Guest Lecture by Kyle Tietz

http://www.ece.iastate.edu/~alexs/classes/

CprE 281: Digital Logic

Minimization

CprE 281: Digital LogicIowa State University, Ames, IACopyright © 2013

Administrative Stuff

• HW4 is out

• It is due on Monday Sep 23 @ 4pm.

• Please write clearly on the first page (in block capital letters) the following three things:

Your First and Last Name Your Student ID Number Your Lab Section Letter

Administrative Stuff

•Exam 1 on Monday Sep 30. Details to follow.

•Homework Office Hours Pratik Mishra TLA M 5:30-7:30pm F 2:00-4:00pm

Recap

Four-variable K-map

x 1 x 2 x 3 x 4 00 01 11 10

00

01

11

10

x 2

x 4

x 1

x 3

m 0

m 1 m 5

m 4 m 12

m 13

m 8

m 9

m 3

m 2 m 6

m 7 m 15

m 14

m 11

m 10

Grouping

• Group with rectangles

• Both sides a power of 2: 1x1, 1x2, 2x1, 2x2, 1x4, 4x1, 2x4, 4x2, 4x4

• Can use same minterm more than once

• Can wrap around edges of map

Recap Example

Terminology

• Literal

A variable, complemented or uncomplemented

Ex. X1

Ex. X2

_

Terminology

• Implicant Product term that indicates the input combinations for

which the function output is 1

Ex. x1 - indicates that x1x2 and x1x2 yield output of 1

Ex. x1x2

x 2

0

1

0 1

1 0

01

x 1

_ _ _ __ _

Terminology

• Prime Implicant Implicant that cannot be combined into another implicant

with fewer literals

Ex.

x1x2x3

0 1

1 1

1 1

1 0

00 01 11 10

0

1

x1x2x3

0 1

1 1

1 1

1 0

00 01 11 10

0

1

Not prime Prime

Terminology

• Essential Prime Implicant Prime implicant that includes a minterm not covered by

any other prime implicant

Ex.

x1x2x3

0 1

1 1

1 1

0 0

00 01 11 10

0

1

Terminology

• Cover Collection of implicants that account for all possible

input valuations where output is 1

Ex. x1’x2x3 + x1x2x3’ + x1x2’x3’

Ex. x1’x2x3 + x1x3’

x1x2x3

0 0

0 1

1 1

0 0

00 01 11 10

0

1

Example

• Number of Implicants? Prime Implicants? Essential Prime Implicants?

x1x2x3

1 1

1 1

0 0

1 0

00 01 11 10

0

1

Why concerned with minimization?

• Simplified function

• Reduce cost of circuit Cost: Gates + Inputs Transistors

CprE 281

0 1

0 1

1 1

0 1

1 1

1 0

1 1

1 1

Example: Minimization in SOP Form

00 01 11 10

00

01

11

10

ZYXW

g= Z’YX’W’ +ZY’X’W’ +Z’YX’W +ZYX’W +ZY’X’W + Z’Y’XW +ZYXW +ZY’XW + Z’Y’XW’ +Z’YXW’ +ZYXW’ +ZY’XW’

CprE 281

0 1

0 1

1 1

0 1

1 1

1 0

1 1

1 1

00 01 11 10

00

01

11

10

ZYXW

g=(Z+Y+X+W). (Z’+Y’+X+W) (Z+Y+X+W’). (Z+Y’+X’+W’)

Example: Minimization in POS Form

CprE 281

Minimization of both SOP and POS Forms

0 1

0 1

1 1

0 1

1 1

1 0

1 1

1 1

00 01 11 10

00

01

11

10

ZYXW

1

2

34

5 1

2

3

4

5

g=ZY’ +XW’ +ZW +Y’X +Z’YX’

0 1

0 1

1 1

0 1

1 1

1 0

1 1

1 1

00 01 11 10

00

01

11

10

ZYXW

1

2

3

g=(Z+Y+X) .(Z+Y’+X’+W’) .(Z’+Y’+X+W)

1

2

3

Cost = 22(5 AND gates, 1 OR gates 16 inputs)

Cost = 18(3 OR gates, 1 AND gates 14 inputs)

Assumption: Complemented formsof primary inputs aregiven at zero cost.

Strategy

1. Generate all prime implicants

2. Find the set of essential prime implicants

3. If set of essential prime implicants covers function, Done!

4. Else, decide which non-essential prime implicants to add to complete minimum-cost cover

Examples

x 1 x 2 x 3 x 4 00 01 11 10

1 1

1 1

1 1

00

01

11

10

x 1 x 2 x 3 x 4 00 01 11 10

1

1 1

1 1

1 1

00

01

11

10

f 1 x 1 x 3 x 1 x 3 x 4 x 1 x 2 x 3 x 5 + + =

x 5 1 = x 5 0 =

Five-variable K-map

CprE 281

K-map for 5-variables functionsF(A,B,C,D,E) = m(2,5,7,8,10,13,15,17,19,21,23,24,29,31)F(A,B,C,D,E) = CE + AB’E + BC’D’E’ + A’C’DE’

CprE 281 Lec 15 23

K-map for 6-variable functions

G(A,B,C,D,E,F)= m(2,8,10,18,24,26,34, 37,42,45,50,53,58,61)

G(A,B,C,D,E,F)= D’EF’ + ADE’F + A’CD’F’

Questions?

THE END