Lecture Slides Kmap

8/12/2019 Lecture Slides Kmap

1/72

Chapter 3

K-Map


2/72

Karnaugh Maps (K-Map)

A K-Map is a graphical representationof a logic functions truth table


3/72

Relationship to Venn Diagrams

a b

ab

ab ab

ab


4/72


a b

0m

2m 1m

3m


5/72


0m

1

m

2m

3mb

a


6/72


0

1

2

3b

a


7/72


8/72

Two-Variable K-Map

0 1

0

1

a

b


9/72

Three-Variable K-Map

ab

c 00 01 11 10

0

1

0m

1m

2m

3m

6m

7m

4m

5m


10/72


ab

c 00 01 11 10

0

1


11/72


ab

c 00 01 11 10

0

1

Edges are adjacent


12/72

Four-variable K-Map

abcd 00 01 11 10

00

01

11

10

0m

1m

2m

3

m

6m

7

m

4m

5m

12m

13m

14m

15m

10m

11m

8m

9m


13/72

Four-variable K-Map

abcd 00 01 11 10

00

01

11

10


14/72

Four-variable K-Map

abcd 00 01 11 10

00

01

11

10

Edges are adjacent

E

dges

are

adja

cent


15/72

Plotting Functions on theK-map

SOP Form


16/72

Canonical SOP Form

Three Variable Example

F ABC ABC ABC ABC

using shorthand notation

6 3 1 5F m m m m

, , 1, 3, 5, 6F A B C m


17/72

Three-Variable K-Map Example

ab

c 00 01 11 10

0

1

, , 1, 3, 5, 6F a b c m

Plot 1s (minterms) of switching function

1 1 1

1


18/72


ab

c 00 01 11 10

0

1

, ,F a b c ab bc

Plot 1s (minterms) of switching function

1 1 1

1 abbc


19/72


20/72

Karnaugh Maps (K-Map)

Simplification of Switching Functions

using K-MAPS


21/72

Terminology/Definition

Literal

A variable or its complement

Logically adjacent terms

Two minterms are logically adjacent ifthey differ in only one variable position

Ex:

abc abcandm6 and m2 are logically adjacent

Note: abc abc a a bc bc Or, logically adjacent terms can be combined


22/72

Terminology/Definition

Implicant Product term that could be used to cover

minterms of a function

Prime Implicant An implicant that is not part of another

implicant

Essential Prime Implicant An implicant that covers at least one minterm

that is not contained in another prime

implicant Cover

A minterm that has been used in at least onegroup


23/72

Guidelines for Simplifying Functions

Each square on a K-map of nvariables has n logically adjacentsquares. (i.e. differing in exactly

one variable) When combing squares, always

group in powers of 2m, wherem=0,1,2,.

In general, grouping 2m variableseliminates m variables.


24/72

Guidelines for Simplifying Functions

Group as many squares as possible.This eliminates the most variables.

Make as few groups as possible.Each group represents a separateproduct term.

You must covereach minterm at

least once. However, it may becovered more than once.


25/72

K-map Simplification Procedure

Plot the K-map

Circle allprime implicants on the K-map

Identify and select all essentialprime implicants for the cover.

Select a minimum subset of the

remaining prime implicants tocomplete the cover.

Read the K-map


26/72

Example

Use a K-Map to simplify thefollowing Boolean expression

, , 1, 2, 3, 5, 6F a b c m


27/72


ab

c 00 01 11 10

0

1

Step 1: Plot the K-map

1 1 1

1

, , 1, 2, 3, 5, 6F a b c m

1


28/72


ab

c 00 01 11 10

0

1

Step 2:Circle ALLPrime Implicants

1 1 1

1

, , 1, 2, 3, 5, 6F a b c m

1


29/72


ab

c 00 01 11 10

0

1

Step 3: Identify Essential Prime Implicants

1 1 1

1

, , 1, 2, 3, 5, 6F a b c m

1

EPI

EPI

PI

PI


30/72


ab

c 00 01 11 10

0

1

Step 4: Select minimum subset of remainingPrime Implicants to complete the cover.

1 1 1

1

, , 1, 2, 3, 5, 6F a b c m

1

EPIPI

EPI


31/72


ab

c 00 01 11 10

0

1

Step 5: Read the map.

1 1 1

1

, , 1, 2, 3, 5, 6F a b c m

1

bcab

bc


32/72

Solution

, ,F a b c ab bc bc ab b c


33/72

Example


, , 2, 3, 6, 7F a b c m


34/72


35/72


36/72


ab

c 00 01 11 10

0

1


EPIEPI

, , 2, 3, 6, 7F a b c m

11

11

Wrong!!

