Principles of Digital Electronics

Principles of Digital Electronics

Dr. Sapana Rane

LULU Publications Inc. NC, USA

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© 2022, Sapana Rane

ISBN 978-1-387-89939-5

LULU Publications Inc. NC, USA

This work is licensed under a Standard Copyright License. All rights reserved.

Index

Chapter 1: Number Systems and  Digital Codes

Chapter 2: Logic Gates and Boolean Algebra

Chapter 3: Boolean Algebra and Karnaugh Maps

Chapter 4: Combinational Circuits

Number Systems and

Digital Codes

UNIT I

Syllabus

Introduction to Decimal, Binary and Hexadecimal number systems and their inter-conversions, binary addition and binary subtraction using 2’s complement, Binary Coded Decimal number, Gray Codes, Gray to Binary and Binary to Gray conversion, Alphanumeric representation in ASCII codes.

The number system is the system of naming or representing numbers. There are various types of number systems like binary, decimal, hexadecimal etc. The computer is a digital circuit which stores the information in the form of numbers and digital codes. This chapter covers the entire concepts of the number system, with their types, conversions and questions. It also deals with binary arithmetic and various digital codes with applications and examples.

1.1      Introduction to Digital Electronics

The signal and systems can be classified into two basic categories as follows:

1.      Analog signals and system

2.      Digital signals and system

1.1.1      Analog Signals and System

Analog signals have infinite number of different magnitudes or values. They vary continuously with time. There are various examples of analog signals such as sine wave, triangular wave etc.

Disadvantages of Analog systems

1.      Less accuracy

2.      Analog systems are more affected by noise.

3.      It is impossible to separate the analog signal, if it is contaminated by noise

1.1.2      Digital Signals

A signal is called as a digital signal if it has only a finite number of predetermined distinct magnitudes. Depending on the number of distinct magnitudes, the digital signals are classified as follows:

1.2      Number System

(Define number system. Define radix or base with reference to number system. Name the number systems supported by computer architecture.)

In any number system, there is an ordered set of symbols known as digits with rules defined for performing arithmetic operations like addition, multiplication etc.

The number system is defined as a system of writing for expressing numbers.

It is the mathematical notation for representing numbers of a given set by using digits or other symbols in a consistent manner.

It provides a unique representation of every number and allows us to perform arithmetic operations like addition, subtraction, multiplication and division.

The value of any digit in a number can be determined by:

The digit itself

Its position in the number

The base of the number system

In number system, a base or radix is the number of different digits or combination of digits and letters that a counting system uses to represent numbers.

For example, the most common base used is the decimal system. The dec means 10, it uses the 10 digits from 0 to 9 to represent the number system.

There are various types of number systems like binary, decimal, octal hexadecimal etc.

In digital electronics, the number system is used for representing the information.

The computer is digital system that recognizes every value the user enters as a number.

The computer architecture supports following number systems so we need to study them and also need to know the conversion technique between them.

Binary number system

Octal number system

Decimal number system

Hexadecimal (hex) number system

1.2.1      The Decimal Number System

GQ.       Write short note on Decimal number System.

GQ.       What is base of Decimal number system?

GQ.       Mention applications of Decimal number system.

Important characteristics of decimal systems are:

1.      The decimal number system has ten digits i.e. 0 through 9

2.      It uses the base of 10.

3.      Each place column number represents a different multiple of 10. These multiples are also called as weighted values. The weighted values of each position are as shown in Table 1.

Table 1: Decimal system with position and corresponding weighted value.

The leftmost digit is called as most significant digit of a number as it has highest weight and the rightmost digit is called as least significant digit of a number as it has lowest weight.

Applications

We use decimals every day, while dealing with money, weight, length etc.

Decimal numbers are used in situations where more precision is required, than the whole numbers can provide.

Ex. 1 : Express the decimal numbers 35 as a sum of the values of each digit (expanded form).

Soln. :

The digit 3 has weight of 10, which is 10¹, as indicated by its position. The digit 5 has a weight of 1 and which is 10⁰, as indicated by its position.

    35      =      (3 × 10¹) + (5 × 10⁰)

=      (3 × 10) + (5 × 1) = 30 + 5

Ex. 2 : Express the decimal number 756.34 as a sum of the values of each digit (expanded form).

Soln. :

The whole number digit 7 has a weight of 100, which is 10², the digit 5 has a weight of 10, which is 10¹, the digit

6 has a weight of 1, which is 10⁰, the fractional digit 3 has a weight of 0.1, which is 10–1, and the fractional digit 4 has a weight of 0.01, which is 10–2.

