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• Introduction to Data Representation
• Computer Science

Data can be anything, including a number, a name, musical notes, or the colour of an image. The way that we stored, processed, and transmitted data is referred to as data representation. We can use any device, including computers, smartphones, and iPads, to store data in digital format. The stored data is handled by electronic circuitry. A bit is a 0 or 1 used in digital data representation.

Data Representation Techniques

## Classification of Computers

Computer scans are classified broadly based on their speed and computing power.

1. Microcomputers or PCs (Personal Computers): It is a single-user computer system with a medium-power microprocessor. It is referred to as a computer with a microprocessor as its central processing unit.

Microcomputer

2. Mini-Computer: It is a multi-user computer system that can support hundreds of users at the same time.

Types of Mini Computers

3. Mainframe Computer: It is a multi-user computer system that can support hundreds of users at the same time. Software technology is distinct from minicomputer technology.

Mainframe Computer

4. Super-Computer: With the ability to process hundreds of millions of instructions per second, it is a very quick computer. They  are used for specialised applications requiring enormous amounts of mathematical computations, but they are very expensive.

Supercomputer

## Types of Computer Number System

Every value saved to or obtained from computer memory uses a specific number system, which is the method used to represent numbers in the computer system architecture. One needs to be familiar with number systems in order to read computer language or interact with the system.

Types of Number System

## 1. Binary Number System

There are only two digits in a binary number system: 0 and 1. In this number system, 0 and 1 stand in for every number (value). Because the binary number system only has two digits, its base is 2.

A bit is another name for each binary digit. The binary number system is also a positional value system, where each digit's value is expressed in powers of 2.

## Characteristics of Binary Number System

The following are the primary characteristics of the binary system:

It only has two digits, zero and one.

Depending on its position, each digit has a different value.

Each position has the same value as a base power of two.

Because computers work with internal voltage drops, it is used in all types of computers.

Binary Number System

## 2. Decimal Number System

The decimal number system is a base ten number system with ten digits ranging from 0 to 9. This means that these ten digits can represent any numerical quantity. A positional value system is also a decimal number system. This means that the value of digits will be determined by their position.

## Characteristics of Decimal Number System

Ten units of a given order equal one unit of the higher order, making it a decimal system.

The number 10 serves as the foundation for the decimal number system.

The value of each digit or number will depend on where it is located within the numeric figure because it is a positional system.

The value of this number results from multiplying all the digits by each power.

Decimal Number System

## Decimal Binary Conversion Table

3. octal number system.

There are only eight (8) digits in the octal number system, from 0 to 7. In this number system, each number (value) is represented by the digits 0, 1, 2, 3,4,5,6, and 7. Since the octal number system only has 8 digits, its base is 8.

## Characteristics of Octal Number System:

Contains eight digits: 0,1,2,3,4,5,6,7.

Also known as the base 8 number system.

Each octal number position represents a 0 power of the base (8).

An octal number's last position corresponds to an x power of the base (8).

Octal Number System

There are sixteen (16) alphanumeric values in the hexadecimal number system, ranging from 0 to 9 and A to F. In this number system, each number (value) is represented by 0, 1, 2, 3, 5, 6, 7, 8, 9, A, B, C, D, E, and F. Because the hexadecimal number system has 16 alphanumeric values, its base is 16. Here, the numbers are A = 10, B = 11, C = 12, D = 13, E = 14, and F = 15.

## Characteristics of Hexadecimal Number System:

A system of positional numbers.

Has 16 symbols or digits overall (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F). Its base is, therefore, 16.

Decimal values 10, 11, 12, 13, 14, and 15 are represented by the letters A, B, C, D, E, and F, respectively.

A single digit may have a maximum value of 15.

Each digit position corresponds to a different base power (16).

Since there are only 16 digits, any hexadecimal number can be represented in binary with 4 bits.

So, we've seen how to convert decimals and use the Number System to communicate with a computer. The full character set of the English language, which includes all alphabets, punctuation marks, mathematical operators, special symbols, etc., must be supported by the computer in addition to numerical data.

## Learning By Doing

1. Which computer is the largest in terms of size?

Minicomputer

Micro Computer

2. The binary number 11011001 is converted to what decimal value?

## Solved Questions

1. Give some examples where Supercomputers are used.

Ans: Weather Prediction, Scientific simulations, graphics, fluid dynamic calculations, Nuclear energy research, electronic engineering and analysis of geological data.

2. Which of these is the most costly?

Mainframe computer

Ans: C) Supercomputer

## FAQs on Introduction to Data Representation

1. What is the distinction between the Hexadecimal and Octal Number System?

The octal number system is a base-8 number system in which the digits 0 through 7 are used to represent numbers. The hexadecimal number system is a base-16 number system that employs the digits 0 through 9 as well as the letters A through F to represent numbers.

2. What is the smallest data representation?

The smallest data storage unit in a computer's memory is called a BYTE, which comprises 8 BITS.

3. What is the largest data unit?

The largest commonly available data storage unit is a terabyte or TB. A terabyte equals 1,000 gigabytes, while a tebibyte equals 1,024 gibibytes.

## Data representation 1: Introduction

This course investigates how systems software works: what makes programs work fast or slow, and how properties of the machines we program impact the programs we write. We discuss both general ideas and specific tools, and take an experimental approach.