We reallyshould drawA circle aroundall four 1s


37/72


ab

c 00 01 11 10

0

1

Step 4: Select Remaining Prime Implicants tocomplete the cover.

EPIEPI

11

11

, , 2, 3, 6, 7F a b c m


38/72


39/72

Solution

, ,F a b c ab ab b

Since we can still simplify the function

this means we did not use the largestpossible groupings.


40/72


ab

c 00 01 11 10

0

1

Step 2:Circle Prime Implicants

11

11

, , 2, 3, 6, 7F a b c m

Right!


41/72


ab

c 00 01 11 10

0

1


EPI

, , 2, 3, 6, 7F a b c m

11

11


42/72


ab

c 00 01 11 10

0

1

Step 5: Read the map.

b

11

11

, , 2, 3, 6, 7F a b c m


43/72

Solution

, ,F a b c b


44/72

Special Cases


45/72


ab

c 00 01 11 10

0

1

1 1 1

1

, , 1F a b c

11

1

1


46/72


ab

c 00 01 11 10

0

1

, , 0F a b c


47/72


ab

c 00 01 11 10

0

1

1

, ,F a b c a b c

1

1

1


48/72


49/72

Example


, , , 0, 2, 3, 6,8,12,13,15F a b c d m


50/72

Four-variable K-Map

abcd 00 01 11 10

00

01

11

10

, , , 0, 2, 3, 6,8,12,13,15F a b c d m

1

1

1

1

11

1

1


51/72

Four-variable K-Map

abcd 00 01 11 10

00

01

11

10

0, 2, 3, 6,8,12,13,15F m

1

1

1

1

11

1

1


52/72

Four-variable K-Map

abcd 00 01 11 10

00

01

11

10

F abd abc acd abd acd

1

1

1

1

11

1

1


53/72

Example


, , , 0, 2, 6,8,12,13,15

3,9,10

F a b c d m

d

D=Dont care (i.e. either 1 or 0)


54/72

Four-variable K-Map

abcd 00 01 11 10

00

01

11

10

1

1

d

1

11

1

1

, , , 0, 2, 6,8,12,13,15 3, 4, 9F a b c d m d

d

d


55/72

Four-variable K-Map

abcd 00 01 11 10

00

01

11

10

1

1

d

1

11

1

1

F ac ad abd

d

d


56/72

Five Variable K-Maps

, , , ,F a b c d e


57/72

Five variable K-map

A=1

A=0

Use two four variable K-maps


58/72

Use Two Four-variable K-Maps

bc

de 00 01 11 10

00

01

11

10

bc

de 00 01 11 10

00

01

11

10

A=0 map A=1 map


59/72

Five variable example

, , , , 5, 7,13,15, 21, 23, 29, 31F a b c d e m


60/72


bc

de 00 01 11 10

00

01

11

10

bc

de 00 01 11 10

00

01

11

10

A=0 map A=1 map

, , , , 5, 7,13,15, 21, 23, 29, 31F a b c d e m

1

1

1

1

1

1

1

1


61/72


bc

de 00 01 11 10

00

01

11

10

bc

de 00 01 11 10

00

01

11

10

A=0 map A=1 map

1

1

1

1

1

1

1

1

1F a ce 2F a ce


62/72

Five variable example

1 2F F F a ce a ce ce


63/72


64/72

K-map Simplification Procedure

Plot the K-map for the function F

Circle allprime implicants on the K-map

Identify and select all essential prime

implicants for the cover. Select a minimum subset of the

remaining prime implicants to completethe cover.

Read the K-map

Use DeMorgans theorem to convert F to Fin POS form


65/72

Example


, , 1, 2, 3, 5, 6F a b c M


66/72


ab

c 00 01 11 10

0

1

Step 1: Plot the K-map of F

11

1

11

, , 1, 2, 3, 5, 6F a b c M


67/72


ab

c 00 01 11 10

0

1

Step 2:Circle ALLPrime Implicants

11

1

11


68/72


ab

c 00 01 11 10

0

1


11

1

11

EPI

EPI

PI

PI


69/72


ab

c 00 01 11 10

0

1

Step 4: Select minimum subset of remainingPrime Implicants to complete the cover.

11

1

11

EPIPI

EPI


70/72


71/72

Solution

F ab bc bc

F ab bc bc

a b b c b c

, , 1, 2, 3, 5, 6F a b c M


72/72

Lecture Slides Kmap

Documents

Transcript of Lecture Slides Kmap