Kmap Slideshare
-
Upload
tech4us -
Category
Technology
-
view
6.141 -
download
4
description
Transcript of Kmap Slideshare
04/12/23 Karnaugh Maps 1
Kulachi Hansraj Model
School
By Ms. Nita Arora, PGT Computer Science
e-Lesson
04/12/23 Karnaugh Maps 2
Subject : Computer Science (083)
Unit : Boolean AlgebraTopic : Minimization of Boolean
Expressions Using Karnaugh Maps (K-Maps)
Category : Senior SecondaryClass : XII
04/12/23 Karnaugh Maps 3
Learning Objectives : After successfully completing this module students
should be able to:
Understand the Need to simplify (minimize) expressions
List Different Methods for Minimization Karnaugh Maps Algebraic method
Use Karnaugh Map method to minimize the Boolean expression
04/12/23 Karnaugh Maps 4
Previous Knowledge : The students should be familiar with the following terms in Boolean Algebra before going through this module on K-MAPS
xx
yy
x+yx+y
Boolean variable, Constants and Operators Postulates of Boolean Algebra Theorems of Boolean Algebra Logic Gates- AND, OR, NOT, NAND, NOR Boolean Expressions and related terms
MINTERM (Product Term) MAXTERM (Sum Term) Canonical Form of Expressions
04/12/23 Karnaugh Maps 5
MinimizationOf
Boolean Expressions
Who Developed it NEED For Minimization Different Methods What is K-Map Drawing a K-Map Minimization Steps Important Links Recap. K-Map Rules
(SOP Exp.)
INDEX
04/12/23 Karnaugh Maps 6
Who Developed K-Maps…
• Name: Maurice Karnaugh, a telecommunications engineer at Bell Labs. While designing digital logic based telephone switching circuits he developed a method for Boolean expression minimization.
• Year : 1950 same year that Charles M. Schulz published his first Peanuts comic.
04/12/23 Karnaugh Maps 7
Boolean expressions are practically implemented in the form of GATES (Circuits).
A minimized Boolean expression means less number of gates which means
Simplified Circuit
MINIMIZATION OF BOOLEAN EXPRESSION
WHY we Need to simplify (minimize) expressions?
04/12/23 Karnaugh Maps 8
MINIMIZATION OF BOOLEAN EXPRESSION
Different methods
Karnaugh Maps
Algebraic Method
04/12/23 Karnaugh Maps 9
Karnaugh MapsWHAT is Karnaugh Map (K-Map)?
A special version of a truth table
Karnaugh Map (K-Map) is a GRAPHICAL display of fundamental terms in a Truth Table.
Don’t require the use of Boolean Algebra theorems and equation
Works with 2,3,4 (even more) input variables (gets more and more difficult with more variables)
NEXT
04/12/23 Karnaugh Maps 10
K-maps provide an alternate way of simplifying logic circuits.
One can transfer logic values from a Truth Table into a K-Map.
The arrangement of 0’s and 1’s within a map helps in visualizing, leading directly to
Simplified Boolean Expression
Karnaugh Maps………(Contd.)
NEXT
04/12/23 Karnaugh Maps 11
Correspondence between the Karnaugh Map and the Truth Table
for the general case of a two Variable Problem
A B0 00 11 01 1
Fabcd
A \ B 0 1
0 a b
1 c d
Truth Table
2 Variable K-Map
Karnaugh Maps………(Contd.)
04/12/23 Karnaugh Maps 12
Drawing a Karnaugh Map (K-Map)
K-map is a rectangle made up of certain number of SQUARES
For a given Boolean function there are 2N squares where N is the number of variables (inputs)
In a K-Map for a Boolean Function with 2 Variables f(a,b) there will be 22=4 squares
Each square is different from its neighbour by ONE Literal
Each SQUARE represents a MAXTERM or MINTERMNEXT
04/12/23 Karnaugh Maps 13
Karnaugh maps consist of a set of 22 squares where 2 is the number of variables
in the Boolean expression being minimized.
Truth Table 2 Variable K-Map
Karnaugh Maps………(Contd.)
A \ B 0 1
0 0 1
1 1 11
A B F
0 0 0
0 1 1
1 0 1
1 1 1
Minterm
A’B’
A’B
A B’
A B
Maxterm
A + B
A + B’
A’ + B
A’ + B’
NEXT
04/12/23 Karnaugh Maps 14
For three and four variable expressions Maps with 23 = 8 and 24 = 16 cells are used.Each cell represents a MINTERM or a MAXTERM
4 Variable K-Map 24 = 16 Cells
Karnaugh Maps………(Contd.)
BCA
00 01 11 10
0
1
A B \ C D 00 01 11 1000
01
11
103 Variable K-Map
23 = 8 Cells
04/12/23 Karnaugh Maps 15
Minimization Steps (SOP Expression with 4 var.)
