digital electronics (b.tech.) chepter 1

 

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 INTRODUCTION ABOUT DIGITAL SYSTEM

A Digital system is an interconnection of digital modules and it is a system that manipulates 

discrete elements of information that is represented internally in the binary form.

Now a day’s digital systems are used in wide variety of industrial and consumer products such as 

automated industrial machinery, pocket calculators, microprocessors, digital computers, digital 

watches, TV games and signal processing and so on.

 Characteristics of Digital Systems

 Digital systems manipulate discrete elements of information.

 Discrete elements are nothing butthedigitssuchas 10 decimaldigitsor 26 letters of alphabets

and so on.

 Digital systems use physical quantities called signals to represent discrete elements.

 In digital systems, the signals have two discrete values and are therefore said to be binary.

 A signal in digital system represents one binary digit called a bit. The bit has a value either 0

or 1.

 Analog systems vs Digital systems

Analog system process information that varies continuously i.e. they process time varying

signals that can take on any values across a continuous range of voltage, current or any physical 

parameter.

Digital systems use digital circuits that can process digital signals which can take either 0 or 1for 

binary system.

 Advantages of Digital system over Analog system

1. Ease of programmability: The digital systems can be used for different applications by 

simply changing the program without additional changes in hardware.

2. Reduction in cost of hardware: The cost of hardware gets reduced by use of digital 

components and this has been possible due to advances in IC technology. With ICs the 

number of components that can be placed in a given area of Silicon are increased which helps 

in cost reduction. High speed Digital processing of data ensures high speed of operation 

which is possible due to advances in Digital Signal Processing.

3. High Reliability: Digital systems are highly reliable one of the reasons for that is use of error 

correction codes.

4. Design is easy: The design of digital systems which require use of Boolean algebra and other

digital techniques is easier compared to analogue designing.

5. Result can be reproduced easily: Since the output of digital systems unlike analogue

systems is independent of temperature, noise, humidity and other characteristics of 

1 Basic of Digital Electronics & Number System

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components the reproducibility of results is higher in digital systems than in analogue

systems.

 Disadvantages of Digital Systems

 Use more energy than analogue circuits to accomplish the same tasks, thus producing more

heat as well.

 Digital circuits are often fragile in that if a single piece of digital data is lost or misinterpreted

the meaning of large blocks of related data can completely change.

 Digital computer manipulates discrete elements of information by means of a binary code.

 Quantization error during analogue signal sampling.

 Number System

Common Number Systems

There are mainly four number systems which are used in digital electronics platform.

1. Decimal number system

 The decimal number system contains ten unique symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

 The base or radix is 10.

 9's and 10's complements are possible for any decimal number.

2. Binary number system

 The binary number system contains two unique symbols 0, 1.

 The base or radix is 2.

 1's and 2's complements are possible for any binary number.

3. Octal number system

 The octal number system contains eight unique symbols 0, 1, 2, 3, 4, 5, 6, 7.

 The base or radix is 8.

 7's and 8's complements are possible for any octal number.

4. Hexadecimal number system

 The hexadecimal number system contains sixteen unique symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 

9, Α, B, C, D, E, F.

 The base or radix is 16.

 15's and 16's complements are possible for any hexadecimal number.

In general, if radix or base of a number system is "r", then there is possibility of r's complement 

and (r - 1)'s complement of a number.

 Decimal to Binary Conversion

 The decimal integer is converted to the binary integer number by successive division by 2, 

and the decimal fraction is converted to the binary fraction number by successive 

multiplication by 2.

 In the successive division-by-2 method, the given decimal integer number is successively 

divided by 2 till the quotient is 0.

 The remainders read from bottom to top give the equivalent binary integer number.

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 In the successive multiplication-by-2 method, the given decimal fraction and the subsequent 

fractions are successively multiplied by 2, till the fraction part of the product is 0 or till the 

desired accuracy is obtained.

 The integers read from top to bottom give the equivalent binary integer number.

 To convert a mixed number to binary, convert the integer and fraction parts separately to 

binary and then combine them.

 Example: (125.6875) 10 =( )2

 Binary to Decimal Conversion

 Binary numbers may be converted to their decimal equivalents by the positional weights 

method. In this method, each binary digit of the number is multiplied by its position weight ( 

2

n

, where n is the weight of the bit) and the product terms are added to obtain the decimal 

number.