• How do computers represent different kinds of information?
• How do data representation choices impact performance and correctness?
• What kind of language is understood by computer processors?
• How is code you write translated to code a processor runs?
• How do hardware and software defend against bugs and attacks?
• How are operating systems interfaces implemented?
• What kinds of computer data storage are available, and how do they perform?
• How can we improve the performance of a system that stores data?
• How can programs on the same computer cooperate and interact?
• What kinds of operating systems interfaces are useful?
• How can a single program safely use multiple processors?
• How can multiple computers safely interact over a network?
• Six problem sets
• Midterm and final
• Starting mid-next week
• Attendance checked for simultaneously-enrolled students
• Rough breakdown: >50% assignments, <35% tests, 15% participation
• Course grading: A means mastery

## Collaboration

Discussion, collaboration, and the exchange of ideas are essential to doing academic work, and to engineering. You are encouraged to consult with your classmates as you work on problem sets. You are welcome to discuss general strategies for solutions as well as specific bugs and code structure questions, and to use Internet resources for general information.

However, the work you turn in must be your own—the result of your own efforts. You should understand your code well enough that you could replicate your solution from scratch, without collaboration.

In addition, you must cite any books, articles, online resources, and so forth that helped you with your work, using appropriate citation practices; and you must list the names of students with whom you have collaborated on problem sets and briefly describe how you collaborated. (You do not need to list course staff.)

## On our programming language

We use the C++ programming language in this class.

C++ is a boring, old, and unsafe programming language, but boring languages are underrated . C++ offers several important advantages for this class, including ubiquitous availability, good tooling, the ability to demonstrate impactful kinds of errors that you should understand, and a good standard library of data structures.

Pset 0 links to several C++ tutorials and references, and to a textbook.

Each program runs in a private data storage space. This is called its memory . The memory “remembers” the data it stores.

Programs work by manipulating values . Different programming languages have different conceptions of value; in C++, the primitive values are integers, like 12 or -100; floating-point numbers, like 1.02; and pointers , which are references to other objects.

An object is a region of memory that contains a value. (The C++ standard specifically says “a region of data storage in the execution environment, the contents of which can represent values”.)

## Objects, values, and variables

Which are the objects? Which are the values?

Variables generally correspond to objects, and here there are three objects, one for each variable i1 , i2 , and i3 . The compiler and operating system associate the names with their corresponding objects. There are three values, too, one used to initialize each object: 61 , 62 , and 63 . However, there are other values—for instance, each argument to the printf calls is a value.

What does the program print?

i1: 61 i2: 62 i3: 63

C and C++ pointer types allow programs to access objects indirectly. A pointer value is the address of another object. For instance, in this program, the variable i4 holds a pointer to the object named by i3 :

There are four objects, corresponding to variables i1 through i4 . Note that the i4 object holds a pointer value, not an integer. There are also four values: 61 , 62 , 63 , and the expression &i3 (the address of i3 ). Note that there are three integer values, but four values overall.

What does this program print?

i1: 61 i2: 62 i3: 63 value pointed to by i4: 63

Here, the expressions i3 and *i4 refer to exactly the same object. Any modification to i3 can be observed through *i4 and vice versa. We say that i3 and *i4 are aliases : different names for the same object.

We now use hexdump_object , a helper function declared in our hexdump.hh helper file , to examine both the contents and the addresses of these objects.

Exactly what is printed will vary between operating systems and compilers. In Docker in class, on my Apple-silicon Macbook, we saw:

But on an Intel-based Amazon EC2 native Linux machine:

The data bytes look similar—identical for i1 through i3 —but the addresses vary.

But on Intel Mac OS X: 103c63020 3d 00 00 00 |=...| 103c5ef60 3e 00 00 00 |>...| 7ffeebfa4abc 3f 00 00 00 |?...| 7ffeebfa4ab0 bc 4a fa eb fe 7f 00 00 |.J......| And on Docker on an Intel Mac: 56499f239010 3d 00 00 00 |=...| 56499f23701c 3e 00 00 00 |>...| 7fffebf8b19c 3f 00 00 00 |?...| 7fffebf8b1a0 9c b1 f8 eb ff 7f 00 00 |........|

A hexdump printout shows the following information on each line.

• An address , like 4000004010 . This is a hexadecimal (base-16) number indicating the value of the address of the object. A line contains one to sixteen bytes of memory starting at this address.
• The contents of memory starting at the given address, such as 3d 00 00 00 . Memory is printed as a sequence of bytes , which are 8-bit numbers between 0 and 255. All modern computers organize their memory in units of 8-bit bytes.
• A textual representation of the memory contents, such as |=...| . This is useful when examining memory that contains textual data, and random garbage otherwise.

## Dynamic allocation

Must every data object be given a name? No! In C++, the new operator allocates a brand-new object with no variable name. (In C, the malloc function does the same thing.) The C++ expression new T returns a pointer to a brand-new, never-before-seen object of type T . For instance:

This prints something like

The new int{64} expression allocates a fresh object with no name of its own, though it can be located by following the i4 pointer.

• i3 and i4 , which are objects corresponding to variables declared local to main , are located very close to one another. In fact they are just 4 bytes part: i3 directly abuts i4 . Their addresses are quite high. In native Linux, in fact, their addresses are close to 2 47 !
• i1 and i2 are at much lower addresses, and they do not abut. i2 ’s location is below i1 , and about 0x2000 bytes away.
• The anonymous storage allocated by new int is located between i1 / i2 and i3 / i4 .

Although the values may differ on other operating systems, you’ll see qualitatively similar results wherever you run ./objects .

What’s happening is that the operating system and compiler have located different kinds of object in different broad regions of memory. These regions are called segments , and they are important because objects’ different storage characteristics benefit from different treatment.

i2 , the const int global object, has the smallest address. It is in the code or text segment, which is also used for read-only global data. The operating system and hardware ensure that data in this segment is not changed during the lifetime of the program. Any attempt to modify data in the code segment will cause a crash.

i1 , the int global object, has the next highest address. It is in the data segment, which holds modifiable global data. This segment keeps the same size as the program runs.

