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 Errors Redundancy Detection Versus Correction Forward Error Correction Versus Retransmission Coding 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, d min

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 CODING In block coding, we divide our message into blocks, each of k bits, called datawords. We add r redundant bits to each block to make the length n = k + r. The resulting n-bit blocks are called codewords. Error Detection Error Correction Hamming Distance Minimum Hamming Distance 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 d min is the smallest Hamming distance between all possible pairs in a set of codewords. d min = 2 d min = 3

To guarantee the detection of up to s errors in all cases, the minimum Hamming distance in a block code must be d min = s + 1. s is the number of detectable error If d min = 2, then s = 1 If d min = 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 d min = 2t + 1. t is the number of correctable error Correctable Error (t) If d min = 2, then s = 1, but t = ½ ~ 0 means able to detect upto 1 error but cannot correct any If d min = 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 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 CODES Cyclic codes are special linear block codes with one extra property. In a cyclic code, if a codeword is cyclically shifted (rotated), the result is another codeword. Cyclic Redundancy Check Hardware Implementation Polynomials Cyclic Code Analysis Advantages of Cyclic Codes Other Cyclic Codes Topics discussed in this section:

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 Division in the CRC decoder for two cases

Figure 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 A polynomial to represent a binary word

Figure CRC division using polynomials Divisor = 1011 Augmented Dataword = CRC Codeword =