 Example: (101011.11)2 = ( )10

= 1 × 2

5

+ 0 × 2

4

+ 1 × 23

+ 0 × 2

2

+ 1 × 21

+ 1 × 2

0

+ 1 × 2-1

+ 1 × 2

-2

= 32 + 0 + 8 + 0 + 2 + 1 + 0.5 + 0.25

= 43.75

Hence, (101011.11)2

= (43.75)10

 Decimal to Octal Conversion

 The decimal integer is converted to the octal integer number by successive division by 8, and 

the decimal fraction is converted to the octal fraction number by successive multiplication by 

8.

 In the successive division-by-8 method, the given decimal integer number is successively 

divided by 8 till the quotient is 0.

 The remainders read from bottom to top give the equivalent octal integer number.

 In the successive multiplication-by-8 method, the given decimal fraction and the subsequent 

fractions are successively multiplied by 8, till the fraction part of the product is 0 or till the 

desired accuracy is obtained.

 The integers read from top to bottom give the equivalent octal integer number.

 To convert a mixed number to octal, convert the integer and fraction parts separately to octal 

and then combine them.

 Example: (125.6875)10 = ( )8

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 Octal to Decimal Conversion

 Octal numbers may be converted to their decimal equivalents by the positional weights 

method.

 In this method, each octal digit of the number is multiplied by its position weight (8", where 

n is the weight of the bit) and the product terms are added to obtain the decimal number.

 Example: (724.25)8 = ( )10

= 7 × 8

2

+ 2 × 8

1

+ 4 × 8

0

+ 2 × 8 -1

+ 5 × 8

-2

= 448 + 16 + 4 + 0.25 + 0.0781

= 468.3281

Hence, (724.25)8 = (468.3281)10

 Decimal to Hexadecimal Conversion

 The decimal integer is converted to the hexadecimal integer number by successive division 

by 16,

 and the decimal fraction is converted to the hexadecimal fraction number by successive

 multiplication by 16.

 In the successive division-by-16 method, the given decimal integer number is successively 

divided by 16 till the quotient is 0.

 The remainders read from bottom to top give the equivalent hexadecimal integer number.

 In the successive multiplication-by-16 method, the given decimal fraction and the subsequent 

fractions are successively multiplied by 16, till the fraction part of the product is 0 or till the 

desired accuracy is obtained.

 The integers read from top to bottom give the equivalent hexadecimal integer number.

 To convert a mixed number to hexadecimal, convert the integer and fraction parts separately 

to hexadecimal and then combine them.

 Example: (2598.675)10 = ( )16

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 Hexadecimal to Decimal Conversion

 Hexadecimal numbers may be converted to their decimal equivalents by the positional 

weights method.

 In this method, each hexadecimal digit of the number is multiplied by its position weight 

(16n

, where n is the weight of the bit) and the product terms are added to obtain the decimal 

number.

 Example: (A0F 9.0EB)16 = ( )10

= 10 × 163

+ 0 × 162

+ 15 × 161

+ 9 × 160

+ 0 × 16-1

+ 14 × 16-2

+ 11 × 163

= 40960 + 0 + 240 + 9 + 0 + 0.0546 + 0.0026 

= 41209.0572

Hence, (A0F9.0EB)16 = (41209.0572)10

 Octal to Binary Conversion

 To convert a given octal number to a binary, just replace each octal digit by its 3-bit binary 

equivalent as per below table.

Octal Number Binary Number

0 000

1 001

2 010

3 011

4 100

5 101

6 110

7 111

 Example: - (367.52)8 = ( )2

= 011 110 111. 101 010 

= 11110111.10101

Hence, (367.52)8

= (11110111.10101)2

 Binary to Octal Conversion

 To convert a binary number to an octal number, starting from the binary point make groups 

of 3 bits each (i.e. from point (".") in binary number, group of 3 bits in left side and group of 

3 bits in right side), if there are not 3 bits available at last, just stuff "0" to make 3 bits’ group.

 Replace each 3-bit binary group by the equivalent octal digit.

 Example: - (110101.101010)2 = ( )8

= 110 101. 101 010

= 65.52 

Hence, (110101.101010)2 = (65.52)8

 Hexadecimal to Binary Conversion

 To convert a given hexadecimal number to a binary, just replace each hexadecimal digit by 

its 4- bit binary equivalent as per below table.