Computer Organization & Articture No. 6 from APCOMS

8/14/2019 Computer Organization & Articture No. 6 from APCOMS

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Lecture 06

Data Representation inComputer Systems


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Signed Binary Numbers

In ordinary arithmetic a negative number is indicated by

minus sign and positive number by plus sign. This is not

possible in computers, because of hardware limitation

computers must represent everything with binary digits.

There are two methods to do this:

The signed magnitude convention uses the left-most bit torepresent the sign (0 for positive and 1 for negative).

The signed complement system negates a number by taking its

complement.

It could be either, 1scomplement representation

or2scomplement representation.


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Signed Magnitude Convention

The signed magnitude convention uses the left-mostbit to represent the sign (0 for positive and 1 fornegative). The user determines whether the number is signed or unsigned If the binary number is signed then the leftmost bit represents

the sign and the rest of the bits represents the number

If the binary number is unsigned then the leftmost bit is the mostsignificant bit of the number

For example: 01001 can be considered as 9 (unsigned binary) or a +9 because

the left most bit is zero. On the other hand, the string of bits 11001 represents binary

equivalent of 25 when considered as an unsigned number or as 9when considered as signed number


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Signed Complement System

The signed Complement System negative number isindicated by its complement (Complement of positive

number)

Positive numbers always start with 0 (plus), its complement

(representing negative number) will always start with 1

Signed complement system can use either1scomplement or2scomplement.

For example:

+9 is represented only as 00001001 but 9 can be represented as:

11110110 Signed 1s complement representation

11110111 Signed 2s complement representation


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Calculations arent useful until their results can be displayed

in a manner that is meaningful to people. We also need to store the results of calculations, and provide ameans for data input.

Thus, human-understandable characters must be converted tocomputer-understandable bit patterns using encoding scheme.

As computers have evolved, character codes have evolved. Larger computer memories and storage devices permit richer

character codes.

The earliest computer coding systems used six bits.

Binary-coded decimal (BCD) was one of these early codes used by

IBM BCD was extended to an 8-bit code, Extended Binary-Coded Decimal

Interchange Code (EBCDIC).

EBCDIC codes supported upperandlowercase alphabeticcharacters, in addition to special characters, such as punctuation andcontrol characters.

EBCDIC and BCD are still in use by IBM mainframes today.

Character Codes


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The 7-bit ASCII (American Standard Code for Information

Interchange) used as a replacement for 6-bit codes.

While BCD and EBCDIC were based upon punched card codes,

ASCII was based upon telecommunications (Telex) codes.

ASCII was the dominant character code outside the IBM mainframe

world.

Many of todays systems embrace Unicode, a 16-bit system that

can encode the characters of every language in the world.

The Java programming language, and some operating systems use

Unicode as their default character code.

The Unicode codespace is divided into six parts. The first part is

for Western alphabet codes, including English, Greek, and

Russian.

Character Codes


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Digital transmission

A computer is design to send information fromone point to another

While designing a system two choices are offered:

Convert information to either a digital or analog signal

Line coding is the process of converting binarydata into digital signal

Voice, data, movies, numbers and pictures are stored

in computer as binary data

Line coding technique convertthe binary data into digital

signal


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Unipolar

Unipoar scheme is simplest, inexpensive to implement butobsolete in use It provides the concept of encoding system

In digital transmission voltage pulses are sent through amedium

Most encoding scheme uses to send one voltage level for zero, andanother for one

The polarity of the pulse decides whether it is positive or negative

Unipolar scheme uses only one polarity, that is assigned toone of the two binary states, normally the 1

The other state is zero voltage

Unipolar encoding uses only one

voltage level.


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Unipolar encoding

Is are encoded as a positive value0s are encoded as zero value

This scheme has two problems

presence of dc component (average amplitude is non zero

Lack of synchronization (if data contains long 1s or 0s)


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Polar

Polar encoding uses two voltage levels, onepositive and one negative

Due to two voltage levels average voltage level

on the line is reduced and the dc component

problem is eliminated

Polar encoding uses two voltage levelsPolar encoding uses two voltage levels

(positive and negative).(positive and negative).


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Polar encoding

Nonreturn to zero (NRZ) the value of the signal is

always either positive or negative, it is further

categorized into:

NRZ-L (NRZ- level) the level of signal depends on the type

of bit that it represents A positive voltage means the bit is a 0

The negative voltage means the bit is a 1

The level of the signal is dependent on the state of the bit

The problem arise when long stream of 0 or 1s and the clockis not synch

In NRZ-L the level of the signal isIn NRZ-L the level of the signal is

dependent upon the state of the bit.dependent upon the state of the bit.


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Polar encoding

NRZ-I (NRZ-invert) an inversion of the voltage level

represents a 1 bit It is a transition between positive and negative, not the

voltage itself, represents a 1 bit

A 0 bit is represented by no change

In NRZ-I the signal is inverted if a 1 isIn NRZ-I the signal is inverted if a 1 is

encountered.encountered.


