 SQUARES
THE 3 VARIABLE K-MAP
• THERE ARE 8 CELLS AS SHOWN:CC
ABAB 00 11
0000
0101
1111
1010
CBA CBA
CBA BCA
CAB ABC
CBA CBA
THE 4-VARIABLE K-MAP
CDCDABAB 0000 0101 1111 1010
0000
0101
1111
1010 DCBA
DCAB
DCBA
DCBA
DCBA
DCAB
DCBA
DCBA
CDBA
ABCD
BCDA
CDBA
DCBA
DABC
DBCA
DCBA
ADJACENT SQUARES
• A LARGER NUMBER OF ADJACENT SQUARES ARE COMBINED, WE OBTAIN A PRODUCT TERM WITH FEWER LITERALS.1 SQUARE = 1 MINTERM = THREE LITERALS.
2 ADJACENT SQUARES = 1 TERM = TWO LITERALS.
4 ADJACENT SQUARES = 1 TERM = ONE LITERAL.
8 ADJACENT SQUARES ENCOMPASS THE ENTIRE MAP AND PRODUCE A FUNCTION THAT IS ALWAYS EQUAL TO 1.
• IT IS OBVIOUSLY TO KNOW THE NUMBER OF ADJACENT SQUARES IS COMBINED IN A POWER OF TWO SUCH AS 1,2,4, AND 8.
8
CELL ADJACENCY
CDCDABAB
0000 0101 1111 1010
0000010111111010
K-MAP SOP MINIMIZATION
• THE K-MAP IS USED FOR SIMPLIFYING BOOLEAN EXPRESSIONS TO THEIR MINIMAL FORM.
• A MINIMIZED SOP EXPRESSION CONTAINS THE FEWEST POSSIBLE TERMS WITH FEWEST POSSIBLE VARIABLES PER TERM.
• GENERALLY, A MINIMUM SOP EXPRESSION CAN BE IMPLEMENTED WITH FEWER LOGIC GATES THAN A STANDARD EXPRESSION.
MAPPING A STANDARD SOP EXPRESSION
FOR AN SOP EXPRESSION IN STANDARD FORM: A 1 IS PLACED ON THE K-
MAP FOR EACH PRODUCT TERM IN THE EXPRESSION.
EACH 1 IS PLACED IN A CELL CORRESPONDING TO THE VALUE OF A PRODUCT TERM.
EXAMPLE: FOR THE PRODUCT TERM , A 1 GOES IN THE 101 CELL ON A 3-VARIABLE MAP.
CBA
CCABAB 00 11
0000
0101
1111
1010
CBA CBA
CBA BCA
CAB ABC
CBA CBA1
CCABAB 00 11
0000
0101
1111
1010
MAPPING A STANDARD SOP EXPRESSION (FULL EXAMPLE)
THE EXPRESSION: CBACABCBACBA
000 001 110 100
1 1
1
1
MAPPING A NONSTANDARD SOP EXPRESSION
• A BOOLEAN EXPRESSION MUST BE IN STANDARD FORM BEFORE YOU USE A K-MAP.• IF ONE IS NOT IN STANDARD FORM, IT MUST BE
CONVERTED.
• YOU MAY USE THE PROCEDURE MENTIONED EARLIER OR USE NUMERICAL EXPANSION.
MAPPING A NONSTANDARD SOP EXPRESSION
NUMERICAL EXPANSION OF A NONSTANDARD PRODUCT TERMASSUME THAT ONE OF THE PRODUCT TERMS IN A
CERTAIN 3-VARIABLE SOP EXPRESSION IS . IT CAN BE EXPANDED NUMERICALLY TO STANDARD
FORM AS FOLLOWS: STEP 1: WRITE THE BINARY VALUE OF THE TWO VARIABLES
AND ATTACH A 0 FOR THE MISSING VARIABLE : 100. STEP 2: WRITE THE BINARY VALUE OF THE TWO VARIABLES
AND ATTACH A 1 FOR THE MISSING VARIABLE : 101. THE TWO RESULTING BINARY NUMBERS ARE THE
VALUES OF THE STANDARD SOP TERMS AND .