756.34      =      (7 × 10²) + (5 × 10¹) + (6 × 10⁰) + (3 × 10–1)

+ (4 × 10–2)

=      (7×100) + (5×10) + (6×1) + (3×0.1) + (4×0.01)

=    700      +  50      +    6    +    0.3    +  0.04

Ex. 3 : Express the decimal number 9846.124 as a sum of the values of each digit(expanded form).

Soln. :

The number digit 9 has a weight of 1000, which is 10³ the number digit 8 has a weight of 100, which is 10², the digit 4 has a weight of 10, which is 10¹, the digit 6 has a weight of 1, which is 10⁰, the fractional digit 1 has a weight of 0.1, which is 10–1, the fractional digit 2 has a weight of 0.01, which is 10–2 and the fractional digit 4 has a weight of 0.001, which is 10–3.

9846.124      =      (9 × 10³) + (8 × 10²) + (4 × 10¹) + (6 × 10⁰)                         + (1 × 10–1) + (2 × 10–2) + (4 × 10–3)

=      (9 × 1000) +  (8 × 100) + (4 × 10) + (6 × 1)

+ (1 × 0.1) + (2 × 0.01) + (4 × 0.001)

=    9000 + 800 + 40 + 6 + 0.1 + 0.02 + 0.004

1.2.2      The Binary Number System

GQ.      Write short note on Binary number System.

GQ.      What is base of Binary number system?

GQ.      Mention applications of Binary number system.

GQ.      Define bit, nibble, byte and word.

GQ.      Write the size of bit, nibble, byte and word.

GQ.      Write the table of 4 bit binary number and its corresponding decimal number.

1.      Binary system is another way to represent quantities.

2.      Binary number system uses only two digits i.e. 0 and 1.

3.      Binary digits (0 and 1) are called as bits.

4.      The leftmost bit in a given binary system is called as most significant bit (MSB) that has highest weight and the rightmost bit in a given binary system is called as least significant bit (LSB) that has lowest weight.

5.      In binary number system, a group of four bits is known as a nibble, and a group of eight bits is known as byte.

Binary number formats

Binary numbers are typically written as a sequence of bits. Computer/Digital system standards defined boundaries for these bits for data presentation. These boundaries are:

In any number base, we may add as many leading zeros as we wish without changing its value. However, we normally add leading zeros to adjust the binary number to a desired size boundary. For example, we can represent the number five as:

The bit

1.      The smallest unit of data is defined as a single bit.

2.      With single bit, it is possible to represent any two distinct items like on or off, true of false etc.

The nibble

1.      A nibble is a combination of four bits.

2.      With nibble, we can represent up to 16 distinct values.

3.      For hexadecimal numbers, the values 0,1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F are represented with four bits.

4.      BCD uses ten different digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) and requires four bits.

5.      The structure of nibble is as shown in fig.1.

6.      Bit ‘b0’ is acting as LSB and bit ‘b3’ is acting as MSB.

MSB                               LSB

Fig. 1: Structure of nibble

The byte

1.      A byte is a combination of 8 binary bits

2.      A byte contains two nibbles.

3.      The structure of a byte is as shown in figure 2.

4.      Bit ‘b0’ is acting as LSB and bit ‘b7’ is acting as MSB.

MSB                                      LSB

    High order nibble             Low order nibble

Fig. 2: Structure of byte

The word

1.      A word is a combination of 16 bits.      2.      It consists of two bytes.

3.      The format of a word is as shown in Fig. 3.      4.      Bit ‘b0’ is acting as LSB and bit ‘b15’ is acting as MSB.

    MSB            LSB

High order byte                              Low order byte

Fig. 3: Format of a word

Any binary number can be converted into its equivalent decimal number using the weights assigned to each bit position as shown in table 2.

Applications

Binary is found in computer system. The computer programming is based on the 2-digit number system used in digital.

Digital encoding is the process of taking data and representing it with discrete bits of information.

These discrete bits consist of the 0s and 1s of the binary system.

Table 2: 4 bit binary numbers and their

corresponding decimal numbers.

1.2.3      Binary to Decimal conversion

GQ.      Write the steps to convert binary to decimal number system.

Binary to decimal conversion is one of a most important operations used in digital electronics and communications.

This conversion is used to observe the value of binary numbers in its equivalent decimal number.

Representing binary in decimal number system is the best way to easily understand such operations.

Steps to convert binary to decimal number system:

1.      Write down the binary number and list the powers of

2 from right to left.

2.      Write the digits of the binary number below their corresponding powers of two.

3.      Multiply the digits in the binary number with their corresponding powers of two.

4.      Write down the final value of each power of two.

5.      Add the final values to get the decimal equivalent of binary.

Examples

1.       Find the equivalent decimal number for the binary (1010)2.

Answer :    (1010)2 =  (10)10

2.       Find the equivalent decimal number for the binary (110101)2

Answer :  (110101)2 =  (53)10

3.       Find the equivalent decimal number for the binary