karnaugh map or kmap

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Digital System Design

Presentation : 1

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Introduction

Name: MD. Wasim Akram

ID: UG 02 22 09 016

Department: CSE

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Topic

Karnaugh Map

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What is Karnaugh Map

•The Karnaugh map (K-map for short), Maurice Karnaugh's 1953 refinement of Edward Veitch's 1952 Veitch diagram, 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, also permitting the rapid identification and elimination of potential race conditions.

•Definition: A Karnaugh map provides a pictorial method of grouping together expressions with common factors and therefore eliminating unwanted variables. The Karnaugh map can also be described as a special arrangement of a truth table.•A visual way to simplify logic expressions•It gives the most simplified form of the expression

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Rules to obtain the most simplified expression

Simplification of logic expression using Boolean algebra is awkward because:

it lacks specific rules to predict the most suitable next step in the simplification process

it is difficult to determine whether the simplest form has been achieved.

A Karnaugh map is a graphical method used to obtained the most simplified form of an expression in a standard form (Sum-of-Products or Product-of-Sums).

The simplest form of an expression is the one that has the minimum number of terms with the least number of literals (variables) in each term.

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Continued

By simplifying an expression to the one that uses the minimum number of terms, we ensure that the function will be implemented with the minimum number of gates.

By simplifying an expression to the one that uses the least number of literals for each terms, we ensure that the function will be implemented with gates that have the minimum number of inputs

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Simplification Process in Bullet Points:

No zeros allowed.

No diagonals.

Only power of 2 number of cells in each group.

Groups should be as large as possible.

Every one must be in at least one group.

Overlapping allowed.

Wrap around allowed.

Fewest number of groups possible.

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Three-Variable K-Maps

C B(0,4)f BA (4,5)f B(0,1,4,5)f A(0,1,2,3)f

BC00

0

01

1

11 10A

1 0 0 0

1 0 0 0

BC00

0

01

1

11 10A

0 0 0 0

1 1 0 0

BC00

0

01

1

11 10A

1 1 1 1

0 0 0 0

BC00

0

01

1

11 10A

1 1 0 0

1 1 0 0

C A(0,4)f CA (4,6)f C A(0,2)f C(0,2,4,6)f

BC00

0

01

1

11 10A

0 1 1 0

0 0 0 0

BC00

0

01

1

11 10A

0 0 0 0

1 0 0 1

BC00

0

01

1

11 10A

1 0 0 1

1 0 0 1

BC00

0

01

1

11 10A

1 0 0 1

0 0 0 0

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Four-Variable K-Maps

DCB(0,8)f DCB(5,13)f DBA(13,15)f DBA(4,6)f

CA(2,3,6,7)f DB)(4,6,12,14f CB)(2,3,10,11f DB(0,2,8,10)f

CD00

00

01

01

11

11

10

10

AB1 0 0 0

0 0 0 0

0 0 0 0

1 0 0 0

CD00

00

01

01

11

11

10

10

AB0 0 0 0

0 1 0 0

0 1 0 0

0 0 0 0

CD00

00

01

01

11

11

10

10

AB0 0 0 0

0 0 0 0

0 1 1 0

0 0 0 0

CD00

00

01

01

11

11

10

10

AB0 0 0 0

1 0 0 1

0 0 0 0

0 0 0 0

CD00

00

01

01

11

11

10

10

AB0 0 1 1

0 0 1 1

0 0 0 0

0 0 0 0

CD00

00

01

01

11

11

10

10

AB0 0 0 0

1 0 0 1

1 0 0 1

0 0 0 0

CD00

00

01

01

11

11

10

10

AB0 0 1 1

0 0 0 0

0 0 0 0

0 0 1 1

CD00

00

01

01

11

11

10

10

AB1 0 0 1

0 0 0 0

0 0 0 0

1 0 0 1

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Design of combinational digital circuits

Steps to design a combinational digital circuit:

From the problem statement derive the truth table

From the truth table derive the unsimplified logic expression

Simplify the logic expression

From the simplified expression draw the logic circuit

Example: Design a 3-input (A,B,C) digital circuit that will give at its output (X) a logic 1 only if the binary number formed at the input has more ones than zeros

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Continued

BCABACX

A B C

0

0

0

0

0

0

0

0

0

0

0

0

1

1

1

1

1

1

1

1

1

1

1

1

X

0

0

0

1

0

1

1

1

Inputs Output

0

1

2

3

4

5

6

7

BC00

0

01

1

11 10A0 0 1 0

0 1 1 1

A B C

X

7) 6, 5, (3,X

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Don't cares Functions

Karnaugh maps also allow easy minimizations of functions whose truth tables include "don't care" conditions (that is, sets of inputs for which the designer doesn't care what the output is) because "don't care" conditions can be included in a circled group in an effort to make it larger. They are usually indicated on the map with a dash or X.

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What are the Advantages of Karnaugh Maps? The primary advantage of a Karnaugh map is the data representation’s

simplicity. The map’s grid structure simplifies the arrangement of similar variables that can group and disperse like terms to identify and resolve potential issues in the design.

Changes in neighboring variables are easily displayed, permitting the engineer to view cause and effect relationships.

Easy and Convenient to implement.

As a result, digital circuit and information theory industries still use Karnaugh maps today.

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Referrence

Wikipedia.com

Digital System Design Book

Digital Design by Morris Mano

Verilog Digital System Design by Zainalabedin Navabi

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Conclusion

To conclude this presentation, I would like to say Thank you Madam and the Class….

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Questions?

????

karnaugh map or kmap

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