Error Detection and Correction

Shashank SrivastavaMotilal Nehru National Institute Of Technology, Allahabad

Error Detection and Correction : Data Link Layer

1Error Detection and CorrectionData can be corrupted during transmission. For reliable communication, errors must be detected and corrected.

Types of errors

Single-bit errorOnly one bit in the data unit changes.Single-bit errors are least likely type of errors.Example: imagine a sender sending at 1 Mbps. This means each bit lasts for 1/ 106 sec = 1 s. For single-bit error to occur, noise must have a duration of only 1 s, which is very rare.

Multiple-bit error

Burst errorMeans two or more bits in the data unit have changed.

The length of the burst is measured from the first corrupted bit to the last corrupted bit.The number of bits affected depends on the data rate and duration of the noise.Example: if we are sending data at 1 Kbps, a noise of 1/100 sec can affect 10 bits.DetectionRedundancy: One technique is to send every data unit twice.Receiving device will do a bit-by-bit comparison between the two versions.The possibility of errors being introduced into exactly same bits are very small, so this method is effective- but it is inefficient and slow.

Second possibility is to append shorter group of bits. This is called redundancy because extra bits are redundant to the information, and discarded as soon as accuracy of the transmission has been determined.

RedundancyIf the received bit stream passes the checking criteria, the data portion ofthe data unit is accepted and the redundant bits are discarded.

Vertical Redundancy Check (Parity Check)Least expensive method.

Even parity : A redundant bit, called a parity bit, is appended to every data unit so that the total number of 1s in the unit becomes even.Odd parity: total number of 1s should become odd.

Even parity VRC conceptPerformance:

VRC can detect all single-bit errors.It can also detect burst errors as long as the total number of bits changed is odd.Longitudinal Redundancy Check (LRC)A block of bits is organized in a table (rows and columns).Then calculate parity bit for each column and create a new row, which are parity bits for the whole block.

Performance:

LRC of n bits can easily detect a burst error of n bits.

A burst error of more than n bits is also detected by LRC with a high probability.Example: if 2 bits in one data unit are damaged, and 2 bits in exactly the same position in another data unit are damaged, the LRC checker will not detect the error.Cyclic Redundancy Check (CRC)Most powerful redundancy checking technique.Unlike VRC and LRC, which are based on addition-, CRC is based on binary division.A CRC or CRC remainder is appended to the end of a data unit so that the resulting data unit becomes exactly divisible by a second, predetermined binary number(divisor).At destination, incoming data unit is divided by the same number.If no remainder- then data unit is intact.If there is a remainder- means data unit is damaged.Note: a CRC must have exactly one bit less than divisor.

CRCA string of n 0s is appended to the data unit. The number of n is one less than the number of bits in the predetermined divisor (which is n+1 bits).The new data unit is divided by the divisor using a process called binary division. The remainder is the CRC.CRC of n bits replaces the appended 0s at the end of data unit. CRC may consist of all 0s.The data unit arrives at receiver- data first , followed by CRC. The receiver treats whole string as a unit and divides by the same divisor.If remainder is zero - data unit is allowed to pass, else it is rejected.

Binary DivisionEach bit of the divisor is subtracted from the corresponding bit of the dividend without disturbing the next higher bit.PolynomialsThe CRC generator(the divisor) is most often represented not as a string of 1s and 0s, but as an algebraic polynomial.

A polynomial selected should have following properties:1) It should not be divisible by x : it guarantees that all the burst errors of a length equal to the degree of polynomial are detected.2) It should be divisible by x+1 : it guarantees that all burst errors affecting an odd number of bits are detected.

Polynomial and Divisor

Standard PolynomialsThe numbers 12, 16, and 32 refers to the size of CRC remainder.The CRC divisors are 13, 17, and 33 bits, respectively.Performance:CRC can detect all burst errors that affect an odd number of bits.CRC can detect all burst errors of length less than or equal to the degree of polynomial.CRC can detect with a very high probability burst errors of length greater than the degree of the polynomial.ChecksumIt is also based on the concept of redundancy.Checksum GeneratorData segment is subdivided into equal segments of n units.

These segments are added together using 1s complement arithmetic in such a way that total is also n-bits long.

The total sum is then complemented and appended to the end of original data unit, called checksum field.

Means if the sum of the data segment is T, checksum is T.

Extended data unit is transmitted across the network.

ChecksumChecksum CheckerThe receiver subdivides the data unit into segments-each of n bits, and adds all segments together.

Then it complements the result.

If total found is zero - data unit is intact.

If total is not zero means data unit contains error.

Data Unit and Checksum AT SENDER: 1 0 0 0 1 0 0 1 Word 1 1 0 1 1 1 0 0 0 Word 2------------------------------------- 1 0 1 0 0 0 0 0 1 Sum using ones complement arithmetic 1 Remove carry and add it back------------------------------------- 0 1 0 0 0 0 1 0 sum 1 0 1 1 1 1 0 1 checksum

The pattern sent is =m+r+1Example: if value of m=7, then r=4.

Hamming codePositioning the redundancy bits: use the positions that are powers of 2. (r1, r2, r4, r8).

Each r bit is the VRC bit (parity bit) for one combination of data bits.

Each data bit may be included in more than one VRC calculation.

R1: bits 1,3,5,7,9,11R2: bits 2,3,6,7,10,11R4: bits 4,5,6,7R8: bits 8,9,10,11

The r1 bit is calculated using all the bit positions whose binary representation includes a 1 in the rightmost position.

The r2 bit is calculated using all bit positions with a 1 in the second position, and so on.

The parity for each combination is the value of the corresponding r bit.The final 11-bit code is sent through transmission line.

Suppose data contains error.

Error Detection-Reassemble the new parity values into a binary number in the order of r position.(r8,r4,r2,r1).Error Correction-Reverse the bit value and correct the error.