1007ICT / 1807ICT / 7611ICT Computer Systems & Networks

3A. Digital Logic and Digital Circuits

Last Section: Data Representation

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Topics Covered:

Representing binary integers

Conversion from binary to decimal

Hexadecimal and octal representations

Binary number operations

One’s complement and two’s complement

Representing characters, images and audio

Lecture Content

Learningobjectives

Digitallogic,Basiclogicgates,Booleanalgebra Combinatoriallogicgates

© . Revised and updated by , , and Wee Lum 3

Learning Objectives

At the end of this lecture you will have:

Gained an understanding of basic logic gates

Learnt the truth tables associated with the basic logic gates

Gained an understanding of combinatorial logic gates

Learnt the truth tables associated with combinatorial logic gates

© . Revised and updated by , , and Wee Lum 4

Digital Logic (Section 2.2)

All digital computers are built from a set of low

Logic Gates.

level digital logic switches or

Gates operate on binary signals that only have one of two values:

Signalsfrom0to2voltsisusedtorepresentabinary0(OFF) Signalsfrom3to5voltsisusedtorepresentabinary1(ON) Signals between 2 and 3 volts represent an invalid state

Three basic logic functions that can be applied to binary signals:

More complex functions can be built from these three basic gates

AND: OR: NOT:

outputtrueifALLinputsaretrue outputtrueifANYinputistrue outputistheinverseoftheinput

© . Revised and updated by , , and Wee Lum 5

Basic Logic Gates (Section 2.4)

Boolean expression

Truth Table

x = a AND b

x = a OR b

© . Revised and updated by , , and Wee Lum 6

Boolean Algebra

There is a basic set of rules about combining simple binary functions.

x OR 0 = x x OR 1 = 1 x OR x = x x OR x = 1 (x)=x

xAND0 = 0 xAND1 = x xANDx = x xANDx = 0

© . Revised and updated by , , and Wee Lum 7

Combinatorial Logic Gates

Next Slide

Symbol Equivalent

Boolean expression

Truth Table

© . Revised and updated by , , and Wee Lum 8

x = a AND b x = a OR b x = a XOR b

Boolean Algebra – 2

This second set of rules are more powerful. OR – form AND – form

(xORy) = xANDy

(xANDy) = xORy

OR – form AND – form

NAND = Theorem

DeMorgan’s

© . Revised and updated by , , and Wee Lum 9

The eXclusive-OR Gate (XOR)

Looking at the truth table we see that the XOR function can be described as:

x = (aANDb)OR(aANDb) x=aXORb

This function can be built in 3 ways: Demorgan’s Theorem

aaa bbb aaa bbb

x = (aANDb)OR(aANDb) x = (aANDb)OR (aANDb) x = (aANDb)AND(aANDb)

© . Revised and updated by , , and Wee Lum 10

© . Revised and updated by , , and Wee Lum 11

Logic Unit

Let’s try to create a “programmable” logic unit that permits us to apply a predefined logic function to a given set of inputs.

Output Select

We need a function that lets us select what operation to perform

AND OR XOR

© . Revised and updated by , , and Wee Lum 12

Have considered:

Operation of basic logic gates

Combinatorial logic gates, Truth tables

© . Revised and updated by , , and Wee Lum 13

Logic unit, Selection logic, Decoder logic

Multiplexing and demultiplexing

© . Revised and updated by , , and Wee Lum 14

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