Theory Lecture 2

7/28/2019 Theory Lecture 2

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MCT-212: DIGITAL LOGIC

DESIGN


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DIGITAL LOGICDESIGNLets start by being literal


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What does each of these words mean?

DIGITAL LOGIC DESIGN

Three words: Digital

Logic

Design


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ANALOG AND DIGITAL QUANTITIES

Analog quantities have

continuous values Digital quantities have a

discrete set of values


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Types of electronic devices or instruments: Analog

Digital


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Combination of analog and digital:

Give 5 examples of digital and analog

devices each from around you.


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DESIGN

A Design is a creative plan or convention for theconstruction of an object or a system.

What is the engineering design process?

Why is design important?One PROBLEM IDENTIFICTION

Two GENERATING POSSIBLE SOLUTIONS

Three SELECTING A SOLUTION

Four CREATING A PROTOTYPE

Five REFINING THE DESIGN


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LOGIC AND BOOLEAN ALGEBRA

What is Logic?

What Boolean Algebra?

What is Logic Design?

What is Digital Design?

What is Circuit Design or

Digital Logic Design?

LOGICBOOLEAN

ALGEBRA

LOGIC

DESIGN


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LOGIC AND BOOLEAN ALGEBRA

English mathematician, Philosopher

and Logician, Goerge Boole (1815-

1864).

Boolean Algebra, developed in 1854by George Boole in his book An

Investigation of the Laws of Thought.

Computer hardware works with

binary numbers, but binaryarithmetic is much more old than the

computers. [Ancient Chinese

(3000B.C), Ancient Greek (2000B.C),

Boolean Algebra (1850)]


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PROPOSITIONAL LOGIC

The ancient Greek philosophers created a system to

formulize arguments, called propositional logic.

A proposition is a statement that could be TRUE or

FALSE.

Propositions could be compounded by means of the

operators AND, OR and NOT.


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PROPOSITIONAL CALCULUS EXAMPLE

We can assign values to propositions, for example:

I will take an umbrella if and only if it is raining OR the

weather forecast is bad

The proposition I will take an umbrella is the result ofthe Boolean combination (OR) between raining and

weather forecast being bad.

I will take an umbrella = it is

raining OR the weather forecast isbad


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DIAGRAMMATIC REPRESENTATION

We can think of the umbrella proposition as a

result that we calculate from the weather forecast

and the fact that it is raining by means of a

logical OR.

OR

Rain

Bad Weather

Forecast

Take Umbrella


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DIAGRAMMATIC REPRESENTATION

Since propositions can only take two values, we

can express all possible outcomes of the

umbrella proposition by a table:


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MORE COMPLEX PROPOSITIONS

We can make our propositions more complex, forexample:

(Take Umbrella ) = ( NOT (Take Car ) ) AND ( (Bad

Forecast) OR (Raining ) )

and as before represent this diagrammatically.


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MORE COMPLEX PROPOSITIONS

ORRaining

Bad Forecast

Take

UmbrellaNOTCarAND


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BOOLEAN ALGEBRA

To perform calculations quickly and efficiently we can usean equivalent, but more succinct notation.

We also need a to have a well-defined semantics for all

the operators, or connectives that we intend to use.

The system we will employ iscalled Boolean Algebra and

satisfies the criteria above.


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FUNDAMENTALS OF BOOLEAN

ALGEBRA

The truth values are replaced by 1 and0:

1 = TRUE

0 = FALSE

Propositions are replaced by variables:

R = it is raining

W = The weather forecast is bad

Operators are replaced by symbols

or' = NOT

or+ = OR or = AND


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FUNDAMENTALS OF BOOLEAN

ALGEBRA

Our previous complex proposition: (Take Umbrella ) = ( NOT (Take Car) ) AND( (Bad Forecast)

OR (Raining ) )

Is formalized by the simpler equation:

U = (C')(W+R)


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LOGIC = BOOLEAN ALGEBRA

Boolean algebra (or Boolean logic) is a logicalcalculus of truth values.

It resembles the algebra of real numbers as

taught in high school, but with the numericoperations of

multiplication xy conjunction xy,

addition x + y disjunction xy negation x complement x

More on Boolean Algebra later on during the semester.


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NUMBER SYSTEMSAND CONVERSIONS

Back to elementary.!


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DECIMAL NUMBER SYSTEM

Any decimal number such as 2610can be representedas:

Two Thousands Plus Six Hundreds Plus One Tens Plus

Zero Units

Or,

2 x + 6 x + 1 x + 0 x

However the convention is to only write the coefficients

and from their position, the power of10 is deduced.

43210. 12


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DECIMAL AND BINARY NUMBER BASES

How many digits does the conventionalnumber system use? What are they?

TEN: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9The decimal number system is knownas base 10 or radix 10.

How many digits does the Binarynumber system use? What are they?

TWO : 1 and 0


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BINARY NUMBER SYSTEM

The binary number system uses only two digits: 1 and 0 Its a base 2 orradix 2 system.

What does this number stand for indecimal system : 101101.01

1x+ 0x+ 1x+ 1x+ 0x+

1x

. 0x

+ 1x

= 45.25


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OCTAL AND HEXADECIMAL SYSTEMS

Octal number system is the base 8 system, whilehexadecimal system is the base 16 system.

How many, and what digits do Octal and

Hexadecimal systems have?

OCTAL(8) : 0,1,2,3,4,5,6,7

HEXADECIMAL(16) : 0, 1, 2, 3, 4, 5, 6,

7, 8, 9, A, B, C, D, E, F


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ARITHMETIC OPERATIONS

Arithmetic operations like multiplication, division,addition and subtraction can be done the same way

like for the decimal (base10) system.

One must take care not to use any digit other than the

ones allowed for that particular base.

Perform the following operations


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NUMBER BASE CONVERSIONS

How to write:

(100)10

(100)8

(100)2

Decimal

Hexade-cimal

Binary

Octal


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COMPLEMENT OF ANUMBER

The what?!


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COMPLEMENT OF A NUMBER

Complements are used in digital numbers to simplify themultiplication and subtraction process.

There are two types of complements for each base-r system:

(r-1)scomplement

rscomplement

For a numberN:

(r-1)scomplement = ()

rs complement =

for N>0; 0 for N=0= (())+

Complement of the complement of a number, is the

number itself!


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ANY QUESTIONS?

Anyone willing to present?

Time allowed : 5 mins

Topic : Any Bonus Points : +3


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REFERENCES

Chapter no 1: Binary Systems Digi tal Lo gic Designby Morris Mano

Chapter no 1: Digital Concepts

Digi tal Fundamentalsby Floyd

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Theory Lecture 2