 # Digital Electronics. Introduction to Number Systems & Codes Digital & Analog systems, Numerical representation, Digital number systems, Binary to Decimal.

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Digital Electronics

Introduction to Number Systems & Codes Digital & Analog systems, Numerical representation, Digital number systems, Binary to Decimal conversions, Decimal to Binary conversions, Hexa-decimal number systems, BCD code, The Gray code, Alphanumeric codes, Parity method for error detection, Octal & Hexa-decimal numbers, Complements. 1 St Cycle

Data can be analog or digital  Analog data refers to information that is continuous  Analog data take on continuous values  Analog signals can have an infinite number of values in a range  Digital data refers to information that has discrete states  Digital data take on discrete values  Digital signals can have only a limited number of values In data communications, we commonly use periodic analog signals and non-periodic digital signals. ANALOG AND DIGITAL

ANALOG & DIGITAL SYSTEMS An Analog system contains devices that manipulate physical quantities that are represented in analog form.  In an analog system, the quantities can vary over a continuous range of values.  For example, the amplitude of the output signal to the speaker in a radio receiver can have any value between zero and its maximum limit.  Other common analog systems are audio amplifiers, magnetic tape recording and playback equipment and a simple light dimmer switch.

 More over, a 1 can be encoded as a positive voltage and a 0 as zero voltage.  A digital signal can have more than two levels.  In this case, we can send more than 1 bit for each level. A Digital system is a combination of devices designed to manipulate logical information or physical quantities that are represented in digital form; that is, the quantities can take on only discrete values.  These devices are most often electronic, but they can also be mechanical, magnetic or pneumatic.  Some of the more familiar digital systems include digital computers, calculator, Digital audio and video equipments and telephone system etc. Continue………………

Digital Signal Analog Signal

Advantages of Digital Techniques  Very much easier to design.  Information storage is easy.  Accuracy and precision are easier to maintain throughout the system.  Operation can be programmed.  Less affected by noise.  More circuits can be fabricated on IC chips. Limitations of Digital Techniques:  The real world is analog  Processing digitized signals takes time.

NUMBER SYSTEMS Many Number systems are in use in digital technology. The most common are-  Decimal  Binary  Octal and  Hexadecimal.

Second Cycle Analysis & synthesis of digital logic circuits Basic logic functions OR operation with OR gates, AND operation with AND gates, NOR operation with NOR gates, Describing logic circuits algebraically, Evaluating logic circuit outputs, Implementing circuits from Boolean expressions, NOR gates & NAND gates. Boolean Algebra Boolean theorems, Demorgan’s Theorems, Universality of NAND gates & NOR gates

DIGITAL LOGIC GATES

 Boolean functions are implemented in digital computer circuits called gates.  A gate is an electronic device that produces a result based on two or more input values. –In reality, gates consist of one to six transistors, but digital designers think of them as a single unit. –Integrated circuits contain collections of gates suited to a particular purpose. Logic gates

 Electronic gates require a power supply.  Gate Input’s are driven by voltage having two nominal values.  The Output of a gate provides two nominal values of voltage.  There is always a Time Delay between an input being applied and the output responding. Point’s Need To Understand

Different Types of Logic gates Digital systems are said to be constructed by using gates. These gates are --  AND GATE  OR GATE  NOT GATE  NAND GATE  Ex-OR GATE  Ex-NOR GATE

Truth Tables  Used to describe the functional behavior of a Boolean expression and/or Logic circuit.  Each row in the truth table represents a unique combination of the input variables.  For n input variables, there are 2 n rows.  The output of the logic function is defined for each row.  Each row is assigned a numerical value, with the rows listed in ascending order.  The order of the input variables defined in the logic function is important.

AND Gate  The AND gate is an electronic circuit that gives a high output only if all its inputs are high. A dot(.) is used to show the AND operation. Truth table 2 Input AND gate ABZ=A.B 000 010 100 111 Circuit Symbol A B AND Z=A.B

Summary of the AND gates  The AND operation is performed the same as ordinary multiplication of 1s and 0s.  An AND gate is a logic circuit that performs the AND operation on the circuit’s input.  An AND gate output will be 1 only for the case when all inputs are 1; for all other cases, the output will be 0.  The expression Z=A.B is read as “Z equals A AND B.”

