Lecture 2

Lecture Lecture 22

Dr Richard Reilly

Dept. of Electronic & Electrical Engineering

Room 153,

Engineering Building

• The main characteristic of a Digital System is its manipulation of discrete elements of information.

• Another term for a digital system would be a discrete information processing system.

BINARY SYSTEMS

1. Most information processing systems are constructed from switches, which are binary devices.

    on-off switches are the basic building blocks of digital systems.

    inherently binary

    Two natural states : on (closed) and off (open).

Why Binary ?Why Binary ?

Off

On

switcharm

terminal

2. The basic decision-making processes required of digital systems are binary.

    Digital systems are often required to make tests.

• Is Condition C1 true ? or Is condition C2 false ?.

    Examples of such decisions are :

    Has button (switch) X been pushed ?,

    Has temperature tmax been reached ?.

    Decisions of this kind are inherently binary because their outcomes are taken from the value-pair {true, false}.

Why Binary ?Why Binary ?

• The values that the two variable take may be called by different names

True and false

Yes and no, etc.

• As engineers it is appropriate to think in terms of voltages and assign the values of 1 and 0 corresponding to voltage levels.

Concept of Binary LogicConcept of Binary Logic

• Binary logic is used to describe, in a mathematical way, the manipulation and processing of binary information

• Binary logic consists of binary variables and logical operations.

Concept of Binary LogicConcept of Binary Logic

Logical Operators: AND Gate Logical Operators: AND Gate

A

BC

C A B .

AND gate Symbol

Function   A B C

Truth-Table   0 0 0

    1 0 0

    0 1 0

    1 1 1

Denote C thus defined : 

read as C = A AND B

OR Gate OR Gate

CA

B

C A B

OR gate Symbol

Function   A B C

Truth-Table   0 0 0

    1 0 1

    0 1 1

    1 1 1

Denote C thus defined : 

read as C = A OR B

Inverter NOT gate

Inverter NOT gateInverter NOT gate

V c c

R

V o

A

  If A = +5v     If A = 0v 

    switch is closed  Vo is 0 v

     switch is open  Vo is +5 v

The truth-table for this operator configuration is

Inverter NOT gateInverter NOT gate

A

10

Vo

01

V c c

R

V o

A

Inverter Inverter

A C

C A

NOT gate (logic inverter) Symbol

Function :   A C  

Truth-Table   0 1  

    1 0  

Denote C thus defined : 

read as C = NOT A

NAND gateNAND gate

V c c

R

V o

A

B

  If A = +5v and B = +5v   If A = 0vand B = +5v  If A = 0vand B = +5v  If A = 0vand B = 0v  

    switches are closed  Vo is 0 v

    Vo is +5 v

     Vo is +5 v

     Vo is +5 v

NAND Gate NAND Gate

A

BC

C A B .

NAND gate  Symbol

Function   A B C

Truth-Table   0 0 1

    1 0 1

    0 1 1

    1 1 0

Denote C thus defined : 

NOR gateNOR gate

 If A = +5v and B = +5v   If A = 0vand B = +5v  If A = 0vand B = +5v  If A = 0vand B = 0v  

   switches are closed  Vo is 0 v

    Vo is 0 v

     Vo is 0 v

     Vo is +5 v

NOR Gate NOR Gate

CA

B

C A B

NOR gate Symbol

Function   A B C

Truth-Table   0 0 1

    1 0 0

    0 1 0

    1 1 0

Denote C thus defined : 

Logical expressions AND, OR and NOT are said to be logically complete, that is using these three operations it is possible to realise any function.

 Logic Gates can have more than two inputs. Thus a three-input AND gate responds when with a logic-1 output if all three input signals are logic-1.

Implementation of Logical Implementation of Logical Functions using switches. Functions using switches.

• The mathematical system of binary logic is better known as Boolean or switching algebra.

• This algebra is conveniently used to describe the operation of complex networks of digital circuits.

• Designers of digital circuits use Boolean Algebra to transform circuit diagrams to algebraic expressions and vice versa.

Implementation of Logical Implementation of Logical Functions using switches. Functions using switches.

George BooleGeorge Boole

• George Boole had little formal education yet was a brilliant scholar.

• Made lasting contribution to mathematics in the areas of differential and difference equations as well as algebra.

• He published in 1854 his work “An Investigation of the Laws of Thought, on which are founded the Mathematical Theories of Logic an Probability”.

• Boole generated a mathematical analysis of logic.

• Boolean algebra like any other deductive mathematical system, may be defined with• a set of elements,

• a set of operators,

• a number of unproved axioms or postulates,

• It is a mathematical analysis of logic

Why do we use Boolean Algebra ?Due to its ability for mathematical analysis of logic to study digital systems.

