Error Detection

Data can be corrupted during transmission.

Some applications require that errors be detected and corrected.

An error-detecting code can detect only 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

Topics discussed in this section:Topics discussed in this section:

Single-bit error

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

Burst error of length 8

A burst error means that 2 or more bits in 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 simple

Block code will be discussed• The ability to detect an error and the ability to correct

an error is two different thing.• Determined by the min Hamming Distance, dmin

To detect or correct errors, we need to send 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, In block coding, we divide our message into blocks, each of k bits, called each of k bits, called datawordsdatawords. We add r redundant . We add r redundant bits to each block to make the length n = k + r. The bits to each block to make the length n = k + r. The resulting n-bit blocks are called resulting n-bit blocks are called codewordscodewords..

Error DetectionError CorrectionHamming DistanceMinimum Hamming Distance

Topics discussed in this section:Topics discussed in this section:

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 two words is the number of differences

between 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 block code 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 errors in all cases, the minimum Hamming distance 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 ~ 0means able to detect upto 1 error but cannot correct any

If dmin = 3, then s = 2, but t = 1means 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 detect an 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 codesCyclic codes are special linear block codes with one are special linear block codes with one extra property. In a cyclic code, if a codeword is extra property. In a cyclic code, if a codeword is cyclically shifted (rotated), the result is another cyclically shifted (rotated), the result is another codeword.codeword.

Cyclic Redundancy CheckHardware ImplementationPolynomialsCyclic Code AnalysisAdvantages of Cyclic CodesOther Cyclic Codes

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Table 10.6 A CRC code with C(7, 4)

Divisor determine the outcome of the codeword, which is pre-agreed between sender and receiver

CRC encoder and decoder

Division in CRC encoder

Figure 10.16 Division in the CRC decoder for two cases

Figure 10.19 The CRC encoder design using shift registers

Divisor = 1011

Simulation and implementation CRC encoder using shift register

General design of encoder and decoder of a CRC code

Figure 10.21 A polynomial to represent a binary word

Figure 10.22 CRC division using polynomials

Divisor = 1011

Augmented Dataword = 1001000

CRC Codeword = 1001110