lecture 9: karnaugh maps - electrical engineeringwlu/teaching/winter/lecture 9.pdf · boolean...

Computer Organization I

Lecture 9: Karnaugh Maps

� What is Karnaugh Map?

� Truth Table and Karnaugh Map

� Logic Function and Karnaugh Map

� Apply Karnaugh Map to Simplify Logic Function

Overview

Objectives

� Understand what is Karnaugh map

� To know how to simplify logic function using Karnaugh map

Boolean Function Optimization- Approach 2: Karnaugh Maps (K-map)

• Why use K-map?

Provides a systematic means to obtain optimum or near optimum

SOP and POS forms for any given functions with small numbers

of variables (usually number of variables no more than 4).

• What is K-map? K-map is an array of squares, in which each square represents a

minterm; given Num. of Variables n, total num. of squares is 2n

y

m1m0

m2 m3

x

0

1

0 1y z

m1m0 m2

m4

m3

m5 m6m7

x

0

1

00 01 11 10


• What is physical adjacence in K-map? – In K-map, each square is physically adjacent to the

squares that are immediately next to it on any of its four side; e.g.

In 2 variable K-map, m1 and m0 are physically adjacent

y

m1m0

m2 m3

x

0

1

0 1

y z

m1m0 m2

m4

m3

m5 m6m7

x

0

1

00 01 11 10


• What is logical adjacence in K-map? – In K-map two squares are logically adjacent if the two squares

differ in only one variable which appears uncomplemented in one square and complemented in the other square; e.g.

In 2 variable K-map, m1 and m0 are logically adjacent

y

m1m0

m2 m3

x

0

1

0 1

y z

m1m0 m2

m4

m3

m5 m6m7

x

0

1

00 01 11 10


• Main feature of K-map? (1) Any two squares are physically adjacent means they are

logically adjacent (i.e.), e.g.

(2) Logically, squares in the top row are adjacent to the

corresponding row in the bottom, and

Similarly, squares in the outer left are logically adjacent to the

corresponding column in the outer right.

y

m1m0

m2 m3

x

0

1

0 1

y z

m1m0 m2

m4

m3

m5 m6m7

x

0

1

00 01 11 10


• Truth Table and K-map? (1) K-map is just a different form of truth table, e.g.

Truth Table of F(A,B) K-map of F(A,B)

In K-map, “1” means the corresponding minterm exists

B

11

0 1

A

0

1

0 1

A B

0 0

0 1

1 0

1 1

F (A,B)

1

1

0

1


• Example: given truth table, to draw K-map?

A B C F(ABC)

0 0 0 1

0 0 1 0

0 1 0 1

0 1 1 0

1 0 0 0

1 0 1 1

1 1 0 0

1 1 1 1

B C

01 1

0

0

1 01

A

0

1

00 01 11 10


• K-map and Logic Function Expression? given K-map, logic function expression can be represented

by the logic sum (i.e. OR) of minterms corresponding to

“1” in K-map. For example:

F(A,B) = m0 + m1 + m3

= AB + AB + AB

= A + AB

= A + B

B

11

0 1

A

0

1

0 1


• K-map and Logic Function Expression (cont.) given logic function expression, to draw K-map. e.g.

If the function is in SOM, i.e. F (A,B,C) = ƩƩƩƩm (4,6,7)

If the function is in SOP, i.e. F (A,B,C) = AB + AC, then convert it

into SOM expression…or

write truth table first and then based on truth table, draw K-map

B C

00 0

1

0

0 11

A

0

1

00 01 11 10


• K-map simplification: combine 2 adjacent squares

CD

00 0

0

0

0 01

AB

0

1

00 01 11 10

00 0

0

1

0 00

1

0

0

0

1

1

A changed

after combination, the function

is simplified into BCD



CD

00 0

0

0

1 01

AB

0

1

00 01 11 10

10 0

0

1

0 00

1

0

0

0

1

1

A changed


is simplified into BD

C changed



CD

00 0

1

0

1 11

AB

0

1

00 01 11 10

11 1

0

1

0 00

1

0

0

0

1

1

A changed


is simplified into B

Both C and D changed


• Combination of adjacent squares for n variables K-map(1) in order to do simplification, the number of adjacent squares

in the circle must be 2i , i = 1,2,…

(2) by combining the 2i adjacent squares in the circle, there are i

variable to be eliminated. So, given n variable, the number of

variable after the 2i combination simplification is (n-i).


• Steps to simplify function using K-map(1) draw K-map based on given logic function

(2) circle all adjacent squares whose value is 1, note: the number

of squares in the circle is 2,4,8…

(3) the size of single circle should be as big as possible, and the

total number of circles should be as less as possible

(4) one square in a circle can be included in the other circles, but

in each circle, it should have at least one square that is circled

only one time

(5) the simplified logic function is the logic sum (OR) of simplified

terms represented by corresponding circles


• Example 1: F (A,B,C) = ƩƩƩƩm (0,1,2,4,6,7),

F (A,B,C) = AB + BC + AB + BC = AB + AB + C

B C

11 1

1

0

0 11

A

0

1

00 01 11 10


• Example 2: F (A,B,C,D) = ƩƩƩƩm (0,2,3,6,7,8,10,11),

F (A,B,C,D) = AC + BC + BD

CD

01 1

0

1

0 11

AB

0

1

00 01 11 10

00 0

1

0

0 11

1

0

0

0

1

1



F (A,B,C,D) = ?

CD

01 0

1

0

1 11

AB

0

1

00 01 11 10

01 1

0

1

0 00

1

0

0

0

1

1



F (A,B,C,D) = ?

CD

10 0

0

0

1 11

AB

0

1

00 01 11 10

11 0

0

1

0 01

1

0

0

0

1

1


• What is “Don’t Care” in K-mapSometimes a function table or map contains entries for which it

is known:

(1) the input values for the minterm will never occur, or

(2) the output value for the minterm is not used, i.e. the value

can be either 0 or 1

In these cases, these entries (in map) or terms (in function) are

defined “don't cares”

By placing “don't cares” into “x” (in map) or “d” in function, we

can simplify the corresponding logic expression.


• Simplifying logic function with “Don’t Care”For example: given logic function F (A,B,C,D) = ƩƩƩƩm

(0,1,5,7,8,11,14) + ƩƩƩƩd(3,9,12,15)

F = BC + AD + CD + ABC

or,

F = BC + AD + CD + ABD

CD

11 0

0

x

1 01

AB

0

1

00 01 11 10

0x 1

1

x

x 01

1

0

0

0

1

1


• Simplifying logic function with “Don’t Care”For example: given logic function F (A,B,C,D) = ƩƩƩƩm

(0,3,4,7,11) + ƩƩƩƩd(8,9,12,13,14,15)

F = CD + CD

CD

01 0

1

1

0 01

AB

0

1

00 01 11 10

xx x

x

x

x 01

1

0

0

0

1

1

Summary

• Applying Karnaugh map to simplify logic function with no more than 4 variables

Thank youQ & A

lecture 9: karnaugh maps - electrical engineeringwlu/teaching/winter/lecture 9.pdf · boolean...

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