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**Introduction to Information Technologies**

Fall 2004 Computer Networks Chapter 10 – Error Detection and Correction History of Internet

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**Errors The physical link is always subject to imperfections**

Noise/interference Limited bandwidth Distortion Errors – a consequence of imperfecton Some bits send at the sender are received with different value at the receiver Sender Receiver Value sent Value received Spring 2006 Computer Networks

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**Types of Errors Single-bit error Burst error**

The value of a single bit in a data unit is changed Does not occur very often in serial data transmission Burst error The value of two or more bits in the data unit is changed Usual type of error in serial data transmission Spring 2006 Computer Networks

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**Errors and Error Effect**

Original Waterfall Image: Spring 2006 Computer Networks

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**Errors and Error Effect (Cont.)**

Waterfall Image: every tenth bit in error Spring 2006 Computer Networks

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**Errors and Error Effect (Cont.)**

Waterfall Image: every other bit in error Spring 2006 Computer Networks

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**Two Strategies for Error Control**

Error detection and correction Enough redundancy (extra bits) is incorporated, so that at the reciever the errors can be not only detected, but also corrected Not applicable to data communication, because too much redundancy is needed Error detection and retransmission There is enough redundancy only to detect the error in a data unit. If an error is descovered the sender is automaticly required to retransmit the data unit Applicabbble to data communication Spring 2006 Computer Networks

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**Error Detection Methods**

Each method involves adding extra bits (redundant bits) to the data unit Three most common methods are Parity checking Cyclic Redundancy Check (CRC) Checksum Spring 2006 Computer Networks

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Parity Checking The sender and the receiver agree in advance whether the data units will have even or odd number of 1s. The sender adds extra bits to create the data units according to the agreement The receiver checks the parity of the 1s according to the agreement. If the data unit received has a number of 1s according to the agreement, it is accepted as correct; otherwise it is rejected as “in error”. Spring 2006 Computer Networks

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**Simple Parity Checking**

The sender adds an additional bit, called parity bit to each data unit Even parity – the parity bit is 0 or 1 depending on which bit will make the total number of 1’s even Odd parity – the parity bit is 0 or 1 depending on which bit will make the total number of 1’s odd The sender and receiver know which scheme they are using The receiver performs parity checking Only a single error or odd number of errors can be detected It is not convenient for use with data transmission Spring 2006 Computer Networks

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Even Parity Concept Spring 2006 Computer Networks

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**Example Assuming even parity, add a single bit to each data unit.**

Solution: 1 Spring 2006 Computer Networks

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**Two-dimensional Parity Check**

Data units are arranged in two-dimensional array Parity bit is added to the rows (each data unit) and to the columns (an extra data unit is created) The receiver checks the parity in the rows and in the columns Improved performance compared to single-parity checking Still not very often used with data transmission Spring 2006 Computer Networks

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**Two-dimensional Parity Concept**

Spring 2006 Computer Networks

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Example Perform a two dimensional parity check on the following data unit by having blocks of 7 bits each. Solution 1 1 1 1 Spring 2006 Computer Networks

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**Cyclic Redundancy Check**

A method for error detection that is often used with data transmission. Based upon treating bit strings as polynomials with coefficients 0 and 1. k bit message is represented as (k-1) degree polynomial Example: has 7 bits It can be represented as a polynomial of 6th degree 1·x6 + 0·x5 + 1·x4 +0·x3 + 1·x2 + 1·x1 + 0·x0 Spring 2006 Computer Networks

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**How CRC Operates? The sender wants to send k bits message**

The sender and the receiver must agree in advance on n+1 bit string called generator polynomial (divisor), G. G can be represented as n-degree polynomial n redundant bits are added to the k bits message. They are called CRC bits. Data bits to be sent CRC bits k bits n bits Spring 2006 Computer Networks

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**How CRC Operates? (Cont.)**

The redundant bits are chosen in such a way that the resulting k+n bit string is exactly divisible (with a reminder=0) by G using modulo 2 arithmetic. The receiver divides the received data together with the CRC bits by G using modulo 2 arithmetic. If the reminder is 0, then the string is considered to be without errors If the reminder is not 0, the data unit is with errors and it is rejected Spring 2006 Computer Networks

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**Generator Polynomial, G**

For the purposes of calculating a CRC, the sender and the reciever need to agree upon a generator polynomial G in advance. The choice of G has impact on what types of errors can be reliably detected. There are a handful generator polynomials that are very good choices for various environments and the exact choice is made as a part of protocol design Spring 2006 Computer Networks

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**Modulo 2 Arithmetic Examples: 1011 XOR 0101 = 1110**

In modulo 2 arithmetic addition and substruction are identical to EXCLUSUVE OR (XOR) operation. Multiplication and division are the same as in base-2 arithmetic without carries in addition or borrows in substraction. 0 XOR 0 = 0 0 XOR 1 = 1 1 XOR 0 = 1 1 XOR 1 = 0 Examples: 1011 XOR 0101 = 1110 1001 XOR 1101 = 0100 Spring 2006 Computer Networks

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**Calculating the CRC Bits**

Let n be the degree of G (G consists of n+1 bits). Append n 0s to the data unit to obtain the extended data unit. Perform the modulo 2 division. The extended data unit is a dividend and G is a divisor. The quotient is not used The reminder is the CRC bits Add the CRC bits to the data unit Spring 2006 Computer Networks

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Example The first bit in the quotient is 1 and one times the divisor results in this The first bit in the quotient is 1 and one times the divisor results in this The number of 0s is one less than the number of bits in G (divisor) Obtained by XOR-ing 1001 and 1101 Obtained by XOR-ing 1000 and 1101 Spring 2006 Computer Networks

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**Another Example The transmitted unit is 11010110111110 Spring 2006**

Computer Networks

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CRC Performance Assuming a good choice of the generator polynomial, G, is made, CRC method shows very good performance Can detect all burst errors that affect odd number of bits Can detect all burst errors of length less than or equal to G Very high probability on detecting errors with length higher than the length of G Spring 2006 Computer Networks

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Checksum To compute the check sum, the sender treats the data unit as a sequence of a certain number of blocks, all with the same number of bits. The sender and receiver agree on how long are the blocks (usually 16 bits) The sender adds the blocks using one’s complement arithmetic and creates an additional block with the same size The additional block is complemented and appended to the data unit as redundancy bits Spring 2006 Computer Networks

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Checksum (Cont.) The receiver divides the received block into blocks with the agreed number of bits. The blocks are added using one’s complement arithmetic The sum is complemented If the result is 0, the data are considered without an error, otherwise the data unit is rejected Spring 2006 Computer Networks

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**Modulo One’s Arithmetic**

The addition starts at the last column from left to right The bits are carried in the respective column before This is repeated for each column If the number of bits in the sum is larger then those in the blocks, they are added to the sum obtained Spring 2006 Computer Networks

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**Example The addition starts in the last column**

Carry from column 5 1 Example Carry from column 4 Carry from column 3 Carry from column 2 The addition starts in the last column The bits are carried in the columns before The 6th and 7th bit are added Carry from column 1 sum checksum Spring 2006 Computer Networks

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**Complement of a Bit String**

A complement of a bit string is obtained when all 0s become 1s and all 1s become 0s Example: Given the bit string , its complement is Spring 2006 Computer Networks

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Error Detection and Correction

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