After a jump, the anonymous new int object pointed to by i4 has the next highest address. This is the heap segment, which holds dynamically allocated data. This segment can grow as the program runs; it typically grows towards higher addresses.

After a larger jump, the i3 and i4 objects have the highest addresses. They are in the stack segment, which holds local variables. This segment can also grow as the program runs, especially as functions call other functions; in most processors it grows down , from higher addresses to lower addresses.

## Introduction to Computer Data Representation

Author(s) : peter fenwick.

DOI: 10.2174/97816080588221140101 eISBN: 978-1-60805-882-2, 2014 ISBN: 978-1-60805-883-9

Back Recommend this Book to your Library Cite as

## Book Web Price: US \$

Cite this Book as:

For Books Peter Fenwick , " Introduction to Computer Data Representation ", Bentham Science Publishers (2014). https://doi.org/10.2174/97816080588221140101

Page: i-ii (2) Author: James R. Goodman DOI: 10.2174/9781608058822114010001

Page: iii-iv (2) Author: Peter Fenwick DOI: 10.2174/9781608058822114010002

## Introduction and Overview

Page: 1-4 (4) Author: Peter Fenwick DOI: 10.2174/9781608058822114010003

## Numbers and Computers

Page: 5-11 (7) Author: Peter Fenwick DOI: 10.2174/9781608058822114010004 PDF Price: \$15

We start with the basic ideas of "numbers" and "counting" and how the concepts and requirements differ with different levels of technical and administrative sophistication. This is followed by a brief summary of the development of calculating devices, to the structure of the "von Neumann" computer. Finally, we introduce the requirements of a computer, the importance of memory and an overview of data storage within that memory.

## Binary and Other Representations

Page: 17-28 (12) Author: Peter Fenwick DOI: 10.2174/9781608058822114010005 PDF Price: \$15

The initial idea of "number" leads quickly to the need to represent numbers and then manipulate numbers. We emphasise the representation as the terms of polynomial in some base and especially "binary numbers" (to base 2) and "decimal numbers" (to base 10). This leads into conversion between bases and the representation of fractions in binary.

## Signed, and Other, Representations

Page: 29-44 (16) Author: Peter Fenwick DOI: 10.2174/9781608058822114010006 PDF Price: \$15

Although the "natural" integers 1,2,3,. . .) are adequate for many, everyday, purposes they do not handle concepts such as debts; the representation must extended to handle negative values. This chapter therefore introduces the three conventional forms of signed binary representation (sign and magnitude, ones complement and two complement). Finally, it describes some other important number representations, especially mixed base (for non-metric measures), but also the more-specialised redundant codings, Gray codes and Zeckendorf representations.

## Basic Arithmetic and Logic

Page: 45-75 (31) Author: Peter Fenwick DOI: 10.2174/9781608058822114010007 PDF Price: \$15

Given that quantities may be represented as bit patterns within the computer, how may these patterns be manipulated to achieve useful results? We look first at the "basic" operations of addition and subtraction in the various binary number representations, extending to addition of multiple-precision values and decimal addition. Then we examine the logical operations, where the bits are "bits", with no numerical significance. Operations here include AND, OR, NOT, XOR, shifting of various types, bit-field operations and, finally, parity.

## Computer Arithmetic

Page: 77-102 (26) Author: Peter Fenwick DOI: 10.2174/9781608058822114010008 PDF Price: \$15

This chapter firstly extends the earlier principles of addition to provide faster adders. It then describes the underlying principles of binary multiplication and division, and especially techniques for fast operations.

## Floating-Point Representations

Page: 103-126 (24) Author: Peter Fenwick DOI: 10.2174/9781608058822114010009 PDF Price: \$15

This chapter extends the earlier representations of integers to the equivalent of the real numbers of scientific calculation. It discusses the basic ideas, especially with reference to the IEEE 754 standard, and contrasting with descriptions of the IBM S/360 and Burroughs B6700 representations. There is extensive discussion of the requirements of ideal floating-point representations and the failings of practical implementations. Special mention is made of the requirements of range, precision and rounding. It concludes with examples of straight-forward calculations which can easily overwhelm many floating-point systems.

## Logarithmic Representations

Page: 127-132 (6) Author: Peter Fenwick DOI: 10.2174/9781608058822114010010 PDF Price: \$15

Some types of calculation emphasise multiplication and division over addition and subtraction. Representing numbers as their logarithms accelerates multiplication and division, but slightly complicates addition and subtraction. This chapter gives a brief overview of logarithmic representations and their arithmetic.

## Characters and Text

Page: 133-157 (25) Author: Peter Fenwick DOI: 10.2174/9781608058822114010011 PDF Price: \$15

Characters are the most-visible aspect of computing. This chapter outlines the development of the EBCDIC codes (from card code), and ASCII (initially from paper tape), and the extension of these codes to include a full range of alphabets, to give UNICODE. Other topics include the collection of characters into text strings, and especially the problems of transmitting binary data over systems designed for handling text. Thus it describes UTF-8 and UTF-7 coding, as well as "punycode", for encoding Internet domain names with arbitrary alphabets.

## Universal (Variable Length) Codes

Page: 159-190 (32) Author: Peter Fenwick DOI: 10.2174/9781608058822114010012 PDF Price: \$15

Text compression requires numbers to represented as compactly as possible, especially the more-frequent values. This chapter describes various compact representations, and especially the "Universal Codes" to represent arbitrarily large values. Many of these codes are seldom mentioned in general literature.

