Dr Richard Reilly

1
Lecture 2

• To insert your company logo on this slide
• From the Insert Menu
• Select Picture
• Locate your logo file
• Click OK
• To resize the logo
• Click anywhere inside the logo. The boxes that
  appear outside the logo are known as resize
  handles.
• Use these to resize the object.
• If you hold down the shift key before using the
  resize handles, you will maintain the proportions
  of the object you wish to resize.

• Dr Richard Reilly
• Dept. of Electronic Electrical Engineering
• Room 153,
• Engineering Building

2
BINARY SYSTEMS

• 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.

3
Why Binary ?

• 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).

4
Why Binary ?

• 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.

5
Concept of Binary Logic

• 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.

6
Concept 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.

7
Logical Operators AND Gate
8
OR Gate
9
Inverter NOT gate

• Inverter NOT gate

10
Inverter NOT gate

• The truth-table for this operator configuration
  is

A
Vo
1 0
0 1
11
Inverter
12
NAND gate
13
NAND Gate
14
NOR gate
15
NOR Gate
16
Implementation of Logical Functions using
switches.

• 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.

17
Implementation of Logical 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.

18
George 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.

19
Boolean Algebra

• 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.

20
Boolean 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

21
Boolean 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.

22
Simplification

• Reducing and simplifying logic networks.
• enabling the designer to simplify the circuitry
  used
• achieving economy of construction
• Reliability of operation
•  

23
Fundamental Concepts of Boolean Algebra

• 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

24
Complementation

• Boolean algebra uses the operation called
  complementation and the symbol of this is
• ? means take the complement of A
• ? means take the complement of AB
• The complement operation can be defined quite
  simply as

25
Boolean Operators

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

26
Boolean Functions

• 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.

27
Evaluate a Boolean Function

• Evaluate

28
Evaluate a Boolean Function

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

29
Boolean Operations AND,OR and NOT
30
Boolean Operations AND,OR and NOT
31
Boolean Operations AND,OR and NOT

• Finally ORing or Logical Addition

32
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 TRUE
• always TRUE
• always FALSE
•  
• We can use the complete set of rules of Boolean
  Algebra to simplify expressions.

33
(No Transcript)
34
Rules of Boolean Algebra
We can extend De Morgans Laws to Example of
the Application of the Rules A truth table
for each expression will verify that both are
equivalent
35
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 0s ? 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 1s ? C is to
be 1
36
A Specific Design Problem
put this into a truth table.
37
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.

38
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.
• ?

39
A Specific Design Problem
simplify Rule 4 ? Rule 18 ? Rule   ?
40
A Specific Design Problem
Check using the Truth-Table Implementatio
n