Error Control CODING

Prepared by :

Ola MashaqiSuhad Malayshe

Mais Masri

Submitted to : Dr. Allam Mousa

Defintion of channel coding Error control coding ,detect, and often correct, symbols which are received in error

The channel encoder separates or segments the incoming bit stream into equal length blocks of L binary digits and maps each L-bit message block into an N-bit code word where N > L

There are M=2L messages and 2L code words of length N bits

•The channel decoder has the task of detecting that there has been a bit error and (if possible) correcting the bit error

ARQ (Automatic-Repeat-Request ) If the channel decoder performs error detection then errors can be detected and a feedback channel from the channel decoder to the channel encoder can be used to control the retransmission of the

code word until the code word is received without detectable errors .

There are two major ARQ techniques stop and wait continuous ARQ

FEC (Forward Error Correction) If the channel decoder performs error correction then errors are not only detected but the bits in error can be identified and corrected (by bit inversion)

There are two major ARQ techniques

• stop and wait, in which each block of data is positively, or negatively, acknowledged by the receiving terminal as being error free before the next data block is transmitted,

• continuous ARQ, in which blocks of data continue to be transmitted without waiting for each previous block to be acknowledged

Error Control Coding (Channel Coding)

particular error control methods : linear group codes, cyclic codes, the Golay code, BCH codes, Reed–Solomon codes and Hamming codes

Block coding VS. convolutional coding

Block coding The (n,k) block code is thecode that convert k bit of themassage signal to n bitcodeword .

It block because ittake number of bits frommassage(information digit)and add redundant bits(parity digit) to it and do soto the rest of the bits.

Convolutional Codingencoding a stream of data

rather than blocks of data .

The sequence of bits in aconvolutional code depends Not only on the current bits of

databut also on previous bits of

data.

Error rate control concepts

How to measure error performance ? Answer is BER : the average rate at which errors occur and is

given by the product PbRbPb: probability of error Rb : bit transmission rate in the channelBUT If BER is too large ! What to do to make it smaller .. • increase transmitter power(not efficient )• Diversity : Frequency diversiy employs two different frequencies to

transmit the same information , time diversity systems the same message is transmitted more than once at different times .

• introduce full duplex transmission: implying simultaneous two-way transmission

• ARQ and FEC

Hamming Distance

The Hamming distance between two code-words is defined as the number of places, bits or digits in which they differ.

•The distance is important factor since it indicates how easy to change one valid code into another.

•The weight of the codeword is defined as the number of ones in the codeword.

•Example) Calculate the hamming distance and weight of the following codeword :11100, 11011

Hamming distance = 3 bit The code word 11100 could changed to 11011The weight of the codeword 1= 3The weight of the codeword 2= 4 The minimum codeword weight =3

(n, k) block codes:

with k information digits going into the coder. n digits coming out after (n −k) redundant parity check digits.

The rate, or efficiency, for this code (R) = k/n Rate is normally in the range 1/2 to unity.

Linear group codes

Group codes contain the all-zeros codeword and have the property referred to as closure .

Advantage : it makes performance calculations with linear group codes particularly easy.

taking any two codewords Ci and Cj , then Ci ⊕ Cj = Ck .

Example

  a contain all-zeros codeword

  b c=d , c d=b , b d=b . ⊕ ⊕ ⊕

Performance prediction

Hamming distances measurer to determine the overall performance of a block code

consideration of each of the codewords with the all-zeroscodewordis sufficient. Example

0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 0 1 1

Dmin = 3 for this (5,2) code.Consider four codewords the weights of these are 0, 3, 3 and 4. the minimum weight in the weight structure (3) is equal to Dmin,

the minimum Hamming distance for the code.

Performance prediction !!!! مراجعةأحذفها يمكن

Consider the probability of the ith codeword (Ci ) beingmisinterpreted as the j th codeword (Cj ). This probability depends onthe distance between these two codewords (Di j ).

Ci Cj Di j is equal to the weight of a third codeword Ck

• Ci being mistaken for Cj is therefore equal to the probability of Ck being mistaken for C0

• The probability of C0 being misinterpretedas Ck depends only on the weight of Ck .

the performance of such a code can be determined completely 1. consideration of C0 2. the weight structure alone

Error detection and correction capability

t :The maximum possible error correcting.

Dmin :minimum Hamming distance

e : is the ability of certain code to detect errors.

t ≤ e.

Error detection and correction capability

Dmin is 3 e=1 , t=1

11001 &11000If any single error occurs in one of the codewords it cantherefore be corrected. Dmin − 1 errors can be detectedthere is no error correction

Longer codes with larger Hamming distancesoffer greater detection and correction capabilityby selecting different t and e

Standerd

The UK Post Office Code Standards Advisory Group(POCSAG) code

k = 21 and n = 32

R ≈ 2/3

Dmin = 6.

3 bit detection 2 bit correction capability.

Syndrome decoding

syndrome is independent of the transmitted codeword and only depends on the error sequence

A decoding table tells a decoder how to correct errors that might have corrupted the code during transmission

Error

location Syndrome

0000000 000

1000000 111

0100000 011

0010000 101

0001000 110

0000100 100

0000010 010

0000001 001

Syndrome decoding

d is a message vector of k digits

G is the k × n generator matrix

c is the n-digit codeword corresponding to the message d,

  d G = c

Where G is the generation matrix Furthermore:

H c = 0where H is the (even) parity check matrix corresponding to G

Syndrome decoding

r = c ⊕ e r is the sequence received after transmitting c . e is an error vector representing the location of the errors

which occur in the received sequence r. syndrome vector s

  s = H r = H (c e⊕ ) = H c H e = 0 H e =He ⊕ ⊕

s is easily calculated

The generator matrix(G) : The generator matrix G for

an (n, k) block code can be used to generate the appropriate

n-digit codeword from any given k-digit data sequence .

Parity check matrix (H) : does not contain any codewords.

(7,4) block block code H matrix .

The right side of G is the transpose of the left hand portion of H.

Parity check section must : must contain at least two ones. rows cannot be identical.

G is the k × n generator matrix . The right side of G is the transpose of the left hand portion of H. .

Example

use this syntax to Produce syndrome decoding table

t = syndtable(h)

 returns a decoding table for an error-correcting binary code having codeword length n and message length

http://www.mathworks.com/help/comm/ref/syndtable.html

Application

• Error control coding, generally, is applied widely in control and communications systems for aerospace applications, in mobile (GSM) cellular telephony and for enhancing security in banking and barcode readers.

Matlab n = 6; k = 4; % Set codeword length and message

length

% for a [6,4] code.

msg = [1 0 0 1 1 0 1 0 1 0 1 1]'; % Message is a binary column.

code = encode(msg,n,k,'cyclic'); % Code will binary column.

msg'

code'

msg consists of 12 entries, which are interpreted as three 4-digit (because k = 4) messages. The resulting vector codecomprises three 6-digit (because n = 6) codewords, which are concatenated to form a

vector of length 18. The parity bits are at the beginning of each codeword