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Chapter 3:. Digital Logic Design. Gate-Level Minimization. Karnaugh Map. Adjacent Squares Number of squares = number of combinations Each square represents a minterm 2 Variables  4 squares 3 Variables  8 squares 4 Variables  16 squares

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Chapter 3

Chapter 3:

Digital Logic Design

Gate-Level

Minimization


Karnaugh map

Karnaugh Map

  • Adjacent Squares

    • Number of squares = number of combinations

      • Each square represents a minterm

      • 2 Variables  4 squares

      • 3 Variables  8 squares

      • 4 Variables  16 squares

    • Each two adjacent squares differ in one variable

      • Two adjacent minterms can be combined together

        Example:F = x y + xy’

        = x ( y + y’ )

        = x


Two variable map

Two-variable Map

Note: adjacent squares horizontally and vertically NOT diagonally


Two variable map1

Two-variable Map

  • Example


Two variable map2

Two-variable Map

  • Example


Three variable map

Three-variable Map


Three variable map1

Three-variable Map

  • Example


Three variable map2

Three-variable Map

  • Example

Extra


Three variable map3

Three-variable Map

  • Example


Three variable map4

Three-variable Map

  • Example


Four variable map

Four-variable Map


Four variable map1

Four-variable Map

  • Example


Four variable map2

Four-variable Map

  • Example

    Simplify: F = A’ B’ C’ + B’ C D’ + A’ B C D’ + A B’ C’


Four variable map3

Four-variable Map

  • Example

    Simplify: F = A’ B’ C’ + B’ C D’ + A’ B C D’ + A B’ C’


Four variable map4

Four-variable Map

  • Example

    Simplify: F = A’ B’ C’ + B’ C D’ + A’ B C D’ + A B’ C’


Four variable map5

Four-variable Map

  • Example

    Simplify: F = A’ B’ C’ + B’ C D’ + A’ B C D’ + A B’ C’


Four variable map6

Four-variable Map

  • Example

    Simplify: F = A’ B’ C’ + B’ C D’ + A’ B C D’ + A B’ C’


Four variable map7

Four-variable Map

  • Example

    Simplify: F = A’ B’ C’ + B’ C D’ + A’ B C D’ + A B’ C’


Five variable map

Five-variable Map


Five variable map1

Five-variable Map

A = 0

A = 1


Implicants

Implicants

Implicant:

Gives F = 1


Prime implicants

Prime Implicants

Prime Implicant:

Can’t grow beyond this size


Essential prime implicants

Essential Prime Implicants

Essential Prime Implicant:

No other choice

Not essential

8 Implicants

5 Prime implicants

4 Essentialprime implicants


Product of sums simplification

Product of Sums Simplification


Don t care condition

Don’t-Care Condition

  • Example

You can only drop one coin at a time.

Used as“don’t care”


Don t care condition1

Don’t-Care Condition

  • Example

A

LogicCircuit

F

B

C

Don’t care what value F may take


Don t care condition2

Don’t-Care Condition

  • Example

A

F

B

C


Don t care condition3

Don’t-Care Condition

  • Example

F (w, x, y, z) = ∑(1, 3, 7, 11, 15)

d (w, x, y, z) = ∑(0, 2, 5)

x = 0

x = 1

x = 0

x = 1


Universal gates

Universal Gates

  • One Type

    • Use as many as you need (quantity), but one type only.

  • Perform Basic Operations

    • AND, OR, and NOT

  • NAND Gate

    • NOT-AND functions

    • OR function can be obtained from AND by Demorgan’s

  • NOR Gate

    • NOT-OR functions (AND by Demorgan’s)


Universal gates1

Universal Gates

  • NAND Gate

    • NOT:

    • AND:

    • OR:

DeMorgan’s


Universal gates2

Universal Gates

  • NOR Gate

    • NOT:

    • OR:

    • AND:

DeMorgan’s


Nand nor implementation

NAND & NOR Implementation

  • Two-Level Implementation


Nand nor implementation1

NAND & NOR Implementation

  • Two-Level Implementation


Nand nor implementation2

NAND & NOR Implementation

  • Multilevel NAND Implementation


Nand nor implementation3

NAND & NOR Implementation

  • Multilevel NOR Implementation


Gate shapes

Gate Shapes

  • AND

  • OR

  • NAND

  • NOR


Other implementations

Other Implementations

  • AND-OR-Invert

  • OR-AND-Invert


Implementations summary

Implementations Summary

  • Sum Of Products:

    • AND-OR

    • AND-OR-Invert = AND-NOR = NAND-AND

  • Products Of Sums

    • OR-AND

    • OR-AND-Invert = OR-NAND = NOR--OR


Exclusive or

Exclusive-OR

  • XOR

    F = xy = x y+ x y

  • XNOR

    F = xy =xy = x y+ x y


Exclusive or1

Exclusive-OR

  • Identities

    • x 0 =x

    • x 1 =x

    • xx= 0

    • xx= 1

    • xy=xy=xy

  • Commutative & Associative

    • xy=yx

    • ( xy ) z=x ( yz ) =xyz


Exclusive or functions

Exclusive-OR Functions

  • Odd Function

    F = xyz

    F = ∑(1, 2, 4, 7)

  • Even Function

    F = xyz

    F = ∑(0, 3, 5, 6)


Parity

Parity

1

0

1

0

1

0

1

0

1

0

0

0

1

0

1

0

1

1

0

1

0

1

1

0

0

0

1

Parity Generator

Parity Checker


Parity generator

Parity Generator

  • Odd Parity

  • Even Parity

1

0

1

0

1

1

0

1

0

Odd number of ‘1’s

1

0

1

0

0

1

0

1

0

Even number of ‘1’s


Parity checker

1

0

1

0

0

Error

Check

Parity Checker

  • Odd Parity

  • Even Parity

1

0

1

0

1

Error

Check


Homework

Homework

  • Mano

    • Chapter 3

      • 3-1

      • 3-3

      • 3-5

      • 3-7

      • 3-9

      • 3-15

      • 3-16

      • 3-18

      • 3-22


Homework1

Homework

  • Mano


Homework2

Homework


Homework3

Homework


Homework4

Homework


Homework5

Homework


Homework6

Homework


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