The process has following steps:
Draw the K-Map for given function as shown
Enter the function values into the K-Map by placing 1's and 0's into the appropriate Cells
A B \ C D 00 01 11 10
00 0 0
0 1
0 3
0 2
01 0 0 0 0
11 1 1 0 0
10 1 1 0 0
0 5
0 4
0 7
0 6
0 0
12 13 15 14
8 9 11 10
1
1 1
1
NEXT
04/12/23 Karnaugh Maps 16
Minimization Steps (SOP Expression)
Form groups of adjacent 1's. Make groups as large as possible.
Group size must be a power of two. i.e. Group of
• 8 (OCTET),
• 4 (QUAD),
• 2 (PAIR) or
• 1 (Single)
A B \ C D 00 01 11 10
00 0 0
0 1
0 3
0 2
01 0 0 0 0
11 1 1 0 0
10 1 1 0 0
0 5
0 4
0 7
0 6
0 0
12 13 15 14
8 9 11 10
NEXT
04/12/23 Karnaugh Maps 17
OCTET REDUCTION ( Group of 8:)
0011
0011
0011
0011W X
YZ 0 0 0 1 1 1 1 0 Y.Z Y.Z Y. Z Y. Z
0 0W.X
0 1W.X
1 1W.X
1 0W.X
OCTET
(m0,m1,m4,m5,m8, m9, m12,m13)
•The term gets reduced by 3 literals i.e. 3 variables change within the group of 8 ( Octets )
NEXT
04/12/23 Karnaugh Maps 18
OCTET REDUCTION ( Group of 8:)
0110
0110
0110
0110W X
YZ 0 0 0 1 1 1 1 0 Y.Z Y.Z Y. Z Y. Z
0 0W.X
0 1W.X
1 1W.X
1 0W.X
OCTET
(m1,m3,m5,m7,m9, m11, m13,m15)
NEXT
04/12/23 Karnaugh Maps 19
OCTET REDUCTION ( Group of 8:)
MAP ROLLING
OCTET(m0,m2,m4,m6,
m8, m10, m12,m14)
1001
1001
1001
1001W X
YZ 0 0 0 1 1 1 1 0 Y.Z Y.Z Y. Z Y. Z
0 0W.X
0 1W.X
1 1W.X
1 0W.X
0 1 3 2
4 57
6
12 13 1514
8 9 11 10
NEXT
04/12/23 Karnaugh Maps 20
OCTET REDUCTION ( Group of 8:)
0000
1111
1111
0000W X
YZ 0 0 0 1 1 1 1 0 Y.Z Y.Z Y. Z Y. Z
0 0W.X
0 1W.X
1 1W.X
1 0W.X
0 1 3 2
4 57
6
12 13 1514
8 9 11 10
OCTET
(m4,m5,m6,m7,m12, m13, m14,m15)
NEXT
04/12/23 Karnaugh Maps 21
OCTET REDUCTION ( Group of 8:)
MAP ROLLING
OCTET(m0,m1,m2,m3
M8,m9,m10,m11)
1111
0000
0000
1111W X
YZ 0 0 0 1 1 1 1 0 Y.Z Y.Z Y. Z Y. Z
0 0W.X
0 1W.X
1 1W.X
1 0W.X
0 1 3 2
4 57
6
12 13 1514
8 9 11 10
04/12/23 Karnaugh Maps 22
QUAD REDUCTION ( Group of 4)
1100
1111
0111
0110
0
WX
YZ
3 2
4 57 6
1
12 13 15 14
8 9 11 10
QUAD (m1,m3,m5,m7)
QUAD(m10,m11,m14,m15)
QUAD(m4,m5,m12,m13)
0 0 0 1 1 1 1 0 Y.Z Y.Z Y. Z Y. Z
0 0W.X
0 1W.X
1 1W.X
1 0W.X
•The term gets reduced by 2 literals i.e. 2 variables change within the group of 4( QUAD )
NEXT
04/12/23 Karnaugh Maps 23
QUAD REDUCTION ( Group of 4)
MAP ROLLING
QUAD (m1,m3,m9,m11)
QUAD(m4,m6,m12,m14)1110
1111
1111
0110
0
WX
YZ
3 2
4 57 6
1
12 13 15 14
8 9 11 10
0 0 0 1 1 1 1 0 Y.Z Y.Z Y. Z Y. Z
0 0W.X
0 1W.X
1 1W.X
1 0W.X
NEXT
04/12/23 Karnaugh Maps 24
QUAD REDUCTION ( Group of 4)
QUAD(m0,m2,m8,m10)
1001
0000
0000
1001
0
WX
YZ
3 2
4 57 6
1
12 13 15 14
8 9 11 10
0 0 0 1 1 1 1 0 Y.Z Y.Z Y. Z Y. Z
0 0W.X
0 1W.X
1 1W.X
1 0W.X
CORNER ROLLING
04/12/23 Karnaugh Maps 25
PAIR REDUCTION ( Group of 2)
YZ
MAP ROLLINGPAIR
(m0,m2)
0000
0000
0110
1001
0
WX
3 2
45 7 6
1
12 13 15 14
8 9 11 10
PAIR(m5,m7)
0 0 0 1 1 1 1 0 Y.Z Y.Z Y. Z Y. Z
0 0W.X
0 1W.X
1 1W.X
1 0W.X
•The term gets reduced by 1 literals i.e. 1 variables change within the group of 2( PAIR )
04/12/23 Karnaugh Maps 26
SINGLE CELL REDUCTION
1100
1101
0000
0010wx
yz00 01 11 10
00
01