## Checksums and Error Control

Page: 191-218 (28) Author: Peter Fenwick DOI: 10.2174/9781608058822114010013 PDF Price: \$15

Much computer data is prone to errors, especially when transmitted in space (or in time, as in data storage). The corresponding need for error detection and correction has been recognised from the very beginning of electrical computation. Error correction is a large and complex subject which is just touched-on here, but error detection is a much simpler, but seldom discussed subject and is the main emphasis of this chapter. After a brief mention of parity checks in textual data transmission, the emphasis is on checksums for verifying strings of decimal digits. Check-digit algorithms include the Luhn, those used in ISBN codes, and the Verhoe and Damm checks, both based on advanced number theory. The chapter concludes with discussion of the more-powerful message checksums, especially those of Fletcher and Adler, and the Cyclic Redundancy checks.

## Miscellaneous Topics

Page: 219-248 (30) Author: Peter Fenwick DOI: 10.2174/9781608058822114010014 PDF Price: \$15

This chapter contains a variety of topics, especially a history of numbers, justification for binary bits, good number representations, the history of `word', `bit' and `byte', a history of character codes, a variety of decimal codes and Roman numbers.

Page: 249-249 (1) Author: Peter Fenwick DOI: 10.2174/9781608058822114010015

## Bibliography

Page: 251-257 (7) Author: Peter Fenwick DOI: 10.2174/9781608058822114010016

Page: 259-264 (6) Author: Peter Fenwick DOI: 10.2174/9781608058822114010017

## Introduction

Introduction to Computer Data Representation introduces readers to the representation of data within computers. Starting from basic principles of number representation in computers, the book covers the representation of both integer and floating point numbers, and characters or text. It comprehensively explains the main techniques of computer arithmetic and logical manipulation. The book also features chapters covering the less usual topics of basic checksums and ‘universal’ or variable length representations for integers, with additional coverage of Gray Codes, BCD codes and logarithmic representations. The description of character coding includes information on both MIME and Unicode formats. Introduction to Computer Data Representation also includes historical aspects of data representation, explaining some of the steps that developers took (and the mistakes they made) that led to the present, well-defined and accepted standards of data representation techniques. The book serves as a primer for advanced computer science graduates and a handy reference for anyone wanting to learn about numbers and data representation in computers.

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## Introduction to Computer Data Representation

Introduction.

Introduction to Computer Data Representation introduces readers to the representation of data within computers. Starting from basic principles of number representation in computers, the book covers the representation of both integer and floating point numbers, and characters or text. It comprehensively explains the main techniques of computer arithmetic and logical manipulation. The book also features chapters covering the less usual topics of basic checksums and ‘universal’ or variable length representations for integers, with additional coverage of Gray Codes, BCD codes and logarithmic representations. The description of character coding includes information on both MIME and Unicode formats. Introduction to Computer Data Representation also includes historical aspects of data representation, explaining some of the steps that developers took (and the mistakes they made) that led to the present, well-defined and accepted standards of data representation techniques. The book serves as a primer for advanced computer science graduates and a handy reference for anyone wanting to learn about numbers and data representation in computers.

• Contributors

## Introduction and Overview

Numbers and computers, binary and other representations, signed, and other, representations, basic arithmetic and logic, computer arithmetic, floating-point representations, logarithmic representations, characters and text, universal (variable length) codes, checksums and error control, miscellaneous topics, concluding comments, bibliography, related books.

## Data Representation in Computer: Number Systems, Characters, Audio, Image and Video

• Post author: Anuj Kumar
• Post published: 16 July 2021
• Post category: Computer Science

• 1 What is Data Representation in Computer?
• 2.1 Binary Number System
• 2.2 Octal Number System
• 2.3 Decimal Number System
• 3.4 Unicode
• 4 Data Representation of Audio, Image and Video
• 5.1 What is number system with example?

## What is Data Representation in Computer?

A computer uses a fixed number of bits to represent a piece of data which could be a number, a character, image, sound, video, etc. Data representation is the method used internally to represent data in a computer. Let us see how various types of data can be represented in computer memory.

Before discussing data representation of numbers, let us see what a number system is.

## Number Systems

Number systems are the technique to represent numbers in the computer system architecture, every value that you are saving or getting into/from computer memory has a defined number system.

A number is a mathematical object used to count, label, and measure. A number system is a systematic way to represent numbers. The number system we use in our day-to-day life is the decimal number system that uses 10 symbols or digits.

The number 289 is pronounced as two hundred and eighty-nine and it consists of the symbols 2, 8, and 9. Similarly, there are other number systems. Each has its own symbols and method for constructing a number.

A number system has a unique base, which depends upon the number of symbols. The number of symbols used in a number system is called the base or radix of a number system.

Let us discuss some of the number systems. Computer architecture supports the following number of systems:

## Binary Number System

Octal number system, decimal number system, hexadecimal number system.

A Binary number system has only two digits that are 0 and 1. Every number (value) represents 0 and 1 in this number system. The base of the binary number system is 2 because it has only two digits.

The octal number system has only eight (8) digits from 0 to 7. Every number (value) represents with 0,1,2,3,4,5,6 and 7 in this number system. The base of the octal number system is 8, because it has only 8 digits.

The decimal number system has only ten (10) digits from 0 to 9. Every number (value) represents with 0,1,2,3,4,5,6, 7,8 and 9 in this number system. The base of decimal number system is 10, because it has only 10 digits.

