introduction to digital logic · 2 algorithmic simplification...

Introduction to Digital Logic

K-Maps

2

Algorithmic Simplification

A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’

Put in sum of products form

Put in sum of minterms form

Find the maximal p-terms

Find the minterms that only have one maximal p-term

Those are the start of the expression

Remove minterms that have that p-term

Find the maximal p-term that occurs the most times. Pick one if tie.Remove minterms that contain that maximal p-term Pick another maximal p-term, add to the expression, remove minterm

3



Find all of the minterms

ABCD ABC'D

A'BCD A'BCD' A'BC'D A'BC'D'

ABC'D' A'BC'D'

AB'CD A'B'CD

AB'CD' A'B'CD'

4



ABCD ABC'D

A'BCD A'BCD' A'BC'D A'BC'D'

ABC'D'

AB'CD A'B'CD

AB'CD' A'B'CD'

5



Find the maximal p-terms

ABCD (BD, CD) ABC'D (BC', BD)

A'BCD (BD, A'B, CD, A'C) A'BCD' (A'C, A'B) A'BC'D (BC', BD, A'B)

A'BC'D' (A'B, BC')

ABC'D' (BC')

AB'CD (CD, B'C) A'B'CD (CD, B'C, A'C)

AB'CD' (B'C) A'B'CD' (B'C, A'C)

6



Find minterms that only have one maximal p-term



A'BC'D' (A'B, BC')

ABC'D' (BC')



7



Write those as the start to the expression



A'BC'D' (A'B, BC')

ABC'D' (BC')



BC' + B'C

8



Remove any minterms that have those maximal p-terms

ABCD (BD, CD)

A'BCD (BD, A'B, CD, A'C) A'BCD' (A'C, A'B)

BC' + B'C

9



Find the maximal p-term that occurs the most times. Pick one if tie.

ABCD (BD, CD)

A'BCD (BD, A'B, CD, A'C) A'BCD' (A'C, A'B)

BC' + B'C

10



Remove minterms that contain that maximal p-term Pick another maximal p-term, add to the expression, remove minterm

ABCD (BD, CD)

BC' + B'C + A'B

11



BC' + B'C + A'B + CD

12

K-Map Simplification


AB00 01

00

01

CD

11 10

11

10

Boxes represent the minterms

Values grouped together

13



AB00 01

00

01

CD

11 10

11

10

1 1

Find all the minterms and put a 1 in those boxes

14



AB00 01

00

01

CD

11 10

11

10

1 1 1 1

1 1

15



AB00 01

00

01

CD

11 10

11

10

1 1 1 1

1 1 1

16



AB00 01

00

01

CD

11 10

11

10

1

1 1 1 1

1 1 1

1

17



AB00 01

00

01

CD

11 10

11

10

1

1 1 1 1

1 1 1

1

18



AB00 01

00

01

CD

11 10

11

10

1 1

1 1 1 1

1 1 1

1 1

All the minterms are represented

19



AB00 01

00

01

CD

11 10

11

10

1 1

1 1 1 1

1 1 1

1 1

Group into 8, 4, 2, 1. The bigger the group, the better.

20



AB00 01

00

01

CD

11 10

11

10

1 1

1 1 1 1

1 1 1

1 1

Represented by maximal p-term BC’

21



AB00 01

00

01

CD

11 10

11

10

1 1

1 1 1 1

1 1 1

1 1

22

K-Map SimplificationA·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’

AB00 01

00

01

CD

11 10

11

10

1 1

1 1 1 1

1 1 1

1 1

23


AB00 01

00

01

CD

11 10

11

10

1 1

1 1 1 1

1 1 1

1 1

24


AB00 01

00

01

CD

11 10

11

10

1 1

1 1 1 1

1 1 1

1 1

25


AB00 01

00

01

CD

11 10

11

10

1 1

1 1 1 1

1 1 1

1 1

All maximal p-terms are represented

26


AB00 01

00

01

CD

11 10

11

10

1 1

1 1 1 1

1 1 1

1 1

Find mniterms represented by only one p-terms. Find 1s only covered by one box.

27


AB00 01

00

01

CD

11 10

11

10

1 1

1 1 1 1

1 1 1

1 1

BC’ + B’C

However, all 1s are not covered

28


AB00 01

00

01

CD

11 10

11

10

1 1

1 1 1 1

1 1 1

1 1

BC’ + B’C + AB + CD

introduction to digital logic · 2 algorithmic simplification...

Documents