Error Control

Error Detection

Data can be corrupted during transmission.

Some applications require that errors bedetected and corrected.

An error-detecting code can detectonly the types of errors for which it is designed;other types of errors may remain undetected.

Types of ErrorsRedundancyDetection Versus CorrectionForward Error Correction Versus RetransmissionCoding

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Single-bit error

In a single-bit error, only 1 bit in the dataunit has changed.

Burst error of length 8

A burst error means that 2 or more bitsin the data unit have changed.

Error Detection

•Two main methods of Error Correction

–Forward Error Correction (FEC) – adding redundancy bits

–Retransmission – resending of data

•FEC is used when the potential error is small.

•Redundancy in FEC is achieved by means of Coding,means adding control bits to data.

•Block codes and Convolutional Codes. Only simpleBlock code will be discussed

•The ability to detect an error and the ability to correctan error is two different thing.

•Determined by the min Hamming Distance, dmin

To detect or correct errors, we need tosend extra (redundant) bits with data.

Using XOR logic of two single bits or two words

BLOCK CODINGBLOCK CODING

In block coding, we divide our message into blocks,each of k bits, called datawords. We add r redundantbits to each block to make the length n = k + r. Theresulting n-bit blocks are called codewords.In block coding, we divide our message into blocks,each of k bits, called datawords. We add r redundantbits to each block to make the length n = k + r. Theresulting n-bit blocks are called codewords.

Error DetectionError CorrectionHamming Distance

Minimum Hamming Distance

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Figure 10.5 Datawords and codewords in block coding

Process of error detection in block coding

A code for error detection; k =2, n=3

A code for error correction; k=2, n=5

The Hamming distance d between twowords is the number of differencesbetween corresponding bits.

•Find the Hamming distance between two pairs of words.

1. The Hamming distance d(000, 011) is 2 because

2. The Hamming distance d(10101, 11110) is 3 because

The minimum Hamming distance dmin

is the smallest Hamming distance

between all possible pairs in a set of codewords.

dmin = 2

dmin = 3

To guarantee the detection of up to

s errors in all cases, the minimum

Hamming distance in a blockcode must be dmin = s + 1.

s is the number of detectable error

If dmin = 2, then s = 1

If dmin = 3, then s = 2

Detectable Error (s)

To guarantee correction of up to t errorsin all cases, the minimum Hammingdistance in a block code must be

dmin = 2t + 1.

t is the number of correctable error

Correctable Error (t)

If dmin = 2, then s = 1, but t = ½ ~ 0

means able to detect upto 1 error but cannot correct any

If dmin = 3, then s = 2, but t = 1

means able to detect upto 2 errors but can only correct 1

Another example of block code using Simple parity-check code C(5, 4)

A simple parity-check code can detectan odd number of errors.

Encoder and decoder for simple parity-check code

An example of block code using Hamming code C(7, 4)

The structure of the encoder and decoder for a Hamming code

10-4 CYCLIC CODES10-4 CYCLIC CODES

Cyclic codes are special linear block codes with oneextra property. In a cyclic code, if a codeword iscyclically shifted (rotated), the result is anothercodeword.Cyclic codes are special linear block codes with oneextra property. In a cyclic code, if a codeword iscyclically shifted (rotated), the result is anothercodeword.

Cyclic Redundancy CheckHardware ImplementationPolynomialsCyclic Code Analysis

Advantages of Cyclic CodesOther Cyclic Codes

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