A Hexadecimal number system has sixteen (16) alphanumeric values from 0 to 9 and A to F. Every number (value) represents with 0,1,2,3,4,5,6, 7,8,9,A,B,C,D,E and F in this number system. The base of the hexadecimal number system is 16, because it has 16 alphanumeric values.

Here A is 10, B is 11, C is 12, D is 13, E is 14 and F is 15 .

## Data Representation of Characters

There are different methods to represent characters . Some of them are discussed below:

The code called ASCII (pronounced ‘􀀏’.S-key”), which stands for American Standard Code for Information Interchange, uses 7 bits to represent each character in computer memory. The ASCII representation has been adopted as a standard by the U.S. government and is widely accepted.

A unique integer number is assigned to each character. This number called ASCII code of that character is converted into binary for storing in memory. For example, the ASCII code of A is 65, its binary equivalent in 7-bit is 1000001.

Since there are exactly 128 unique combinations of 7 bits, this 7-bit code can represent only128 characters. Another version is ASCII-8, also called extended ASCII, which uses 8 bits for each character, can represent 256 different characters.

For example, the letter A is represented by 01000001, B by 01000010 and so on. ASCII code is enough to represent all of the standard keyboard characters.

It stands for Extended Binary Coded Decimal Interchange Code. This is similar to ASCII and is an 8-bit code used in computers manufactured by International Business Machines (IBM). It is capable of encoding 256 characters.

If ASCII-coded data is to be used in a computer that uses EBCDIC representation, it is necessary to transform ASCII code to EBCDIC code. Similarly, if EBCDIC coded data is to be used in an ASCII computer, EBCDIC code has to be transformed to ASCII.

ISCII stands for Indian Standard Code for Information Interchange or Indian Script Code for Information Interchange. It is an encoding scheme for representing various writing systems of India. ISCII uses 8-bits for data representation.

It was evolved by a standardization committee under the Department of Electronics during 1986-88 and adopted by the Bureau of Indian Standards (BIS). Nowadays ISCII has been replaced by Unicode.

Using 8-bit ASCII we can represent only 256 characters. This cannot represent all characters of written languages of the world and other symbols. Unicode is developed to resolve this problem. It aims to provide a standard character encoding scheme, which is universal and efficient.

It provides a unique number for every character, no matter what the language and platform be. Unicode originally used 16 bits which can represent up to 65,536 characters. It is maintained by a non-profit organization called the Unicode Consortium.

The Consortium first published version 1.0.0 in 1991 and continues to develop standards based on that original work. Nowadays Unicode uses more than 16 bits and hence it can represent more characters. Unicode can represent characters in almost all written languages of the world.

## Data Representation of Audio, Image and Video

In most cases, we may have to represent and process data other than numbers and characters. This may include audio data, images, and videos. We can see that like numbers and characters, the audio, image, and video data also carry information.

We will see different file formats for storing sound, image, and video .

Multimedia data such as audio, image, and video are stored in different types of files. The variety of file formats is due to the fact that there are quite a few approaches to compressing the data and a number of different ways of packaging the data.

For example, an image is most popularly stored in Joint Picture Experts Group (JPEG ) file format. An image file consists of two parts – header information and image data. Information such as the name of the file, size, modified data, file format, etc. is stored in the header part.

The intensity value of all pixels is stored in the data part of the file. The data can be stored uncompressed or compressed to reduce the file size. Normally, the image data is stored in compressed form. Let us understand what compression is.

Take a simple example of a pure black image of size 400X400 pixels. We can repeat the information black, black, …, black in all 16,0000 (400X400) pixels. This is the uncompressed form, while in the compressed form black is stored only once and information to repeat it 1,60,000 times is also stored.

Numerous such techniques are used to achieve compression. Depending on the application, images are stored in various file formats such as bitmap file format (BMP), Tagged Image File Format (TIFF), Graphics Interchange Format (GIF), Portable (Public) Network Graphic (PNG).

What we said about the header file information and compression is also applicable for audio and video files. Digital audio data can be stored in different file formats like WAV, MP3, MIDI, AIFF, etc. An audio file describes a format, sometimes referred to as the ‘container format’, for storing digital audio data.

For example, WAV file format typically contains uncompressed sound and MP3 files typically contain compressed audio data. The synthesized music data is stored in MIDI(Musical Instrument Digital Interface) files.

Similarly, video is also stored in different files such as AVI (Audio Video Interleave) – a file format designed to store both audio and video data in a standard package that allows synchronous audio with video playback, MP3, JPEG-2, WMV, etc.

## FAQs About Data Representation in Computer

What is number system with example.

Let us discuss some of the number systems. Computer architecture supports the following number of systems: 1. Binary Number System 2. Octal Number System 3. Decimal Number System 4. Hexadecimal Number System

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## Course Computer Architecture

Digital computers store and process information in binary form as digital logic has only two values "1" and "0" or in other words "True or False" or also said as "ON or OFF". This system is called radix 2. We human generally deal with radix 10 i.e. decimal. As a matter of convenience there are many other representations like Octal (Radix 8), Hexadecimal (Radix 16), Binary coded decimal (BCD), Decimal etc.

Every computer's CPU has a width measured in terms of bits such as 8 bit CPU, 16 bit CPU, 32 bit CPU etc. Similarly, each memory location can store a fixed number of bits and is called memory width. Given the size of the CPU and Memory, it is for the programmer to handle his data representation. Most of the readers may be knowing that 4 bits form a Nibble, 8 bits form a byte. The word length is defined by the Instruction Set Architecture of the CPU. The word length may be equal to the width of the CPU.