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Return to Zero (RZ)

With the signal containing a stream of 1s and 0s, the

receiver may looses the track of information contain in thatsignal That is why a synchronization method was introduced as NRZ-I

To ensure synchronization, there must be a signal change

for each bit The receiver can use these changes to buildup, updates andsynchronize its clock

To accommodate change in each bit three signal values arerequired

Positive

Negative

zero


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Return to Zero (RZ)

With Return to Zero (RZ) encoding, the three

values to the signal can be assigned

With RZ signal changes not between bits but

during each bit

Like NRZ-L, a positive voltage means a 1 and anegative voltage means 0

With RZ , half way through each bit interval, the signal

return to zero

1 bit is represented by positive to zero 0 bit is represented by negative to zero


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RZ encoding

1 bit is represented by positive to zero0 bit is represented by negative to zero

Disadvantage: It requires two signal changes to encode 1 bit

and therefore occupies more bandwidth

Advantage: More effective


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Manchester encoding

Also known as Manchester Phase Encoding

(MPE)

It uses an inversion at the middle of each bit

interval for synchronization and bit representation

A negative to positive transition represents binary 1 A positive to negative transition represents binary 0

By using the signal transition for dual purpose, this

encoding scheme has the same level of

synchronization as of RZ, but with two levels of

amplitude


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Manchester encoding

A negative to positive transition represents binary 1A positive to negative transition represents binary 0


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It is physically impossible for any data recording or

transmission medium to be 100% perfect 100% of the timeover its entire expected useful life.

As more bits are packed onto a square centimeter of disk storage, ascommunications transmission speeds increase, the likelihood of errorincreases.

Thus, error detection and correction is critical to accurate datatransmission, storage and retrieval.

Types of errors Single bit Error

Only one bit of a given data unit byte, character, or packet may

changed from 1 to 0 or vice versa Single bit error are least likely in serial transmission

Noise must be of the duration of that bit which is rare

Single bit error are most likely in parallel transmission One of the line in the set is noisy will generate noise to one of the bit

The effect of single bit change is shown

Error Detection and Correction


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Type of Error

Burst error: two or more bits in the data has changed

Effect is shown in the figure Burst error does not necessarily occur in the consecutive bits

Length of the burst is measured from the first corrupted bit to last corruptedbit

Most likely occur in serial transmission as the duration of the noise is

normally longer than the bit on the medium Number of affected bits depends on the data rate and duration of noise


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Error Detection

The goal of error checking is

To correct the error For correction the error must first be detected

Error detection the first step to correct error

Redundancy Sending the data unit twice is one of the error detection method

The receiving device will receive both the units and a bit by bitcomparison is done

The discrepancies would be discarded

Time consuming, double effort in transmission

Instead of completely repeating all the bits, If some extra bitare appended with the data unit, this will address time andeffort This extra information attached with the data unit is known as the

redundancy

As the extra bits are redundant and may be discarded


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Error detection uses the concept of redundancy,

which means adding extra bits for detecting

errors at the destination.

Redundancy


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Types of redundancy checks

Parity Check The most common and economical method

a parity bit is added to every data unit so that the total number of

1s is even (or odd for odd-parity).

Two sub parts are

Simple Parity Check:

One redundant bit Parity bit is added to the data unit, to make the

total number of 1s in the data unit even or odd

Two dimensional Parity Check:

A block of bits are organized in a table parity bit is calculated on rowsand column of the table

CRC:

Based on binary division, the remainder is appended to

the end of the data


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Even-parity concept


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Example 1Example 1

Suppose the sender wants to send the word world. In ASCII the

five characters are coded as

1110111 1101111 1110010 1101100 1100100

The following shows the actual bits sent

11101110 11011110 11100100 11011000 11001001

Example 2Example 2

Now suppose the word world is received by the receiver without

being corrupted in transmission.

11101110 11011110 11100100 11011000 11001001

The receiver counts the 1s in each character and comes up with

even numbers (6, 6, 4, 4, 4). The data are accepted.


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Example 3Example 3

Now suppose the word world is corrupted during

transmission.

11111110 11011110 11101100 11011000

11001001

The receiver counts the 1s in each character and comes up

with even and odd numbers (7, 6, 5, 4, 4). The receiver

knows that the data are corrupted, discards them, and asks

for retransmission.

Simple parity check can detect all single-bitSimple parity check can detect all single-bit

errors. It can detect burst errors only if the totalerrors. It can detect burst errors only if the total

number of errors in each data unit is odd.number of errors in each data unit is odd.


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Two dimensional Parity Check

A block of bits is organized in a table (rows and columns)

First calculate the parity bit for each data unit

Organize the data units into a table

Calculate the parity for each column to create a new row with one

parity bit these will be the parity bit for whole block

This method is useful to detect burst error A redundancy of n bit can detect a burst error of n bits


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Two-dimensional parity


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Example 4Example 4

Suppose the following block is sent:

10101001 00111001 11011101 11100111 10101010

However, it is hit by a burst noise of length 8, and some bits are

corrupted.

10100011 10001001 11011101 11100111 10101010

When the receiver checks the parity bits, some of the bits do not

follow the even-parity rule and the whole block is discarded.

10100011 10001001 11011101 11100111 10101010In two-dimensional parity check, a block of bits is

divided into rows and a redundant row of bits is

added to the whole block.


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Cyclic Redundancy Check (CRC)

Parity is based on binary addition, the CRC is based on

binary division

Instead of adding bits to achieve a desired parity, the CRCremainder of the division is appended to the end of the data unit

CRC remainder is a result of a binary division of dividend with a

predetermined divisor

The extra zeros (1 less than the divisor) are appended to the data

unit The remainder is appended to the data unit before

At the Receiver end the data unit is again divided with the same

divisor

If 0 remainder no error

If remainder is

non zero shows

the error

Bi di i i i CRC t


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Binary division in a CRC generator

Bi di i i i CRC h k


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Binary division in CRC checker

Computer Organization & Articture No. 6 from APCOMS

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