OR Gate  The OR gate is an electronic circuit that gives a high output. One or more of its inputs are high. A plus (+) is used to show the OR operation. Truth table 2 Input OR gate ABZ=A+B 000 011 101 111 A B OR Z=A+B Circuit Symbol

Summary of the OR gates  The OR operation produces a result (output) of 1 whenever any input is a1. Otherwise the output is 0.  An OR gate is a logic circuit that performs an OR operation on the circuit’s input.  The expression Z=A+B is read as “Z equals A OR B.”

NOT Gate  The NOT gate is an electronic circuit that produces an inverted version of the input at its output.  More commonly called an Inverter. Truth table Circuit Symbol NOT GATE AA 01 10 A A

NAND Gate  The NAND gate which is to NOT - AND gate. Truth table Circuit Symbol A.B A B NAND 2 Input NAND gate ABA.B 001 011 101 110

NOR Gate  The NOR gate which is to NOT - OR gate. Truth table 2 Input NOR gate ABA.B 001 010 100 110 Circuit Symbol A+B A B NOR

EX-OR Gate  The Exclusive –OR gate is a circuit which will give a high output.An encircled plus sign(  ) is used to show the EX-OR operation. Truth table Circuit Symbol A  B A B EX-OR 2 Input EXOR gate ABA BA B 000 011 101 110

EX-NOR Gate  The Exclusive –NOR gate is circuit does the opposite to the EX-NOR gate. The symbol is an EX-NOR gate with a small circle on the output. The small circle represents inversion. Truth table 2 Input EXNOR gate ABA BA B 001 010 100 111 Circuit Symbol A  B A B EX-NOR

 NAND and NOR are known as universal gates because they are inexpensive to manufacture and any Boolean function can be constructed using only NAND or only NOR gates. Universal Gates

Boolean Expressions Boolean expressions are composed of Literals – variables and their complements Logical operations Examples F = A.B'.C + A'.B.C' + A.B.C + A'.B'.C' F = (A+B+C').(A'+B'+C).(A+B+C) F = A.B'.C' + A.(B.C' + B'.C) literals logic operations

Boolean Expressions Boolean expressions are realized using a network (or combination) of logic gates.  Each logic gate implements one of the logic operations in the Boolean expression  Each input to a logic gate represents one of the literals in the Boolean expression f A B logic operations literals

Boolean Expressions Boolean expressions are evaluated by Substituting a 0 or 1 for each literal. Calculating the logical value of the expression. A Truth Table specifies the value of the Boolean expression for every combination of the variables in the Boolean expression. For an n-variable Boolean expression, the truth table has 2 n rows (one for each combination).

Boolean Algebra George Boole developed an algebraic description for processes involving logical thought and reasoning.  Became known as Boolean Algebra Claude Shannon later demonstrated that Boolean Algebra could be used to describe switching circuits.  Switching circuits are circuits built from devices that switch between two states (e.g. 0 and 1).  Switching Algebra is a special case of Boolean Algebra in which all variables take on just two distinct values Boolean Algebra is a powerful tool for analyzing and designing logic circuits.

Basic Laws and Theorems

DeMorgan's Laws Can be stated as follows:  The complement of the product (AND) is the sum (OR) of the complements. (X.Y)' = X' + Y'  The complement of the sum (OR) is the product (AND) of the complements. (X + Y)' = X'. Y' Easily generalized to n variables. Can be proven using a Truth table

Proving DeMorgan's Law (X. Y)' = X' + Y'

DeMorgan's Theorems x 1 x 2 x 1 x 2 x 1 x 2 x 1 x 2 += (a) x 1 x 2 + x 1 x 2 = (b) x 1 x 2 x 1 x 2 x 1 x 2 x 1 x 2

Importance of Boolean Algebra Boolean Algebra is used to simplify Boolean expressions. –Through application of the Laws and Theorems discussed Simpler expressions lead to simpler circuit realization, which, generally, reduces cost, area requirements, and power consumption. The objective of the digital circuit designer is to design and realize optimal digital circuits.

Algebraic Simplification Justification for simplifying Boolean expressions: –Reduces the cost associated with realizing the expression using logic gates. –Reduces the area (i.e. silicon) required to fabricate the switching function. –Reduces the power consumption of the circuit. In general, there is no easy way to determine when a Boolean expression has been simplified to a minimum number of terms or minimum number of literals. –No unique solution

Algebraic Simplification Boolean (or Switching) expressions can be simplified using the following methods: 1.Multiplying out the expression. 2.Factoring the expression. 3.Combining terms of the expression. 4.Eliminating terms in the expression. 5.Eliminating literals in the expression 6.Adding redundant terms to the expression.

Examples

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