The memory simply stores information as a binary pattern of 1's and 0's. It is to be interpreted as what the content of a memory location means. If the CPU is in the Fetch cycle, it interprets the fetched memory content to be instruction and decodes based on Instruction format. In the Execute cycle, the information from memory is considered as data. As a common man using a computer, we think computers handle English or other alphabets, special characters or numbers. A programmer considers memory content to be data types of the programming language he uses. Now recall figure 1.2 and 1.3 of chapter 1 to reinforce your thought that conversion happens from computer user interface to internal representation and storage.

• Data Representation in Computers

Information handled by a computer is classified as instruction and data. A broad overview of the internal representation of the information is illustrated in figure 3.1. No matter whether it is data in a numeric or non-numeric form or integer, everything is internally represented in Binary. It is up to the programmer to handle the interpretation of the binary pattern and this interpretation is called Data Representation . These data representation schemes are all standardized by international organizations.

Choice of Data representation to be used in a computer is decided by

• The number types to be represented (integer, real, signed, unsigned, etc.)
• Range of values likely to be represented (maximum and minimum to be represented)
• The Precision of the numbers i.e. maximum accuracy of representation (floating point single precision, double precision etc)
• If non-numeric i.e. character, character representation standard to be chosen. ASCII, EBCDIC, UTF are examples of character representation standards.
• The hardware support in terms of word width, instruction.

Before we go into the details, let us take an example of interpretation. Say a byte in Memory has value "0011 0001". Although there exists a possibility of so many interpretations as in figure 3.2, the program has only one interpretation as decided by the programmer and declared in the program.

• Fixed point Number Representation

Fixed point numbers are also known as whole numbers or Integers. The number of bits used in representing the integer also implies the maximum number that can be represented in the system hardware. However for the efficiency of storage and operations, one may choose to represent the integer with one Byte, two Bytes, Four bytes or more. This space allocation is translated from the definition used by the programmer while defining a variable as integer short or long and the Instruction Set Architecture.

In addition to the bit length definition for integers, we also have a choice to represent them as below:

• Unsigned Integer : A positive number including zero can be represented in this format. All the allotted bits are utilised in defining the number. So if one is using 8 bits to represent the unsigned integer, the range of values that can be represented is 28 i.e. "0" to "255". If 16 bits are used for representing then the range is 216 i.e. "0 to 65535".
• Signed Integer : In this format negative numbers, zero, and positive numbers can be represented. A sign bit indicates the magnitude direction as positive or negative. There are three possible representations for signed integer and these are Sign Magnitude format, 1's Compliment format and 2's Complement format .

Signed Integer – Sign Magnitude format: Most Significant Bit (MSB) is reserved for indicating the direction of the magnitude (value). A "0" on MSB means a positive number and a "1" on MSB means a negative number. If n bits are used for representation, n-1 bits indicate the absolute value of the number. Examples for n=8:

Examples for n=8:

0010 1111 = + 47 Decimal (Positive number)

1010 1111 = - 47 Decimal (Negative Number)

0111 1110 = +126 (Positive number)

1111 1110 = -126 (Negative Number)

0000 0000 = + 0 (Postive Number)

1000 0000 = - 0 (Negative Number)

Although this method is easy to understand, Sign Magnitude representation has several shortcomings like

• Zero can be represented in two ways causing redundancy and confusion.
• The total range for magnitude representation is limited to 2n-1, although n bits were accounted.
• The separate sign bit makes the addition and subtraction more complicated. Also, comparing two numbers is not straightforward.

Signed Integer – 1’s Complement format: In this format too, MSB is reserved as the sign bit. But the difference is in representing the Magnitude part of the value for negative numbers (magnitude) is inversed and hence called 1’s Complement form. The positive numbers are represented as it is in binary. Let us see some examples to better our understanding.

1101 0000 = - 47 Decimal (Negative Number)

1000 0001 = -126 (Negative Number)

1111 1111 = - 0 (Negative Number)

• Converting a given binary number to its 2's complement form

Step 1 . -x = x' + 1 where x' is the one's complement of x.

Step 2 Extend the data width of the number, fill up with sign extension i.e. MSB bit is used to fill the bits.

Example: -47 decimal over 8bit representation

As you can see zero is not getting represented with redundancy. There is only one way of representing zero. The other problem of the complexity of the arithmetic operation is also eliminated in 2’s complement representation. Subtraction is done as Addition.

More exercises on number conversion are left to the self-interest of readers.

• Floating Point Number system

The maximum number at best represented as a whole number is 2 n . In the Scientific world, we do come across numbers like Mass of an Electron is 9.10939 x 10-31 Kg. Velocity of light is 2.99792458 x 108 m/s. Imagine to write the number in a piece of paper without exponent and converting into binary for computer representation. Sure you are tired!!. It makes no sense to write a number in non- readable form or non- processible form. Hence we write such large or small numbers using exponent and mantissa. This is said to be Floating Point representation or real number representation. he real number system could have infinite values between 0 and 1.

Representation in computer

Unlike the two's complement representation for integer numbers, Floating Point number uses Sign and Magnitude representation for both mantissa and exponent . In the number 9.10939 x 1031, in decimal form, +31 is Exponent, 9.10939 is known as Fraction . Mantissa, Significand and fraction are synonymously used terms. In the computer, the representation is binary and the binary point is not fixed. For example, a number, say, 23.345 can be written as 2.3345 x 101 or 0.23345 x 102 or 2334.5 x 10-2. The representation 2.3345 x 101 is said to be in normalised form.

Floating-point numbers usually use multiple words in memory as we need to allot a sign bit, few bits for exponent and many bits for mantissa. There are standards for such allocation which we will see sooner.