C
C
BA
CBA CBA
K-MAP SIMPLIFICATION OF SOP EXPRESSIONS
• AFTER AN SOP EXPRESSION HAS BEEN MAPPED, WE CAN DO THE PROCESS OF MINIMIZATION:
• GROUPING THE 1S• DETERMINING THE MINIMUM SOP EXPRESSION FROM THE
MAP
GROUPING THE 1S
• YOU CAN GROUP 1S ON THE K-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.
GROUPING THE 1S (EXAMPLE)
CCABAB 00 11
0000 11
0101 11
1111 11 11
1010
CCABAB 00 11
0000 11 11
0101 11
1111 11
1010 11 11
GROUPING THE 1S (EXAMPLE)CDCD
ABAB 0000 0101 1111 1010
0000 11 11
0101 11 11 11 11
1111
1010 11 11
CDCDABAB 0000 0101 1111 1010
0000 11 11
0101 11 11 11
1111 11 11 11
1010 11 11 11
DETERMINING THE MINIMUM SOP EXPRESSION FROM THE
MAP• THE FOLLOWING RULES ARE APPLIED TO FIND THE MINIMUM
PRODUCT TERMS AND THE MINIMUM SOP EXPRESSION:1. GROUP THE CELLS THAT HAVE 1S. EACH GROUP OF
CELL CONTAINING 1S CREATES ONE PRODUCT TERM COMPOSED OF ALL VARIABLES THAT OCCUR IN ONLY ONE FORM (EITHER COMPLEMENTED OR COMPLEMENTED) WITHIN THE GROUP. VARIABLES THAT OCCUR BOTH COMPLEMENTED AND UNCOMPLEMENTED WITHIN THE GROUP ARE ELIMINATED CALLED CONTRADICTORY VARIABLES.
DETERMINING THE MINIMUM SOP EXPRESSION FROM THE
MAP2. DETERMINE THE MINIMUM PRODUCT TERM FOR EACH GROUP.• FOR A 3-VARIABLE MAP:
1. A 1-CELL GROUP YIELDS A 3-VARIABLE PRODUCT TERM2. A 2-CELL GROUP YIELDS A 2-VARIABLE PRODUCT TERM3. A 4-CELL GROUP YIELDS A 1-VARIABLE PRODUCT TERM4. AN 8-CELL GROUP YIELDS A VALUE OF 1 FOR THE EXPRESSION.
• FOR A 4-VARIABLE MAP:1. A 1-CELL GROUP YIELDS A 4-VARIABLE PRODUCT TERM2. A 2-CELL GROUP YIELDS A 3-VARIABLE PRODUCT TERM3. A 4-CELL GROUP YIELDS A 2-VARIABLE PRODUCT TERM4. AN 8-CELL GROUP YIELDS A A 1-VARIABLE PRODUCT TERM5. A 16-CELL GROUP YIELDS A VALUE OF 1 FOR THE EXPRESSION.
DETERMINING THE MINIMUM SOP EXPRESSION FROM THE
MAP3. WHEN ALL THE MINIMUM PRODUCT TERMS ARE DERIVED
FROM THE K-MAP, THEY ARE SUMMED TO FORM THE MINIMUM SOP EXPRESSION.
DETERMINING THE MINIMUM SOP EXPRESSION FROM THE MAP (EXAMPLE)
CDCDABAB 0000 0101 1111 1010
0000 11 110101 11 11 11 111111 11 11 11 111010 11
BCA
DCA
DCACAB
DETERMINING THE MINIMUM SOP EXPRESSION FROM THE MAP (EXERCISES)
CBABCAB ACCAB
CCABAB 00 11
0000 11
0101 11
1111 11 11
1010
CCABAB 00 11
0000 11 11
0101 11
1111 11
1010 11 11
DETERMINING THE MINIMUM SOP EXPRESSION FROM THE MAP
(EXERCISES)
DBACABA CBCBAD
CDCDABAB 0000 0101 1111 1010
0000 11 11
0101 11 11 11 11
1111
1010 11 11
CDCDABAB 0000 0101 1111 1010
0000 11 11
0101 11 11 11
1111 11 11 11
1010 11 11 11
MAPPING DIRECTLY FROM A TRUTH TABLEI/PI/P O/PO/P
AA BB CC XX00 00 00 1100 00 11 0000 11 00 0000 11 11 0011 00 00 1111 00 11 0011 11 00 1111 11 11 11
CCABAB 00 11
0000
0101
1111
1010
1
11
1