11
10
SINGLE CELL (m1)
SINGLE CELL (m12)
QUAD
(m10,m11,m14,m15)
•The term is not reduced in a single cell
04/12/23 Karnaugh Maps 27NEXT
Karnaugh Maps - Rules of
Simplification(SOP Expression)
04/12/23 Karnaugh Maps 28
• Groups may not include any cell containing a zero
NEXT
Karnaugh Maps - Rules of Simplification (SOP Expression)
04/12/23 Karnaugh Maps 29
•Groups may be horizontal or vertical, but not diagonal.
NEXT
Karnaugh Maps - Rules of Simplification (SOP Expression)
04/12/23 Karnaugh Maps 30
• Groups must contain 1, 2, 4, 8, or in general 2n cells. • That is if n = 1, a group will contain two 1's since 21 = 2.• If n = 2, a group will contain four 1's since 22 = 4.
NEXT
Karnaugh Maps - Rules of Simplification (SOP Expression)
04/12/23 Karnaugh Maps 31
•Each group should be as large as possible.
NEXT
Karnaugh Maps - Rules of Simplification (SOP Expression)
04/12/23 Karnaugh Maps 32
•Each cell containing a 1 must be in at least one group.
NEXT
Karnaugh Maps - Rules of Simplification (SOP Expression)
04/12/23 Karnaugh Maps 33
•Groups may overlap.
NEXT
Karnaugh Maps - Rules of Simplification (SOP Expression)
04/12/23 Karnaugh Maps 34
• Groups may wrap around the table. • The leftmost cell in a row may be grouped with the rightmost cell and • The top cell in a column may be grouped with the bottom cell.
NEXT
Karnaugh Maps - Rules of Simplification (SOP Expression)
04/12/23 Karnaugh Maps 35
• There should be as few groups as possible, as long as this does not contradict any of the previous rules.
NEXT
Karnaugh Maps - Rules of Simplification
(SOP Expression)
04/12/23 Karnaugh Maps 36
1. No 0’s allowed in the groups. 2. No diagonal grouping allowed.3. Groups should be as large as possible. 4. Only power of 2 number of cells in each
group. 5. Every 1 must be in at least one group. 6. Overlapping allowed. 7. Wrap around allowed. 8. Fewest number of groups are considered. 9. Redundant groups ignored
Karnaugh Maps - Rules of Simplification
(SOP Expression)
04/12/23 Karnaugh Maps 37
Minimization Steps (SOP Expression)
Select the least number of groups that cover all the 1's.
1100
1101
0111
0110
0
wx
yz00 01 11 10
00
01
11
10
3 2
4 57 6
1
12 13 15 14
8 9 11 10
Ensure that every 1 is in a group.1's can be part of more than one group.
Eliminate Redundant Groups
NEXT
04/12/23 Karnaugh Maps 38
Example: Reduce f(wxyz)=Σ(1,3,4,5,7,10,11,12,14,15)
PAIR (m4,m5)REDUNDANTGROUP1100
1101
0111
0110
0
wx
yz00 01 11 10
00
01
11
10
3 2
4 57 6
1
12 13 15 14
8 9 11 10
QUAD (m1,m3,m5,m7)
QUAD(m10,m11,m14,m15)
QUAD(m3,m7,m11,m15)REDUNDANT Group
PAIR (m4,m12)
Minimized Expression : xy’z’ + wy + w’z
04/12/23 Karnaugh Maps 39
Happy Learning……….
04/12/23 Karnaugh Maps 40
References
For K-Map Minimizer Downloadhttp://karnaugh.shuriksoft.com
Thomas C. Bartee, DIGITAL COMPUTER FUNDAMENTALS, McGraw Hill International.
Computer Science (Class XII)By Sumita Arora
http://www.ee.surrey.ac.uk/Projects/Labview/minimisation/karrules.html
04/12/23 Karnaugh Maps 41
The End