Boolean AlgebraBoolean Algebra

• In Boolean algebra a proposition is either true or false (no in-between state possible), these proposition are denoted by letters (usually at start of the alphabet)

e.g. A. The grass is green TRUE

B. 3 is an even number FALSE

• We can combine these propositions to get Boolean Functions denoted by letters (from the end of the alphabet).

 e.g. Z = A AND B FALSE

Boolean AlgebraBoolean Algebra

• Several advantages for having a mathematical method for description of the internal workings of a computer.

• more convenient to calculate using expressions that represent switching circuits then it is to use schematic or even logical expressions

• just as an ordinary algebraic expression may be simplified by means of basic theorems, the expression describing a given switching circuit network may be reduced or simplified.

Boolean AlgebraBoolean Algebra

• Reducing and simplifying logic networks.

enabling the designer to simplify the circuitry used

achieving economy of construction

Reliability of operation

SimplificationSimplification

• When a variable is used in an algebraic formula, it is generally assumed that the variable may take on any numerical value.

•  However a variable in Boolean equations has a unique characteristic .

• it may assume only one of two possible states.

these states can be represented by the symbols 0 and 1. i.e. T or F

Fundamental Concepts of Boolean Algebra

zyx 52 assume x,y and z range through the entire field of real numbers

• Boolean algebra uses the operation called complementation and the symbol of this is

means “take the complement of A”

means “take the complement of A+B”

The complement operation can be defined quite simply as

ComplementationComplementation

A

A

BA

0

1

1

0

As we have seen the complementation operation is physically realised by a gate or circuit called an inverter.

Boolean OperatorsBoolean Operators

A

A B

A B

NOT A Complement of A

A OR B Logical Sum, True if either A OR B true

A AND B

Logical Product, True if both A AND B true

Examples of Boolean Functions

• To study a logical expression, it is very useful to construct a table of values for the variables.

then evaluate the expression for each possible combination of variables.

Boolean FunctionsBoolean Functions

BCAZ

CDABY

Evaluate

Evaluate a Boolean Evaluate a Boolean FunctionFunction

CBA

• List all possible versions of the input variables in a Truth Table

Evaluate a Boolean Evaluate a Boolean FunctionFunction

CBA A

00001111

B

00110011

C

01010101

Boolean Operations : AND,OR and Boolean Operations : AND,OR and NOTNOT

A

00001111

B

00110011

C

01010101

10101010

C

CBA

Boolean Operations : AND,OR and Boolean Operations : AND,OR and NOTNOT

A

00001111

B

00110011

C

01010101

10101010

00100010

C BC

CBA

Finally ORing or Logical Addition

Boolean Operations : AND,OR and Boolean Operations : AND,OR and NOTNOT

C BC A BCA

00001111

B

00110011

C

01010101

10101010

00100010

00101111

Rules of Boolean Algebra

• We represent FALSE with 0 and TRUE with 1.

• If we have a large number of propositions and a complicated Boolean function we may be able to simplify it using the concept of tautology (redundancy).

e.g. always TRUEalways TRUEalways FALSE

 We can use the complete set of rules of Boolean Algebra

to simplify expressions.

AAZ ABAZ

AAZ

1.      

2.      

3.      

4.      

5.      

6.      

7.      

8.      

9.      

10.     Commutative Laws

11.     

12.     Associative Laws

13.     

14.     

Distributive Law

15.      

16.      

17.      

18.      

19.     De Morgan’s Laws

20    

0 A A

1 1 A

A A A

A A 10 0 A

1 A A

A A A

A A 0

A AA B B A

A B B A

A B C A B C A B C ( ) ( )

A BC AB C ABC

A B C AB AC

A AC A A A B A

A AB A B

AB BC BC AB C

A B A B

A B A B

Rules of Boolean Algebra

We can extend De Morgan’s Laws to

Example of the Application of the Rules

A truth table for each expression will verify that both are equivalent

...... CBAABC

...... CBACBA

BCBA

BCABCCBABA

BCBBACBA

BCACBAZ

 Rule 4Rule 14

Rule 15

ABCCBA 11

A Specific Design Problem

A logical network has two inputs, A and B and output C. The relationship between the inputs and outputs is as

follows :     When A and B are 0’s C is to be 1    When A is 0 and B is 1 C is to be 0    When A is 1 and B is 0 C is to be 1    When A and B are 1’s C is to be 1

A Specific Design Problem

put this into a truth table.

A

0011

B

0101

C

1011

A Specific Design Problem

• Now add a new column for the product terms : • will contain each of the input variables for each row, • with the letter complemented when input value for the

variable is 0 and

• not complemented when the input value is 1.

A

0011

B

0101

C

1011

Product Terms

AB

AB

AB

AB

A Specific Design Problem

• When the product term is equal to 1

product term is removed and used as a sum-of -products expansion

 in this case 1st, 2nd and 4th rows are selected.

ABBABAC

A Specific Design Problem

simplifyRule 4

Rule 18 Rule :  

BBABAC

ABAC BABAA

BAC

A Specific Design Problem

Check using the Truth-Table :

Implementation :

A

0011

B

0101

1010

1011

B A B

CAB