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Lecture 2 Dr Richard Reilly Dept. of Electronic & Electrical Engineering Room 153, Engineering Building 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.

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 ?

2. The basic decision-making processes required of digital systems are binary.  Digital systems are often required to make tests. Is Condition C 1 true ? or Is condition C 2 false ?.  Examples of such decisions are :  Has button (switch) X been pushed ?,  Has temperature t max been reached ?.  Decisions of this kind are inherently binary because their outcomes are taken from the value-pair {true, false}. 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 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 Logic

Logical Operators: AND Gate AND gate Symbol Function ABC Truth-Table 000 100 010 111 Denote C thus defined : read as C = A AND B

OR Gate OR gate Symbol Function ABC Truth-Table 000 101 011 111 Denote C thus defined : read as C = A OR B

Inverter NOT gate If A = +5v If A = 0v  switch is closed  V o is 0 v  switch is open  V o is +5 v

The truth-table for this operator configuration is Inverter NOT gate A 1010 VoVo 0101

Inverter NOT gate (logic inverter) Symbol Function : AC Truth-Table 01 10 Denote C thus defined : read as C = NOT A

NAND gate If A = +5v and B = +5v If A = 0v and B = +5v If A = 0v and B = +5v If A = 0v and B = 0v  switches are closed  V o is 0 v  V o is +5 v  V o is +5 v  V o is +5 v

NAND Gate NAND gate Symbol Function ABC Truth-Table 001 101 011 110 Denote C thus defined :

NOR gate If A = +5v and B = +5v If A = 0v and B = +5v If A = 0v and B = +5v If A = 0v and B = 0v  switches are closed  V o is 0 v  V o is 0 v  V o is 0 v  V o is +5 v

NOR Gate NOR gate Symbol Function ABC Truth-Table 001 100 010 110 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 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 Functions using switches.

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.

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 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 greenTRUE B. 3 is an even numberFALSE We can combine these propositions to get Boolean Functions denoted by letters (from the end of the alphabet). e.g.Z = A AND BFALSE 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. Boolean Algebra

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

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  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 Complementation 1 0

As we have seen the complementation operation is physically realised by a gate or circuit called an inverter. Boolean Operators NOT AComplement of A A OR BLogical 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 Functions

Evaluate Evaluate a Boolean Function

List all possible versions of the input variables in a Truth Table Evaluate a Boolean Function A 0000111100001111 B 0011001100110011 C 0101010101010101

Boolean Operations : AND,OR and NOT A 0000111100001111 B 0011001100110011 C 0101010101010101 1010101010101010

A 0000111100001111 B 0011001100110011 C 0101010101010101 1010101010101010 0010001000100010

Finally ORing or Logical Addition Boolean Operations : AND,OR and NOT A 0000111100001111 B 0011001100110011 C 0101010101010101 1010101010101010 0010001000100010 0010111100101111

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.

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 

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 Rule 4 Rule 14 Rule 15

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 00110011 B 01010101 C 10111011

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 00110011 B 01010101 C 10111011 Product Terms

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  1 st, 2 nd and 4 th rows are selected. 

A Specific Design Problem simplify Rule 4  Rule 18  Rule : 

A Specific Design Problem Check using the Truth-Table : Implementation : A 00110011 B 01010101 10101010 10111011

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