[SOLVED] CS 1007ICT / 1807ICT / 7611ICT Computer Systems & Networks

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
Ones complement and twos 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
DeMorgans
. 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: Demorgans 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
Lets 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

1007ICT / 1807ICT / 7611ICT Computer Systems & Networks
3B. Digital Logic and Digital Circuits

Last Lecture:
Topics Covered:
Digital logic, Basic logic gates, Boolean algebra
Combinatorial logic gates

Lecture Content
Learning objectives
Logic unit, Selection logic, Decoder logic Multiplexing and demultiplexing
Half and Full adders
. Revised and updated by,, and Wee Lum 3

Learning Objectives
At the end of this lecture you will have gained an understanding of:
Selection logic
Decoder logic
Multiplexors
Demultiplexors
Half and Full adders
. Revised and updated by,, and Wee Lum 4

Logic Unit
Lets 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 5

Selection Logic
SaywehaveanumberofinputsA,B,C,Dandwe want to select one of them to use in a logic function.
Weneedaspecialfunctiontoswitchtheselected input into the output X based on the selection.
WecanuseinputsS0toS3toselectbetweenA-D
S0 S1 S2 S3
. Revised and updated by,, and Wee Lum 6

Decoder Logic
Justsaywewanttoselectbetweenoneof100s of possible inputs we need 100s of selection inputs as well which gets too complicated.
DecoderswithNinputsallowustoenableany one of 2N possible selection lines.
Basicallyadecodertakesabinarycodednumber and enables the output representing the number
2 to 4 Decoder
x0 = s1ANDs2 x1 = s1ANDs2 x2 = s1ANDs2 x3 = s1ANDs2
2N outputs
. Revised and updated by,, and Wee Lum 7

Multiplexing
Combiningtheselectoranddecoderwecan
create what is called a
IngeneralifwehaveNinputswewanttoswitch between we need to have log2(N) selection lines.
Multiplexor
S0 S1 selection
. Revised and updated by,, and Wee Lum 8

Demultiplexing
We can also switch in the opposite direction to send one input A into one of many different outputs (eg X1..Xn)
Input X1 A X2
outputs X4
(s0,s1) Select
S0 S1 selection
. Revised and updated by,, and Wee Lum 9

Half-Adders
In addition to logic functions we can also create maths functions.
The simplest math function is the half-adder which can add 2 digits (bits) to give a sum and a carry bit
0011A 0 1 0 1B+ 0 1 1 10
Sum = A XOR B Carry = A AND B
. Revised and updated by,, and Wee Lum 10

Full-Adders
Ifwewanttoaddmorethan1bitvaluestogether we need to deal with the carry.
Full-addersacceptthetwoinputstobeaddedplus the carry from a previous stage.
Thecircuitmustbeabletodealwithpropagation delays
00 00 01 01 00 10 11 11A 00 01 00 01 10 01 10 11B+ 00 01 01 10 10 11 101 110
A0 B0 A1 B1 A2 B2 A3 B3
Full Adder
Full Adder
Full Adder
Full Adder

S0 S1 4 bit adder S2 S3 . Revised and updated by,, and Wee Lum-Adders
The logic to perform add with carry combines two half adders together.
Sum =AXORBXOR= (AANDB)OR(CinAND(AXORB))
. Revised and updated by,, and Wee Lum 12

Subtraction
ABisthesameasA+(-B)
IfweconvertBtothenegativeequivalentofits value, we can use the basic adder as it is. We can use an XOR gate to do this.
However,convertingBtoBusingtwos complement also requires adding 1
A B0 Sign bit 0
+0 if positive +1 if negative
Full Adder
. Revised and updated by,, and Wee Lum 13

Have considered: Selection logic Decoder logic
Multiplexors
Demultiplexors
Half and Full adders
. Revised and updated by,, and Wee Lum 14

Arithmetic logic unit
Binary multiplication and division
Shifting
Sequential Logic
Data latches, S-R Latch
Clocks and synchronisation
Registers, Buses, Computer memory Processors and Memory Organisation
. Revised and updated by,, and Wee Lum 15

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