• IEEE 754 Floating Point Representation

We have two standards known as Single Precision and Double Precision from IEEE. These standards enable portability among different computers. Figure 3.3 picturizes Single precision while figure 3.4 picturizes double precision. Single Precision uses 32bit format while double precision is 64 bits word length. As the name suggests double precision can represent fractions with larger accuracy. In both the cases, MSB is sign bit for the mantissa part, followed by Exponent and Mantissa. The exponent part has its sign bit.

It is to be noted that in Single Precision, we can represent an exponent in the range -127 to +127. It is possible as a result of arithmetic operations the resulting exponent may not fit in. This situation is called overflow in the case of positive exponent and underflow in the case of negative exponent. The Double Precision format has 11 bits for exponent meaning a number as large as -1023 to 1023 can be represented. The programmer has to make a choice between Single Precision and Double Precision declaration using his knowledge about the data being handled.

The Floating Point operations on the regular CPU is very very slow. Generally, a special purpose CPU known as Co-processor is used. This Co-processor works in tandem with the main CPU. The programmer should be using the float declaration only if his data is in real number form. Float declaration is not to be used generously.

• Decimal Numbers Representation

Decimal numbers (radix 10) are represented and processed in the system with the support of additional hardware. We deal with numbers in decimal format in everyday life. Some machines implement decimal arithmetic too, like floating-point arithmetic hardware. In such a case, the CPU uses decimal numbers in BCD (binary coded decimal) form and does BCD arithmetic operation. BCD operates on radix 10. This hardware operates without conversion to pure binary. It uses a nibble to represent a number in packed BCD form. BCD operations require not only special hardware but also decimal instruction set.

• Exceptions and Error Detection

All of us know that when we do arithmetic operations, we get answers which have more digits than the operands (Ex: 8 x 2= 16). This happens in computer arithmetic operations too. When the result size exceeds the allotted size of the variable or the register, it becomes an error and exception. The exception conditions associated with numbers and number operations are Overflow, Underflow, Truncation, Rounding and Multiple Precision . These are detected by the associated hardware in arithmetic Unit. These exceptions apply to both Fixed Point and Floating Point operations. Each of these exceptional conditions has a flag bit assigned in the Processor Status Word (PSW). We may discuss more in detail in the later chapters.

• Character Representation

Another data type is non-numeric and is largely character sets. We use a human-understandable character set to communicate with computer i.e. for both input and output. Standard character sets like EBCDIC and ASCII are chosen to represent alphabets, numbers and special characters. Nowadays Unicode standard is also in use for non-English language like Chinese, Hindi, Spanish, etc. These codes are accessible and available on the internet. Interested readers may access and learn more.

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• Engineering Mathematics
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## Computer Organization and Architecture Tutorial

• Basic Computer Instructions
• What is Computer
• Issues in Computer Design
• Difference between assembly language and high level language
• Difference between Memory based and Register based Addressing Modes
• Computer Organization | Von Neumann architecture
• Harvard Architecture
• Interaction of a Program with Hardware
• Simplified Instructional Computer (SIC)
• Instruction Set used in simplified instructional Computer (SIC)
• Instruction Set used in SIC/XE
• RISC and CISC in Computer Organization
• Vector processor classification
• Essential Registers for Instruction Execution
• Introduction of Single Accumulator based CPU organization
• Introduction of Stack based CPU Organization
• Machine Control Instructions in Microprocessor
• Very Long Instruction Word (VLIW) Architecture
• Input and Output Systems
• Computer Organization | Different Instruction Cycles
• Machine Instructions
• Computer Organization | Instruction Formats (Zero, One, Two and Three Address Instruction)
• Register content and Flag status after Instructions
• Debugging a machine level program
• Vector Instruction Format in Vector Processors
• Vector instruction types
• Instruction Design and Format
• Introduction of ALU and Data Path
• Computer Arithmetic | Set - 1
• Computer Arithmetic | Set - 2
• Difference between 1's Complement representation and 2's Complement representation Technique
• Restoring Division Algorithm For Unsigned Integer
• Non-Restoring Division For Unsigned Integer
• Computer Organization | Booth's Algorithm
• How the negative numbers are stored in memory?
• Microprogrammed Control
• Computer Organization | Micro-Operation
• Microarchitecture and Instruction Set Architecture
• Types of Program Control Instructions
• Difference between CALL and JUMP instructions
• Computer Organization | Hardwired v/s Micro-programmed Control Unit
• Implementation of Micro Instructions Sequencer
• Performance of Computer in Computer Organization
• Introduction of Control Unit and its Design
• Computer Organization | Amdahl's law and its proof
• Subroutine, Subroutine nesting and Stack memory
• Different Types of RAM (Random Access Memory )
• Random Access Memory (RAM) and Read Only Memory (ROM)
• 2D and 2.5D Memory organization

## Input and Output Organization

• Priority Interrupts | (S/W Polling and Daisy Chaining)
• I/O Interface (Interrupt and DMA Mode)
• Direct memory access with DMA controller 8257/8237
• Computer Organization | Asynchronous input output synchronization
• Programmable peripheral interface 8255
• Synchronous Data Transfer in Computer Organization
• Introduction of Input-Output Processor
• MPU Communication in Computer Organization
• Memory mapped I/O and Isolated I/O
• Memory Organization
• Introduction to memory and memory units
• Memory Hierarchy Design and its Characteristics
• Register Allocations in Code Generation
• Cache Memory
• Cache Organization | Set 1 (Introduction)
• Multilevel Cache Organisation
• Difference between RAM and ROM
• What's difference between CPU Cache and TLB?
• Introduction to Solid-State Drive (SSD)
• Read and Write operations in Memory
• Instruction Level Parallelism
• Computer Organization and Architecture | Pipelining | Set 1 (Execution, Stages and Throughput)
• Computer Organization and Architecture | Pipelining | Set 3 (Types and Stalling)
• Computer Organization and Architecture | Pipelining | Set 2 (Dependencies and Data Hazard)
• Last Minute Notes Computer Organization

## COA GATE PYQ's AND COA Quiz

• Computer Organization and Architecture
• Digital Logic & Number representation
• Number Representation
• Microprocessor
• GATE CS Preparation

Computer Organization and Architecture is used to design computer systems. Computer Architecture is considered to be those attributes of a system that are visible to the user like addressing techniques, instruction sets, and bits used for data, and have a direct impact on the logic execution of a program, It defines the system in an abstract manner, It deals with What does the system do.

Whereas, Computer Organization is the way in which a system has to structure and It is operational units and the interconnections between them that achieve the architectural specifications, It is the realization of the abstract model, and It deals with How to implement the system.

In this Computer Organization and Architecture Tutorial, you’ll learn all the basic to advanced concepts like pipelining, microprogrammed control, computer architecture, instruction design, and format.

## Recent Articles on Computer Organisation

• Computer Arithmetic
• Miscellaneous

Basic Computer Instructions :

• A simple understanding of Computer
• Computer System Level Hierarchy
• Computer Architecture and Computer Organization
• Timing diagram of MOV Instruction in Microprocessor
• Assembly language and High level language
• Memory based Vs Register based addressing modes
• Von Neumann architecture
• Data Transfer instructions in AVR microcontroller
• Arithmetic instructions in AVR microcontroller
• Conditional Branch Instructions in AVR Microcontroller
• CALL Instructions and Stack in AVR Microcontroller
• Branch Instructions in AVR Microcontroller
• Logical Instructions in AVR Microcontroller
• Data Manipulation Instructions

Instruction Design and Format :

• Different Instruction Cycles
• Instruction Formats (Zero, One, Two and Three Address Instruction)

Computer Arithmetic :

• Computer Arithmetic | ALU and Data Path
• Computer Arithmetic | Set 1
• Computer Arithmetic | Set 2
• Difference between 1’s complement and 2’s complement
• Booth’s Algorithm

Microprogrammed Control :

• Micro-Operation
• Hardwired v/s Micro-programmed Control Unit

Memory Organization :

• What’s difference between CPU Cache and TLB?
• Different Types of RAM
• Types of computer memory (RAM and ROM)
• Introduction to solid-state drive (SSD)

Input and Output Systems :

• Asynchronous input output synchronization

Pipelining :

• Execution, Stages and Throughput
• Types and Stalling
• Dependencies and Data Hazard

IEEE Number Statndards

Miscellaneous :

• Generations of computer
• Introduction to quantum computing
• Conventional Computing vs Quantum Computing
• Flynn’s taxonomy
• Clusters In Computer Organisation
• Program for Binary To Decimal Conversion
• Program for Decimal to Binary Conversion
• Program for decimal to octal conversion
• Program for octal to decimal conversion
• Program for hexadecimal to decimal

• ‘Quizzes’ on Computer Organization and Architecture !
• ‘Practice Problems’ on Computer Organization and Architecture !

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## Computer Science > Computer Vision and Pattern Recognition

Abstract: While recent advances in 3D-aware Generative Adversarial Networks (GANs) have aided the development of near-frontal view human face synthesis, the challenge of comprehensively synthesizing a full 3D head viewable from all angles still persists. Although PanoHead proves the possibilities of using a large-scale dataset with images of both frontal and back views for full-head synthesis, it often causes artifacts for back views. Based on our in-depth analysis, we found the reasons are mainly twofold. First, from network architecture perspective, we found each plane in the utilized tri-plane/tri-grid representation space tends to confuse the features from both sides, causing "mirroring" artifacts (e.g., the glasses appear in the back). Second, from data supervision aspect, we found that existing discriminator training in 3D GANs mainly focuses on the quality of the rendered image itself, and does not care much about its plausibility with the perspective from which it was rendered. This makes it possible to generate "face" in non-frontal views, due to its easiness to fool the discriminator. In response, we propose SphereHead, a novel tri-plane representation in the spherical coordinate system that fits the human head's geometric characteristics and efficiently mitigates many of the generated artifacts. We further introduce a view-image consistency loss for the discriminator to emphasize the correspondence of the camera parameters and the images. The combination of these efforts results in visually superior outcomes with significantly fewer artifacts. Our code and dataset are publicly available at this https URL .

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#### IMAGES

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5. Introduction to Computer Data Representation:

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Introduction to Computer Data Representation introduces readers to the representation of data within computers. Starting from basic principles of number representation in computers, the book covers the representation of both integer and floating point numbers, and characters or text. It comprehensively explains the main techniques of computer arithmetic and logical manipulation.

7. PDF Lecture Notes on Data Representation

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We also cover the basics of digital circuits and logic gates, and explain how they are used to represent and process data in computer systems. Our guide includes real-world examples and case studies to help you master data representation principles and prepare for your computer science exams. Check out the links below:

10. Book

Introduction to Computer Data Representation also includes historical aspects of data representation, explaining some of the steps that developers took (and the mistakes they made) that led to the present, well-defined and accepted standards of data representation techniques. The book serves as a primer for advanced computer science graduates ...

11. Introduction to Computer Data Representation

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17. Introduction to Computer Data Representation

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