BB Channel Coding

8/13/2019 BB Channel Coding

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ESDE505 Digital CommunicationsPresented by Dr. R. Ramirez-Iniguez1

ESDE505 Digital Communications

Dr. Roberto Ramirez-IniguezM206

[email protected]. 3527


ESDE505 Digital Communications

Error Control CodingChapter 10 (Glover/Grant)

Pages: 347-375


Introduction

Hamming Distance and codeword weight

(n,k) Block Codes

Probability of error in n-digit codewords

Linear group codes

Error Control Coding

Lecture contents


Nearest neighbour decoding of block codes

Syndrome decoding

Cyclic codes

Encoding of convolutional codes

Viterbi decoding of convolutional codes

Summary


Lecture contents (contd.)


the resources available to modify the characteristics andquality of the transmission are:

Signal power Time Bandwidth

The quality of transmission is directly related to theprobability of bit error P b at the receiver

Shannon-Hartley law for the capacity of the channel shows: how bandwidth and signal power can be traded

theoretical limit for the transmission rate


Introduction


to achieve the limit indicated by the Shannon-Hartleylaw, an appropriate coding scheme must be found

the objective of the design engineer is to achievethe desired data rate within the bandwidth andpower constraints

error control coding can be used to achieve a requiredBER within the bandwidth and power constraints


Introduction (contd.)


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Error control coding is used to detect and oftencorrect, symbols which are received in error

Error detection is the initial step prior to errorcorrection

In situations where ARQ techniques are inconvenient,FECC can be used

FECC incorporate extra information (redundandy) intothe transmitted data to avoiding the need ofretransmission

Introduction to Error Control Coding

Introduction (contd.)


Introduction to Error Control Coding

Group

BCH

hamming

FECCARQStop&wait Continuous ARQ

Go-back-NSelectiverepeat

Block codes Convolutionalcodes

Polynomial generated

RS Binary BCH

Others

Others

Golay

ARQ: Auto Repeat RequestFECC: Forward Error Correction CodingBCH: Bose-Chaudhuri-HocquenghemRS: Reed Solomon

Error Control Codes


Error rate control concepts


the normal measure of error performance is bit errorrate (BER) or the probability of bit error (Pb)

the BER rate is the average rate at which an erroroccurs:

There is a variety of techniques to reduce the error rateof a particular system

bb R P BER =


1) increasing the transmitter power 2) using diversity3) introducing full duplex

transmission4) using ARQ techniques

5) using FECC


Ways to reduce the BER


the three main typed of diversity are: space diversity Frequency diversityTime diversity

full-duplex transmission consists in echoing theinformation back to the receiver


Ways to reduce the BER (contd.)


in ARQ a simple error detecting code is used if an error is detected, a request is sent via a feedbackchannel to retransmit the block

the two ARQ techniques are:

Stop and wait (each block of data is positively ornegatively acknowledged beforethe next data block is transmitted)

Continuous (blocks of data continue to be

transmitted without waiting for eachprevious block to be acknowledged)


Ways to reduce the BER (contd.)


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The continuous ARQ techniques are:

go-back-N * here data blocks carry a sequencereference number * each acknowledgement signal containsthe reference number of a data block* when a negative acknowledgement is

received, all data blocks starting from thereference number in the lastacknowledgement signal are retransmitted

selective repeat * only the blocks of dataacknowledged negatively areretransmitted



FECC introduces redundancy through data check bitsinterleaved with the information bits

it relies on the number of errors in a long block of databeing close to the statistical average

it requires no channel return

PB can be reduced at the expense of increasing thetransmission delay, by using FECC with a sufficient longblock or constraint length


Forward Error Correction Coding


Threshold phenomenon

in FECC, if the SNR is abovea certain value, the error rateis virtually zero

below this value, the systemperformance degrades rapidlyuntil the coded system isactually poorer than thecorresponding uncodedsystem

a coding gain can be definedfor a given P b



Compact disk players Computers data storage and retrieval Digital audio video systems Aerospace applications Mobile GSM cellular telephony

banking Barcode readers

Applications of Error Control



The Hamming distance between two codewords isdefined as the number of places, bits or digits inwhich they differ

it determines how easy it is to alter one validcodeword into another

The weight of a binary codeword is defined as thenumber of ones which it contains

Hamming distance and codeword weight



Calculate the hamming distance between 2codewords 11100 and 11011 and find the mincodeword weight.

Hamming distance and codeword weight (contd.)


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Calculate the hamming distance between 2codewords 11100 and 11011 and find the mincodeword weight.

Hamming distance and codeword weight (contd.)


the n-digit codeword isthus made up of kinformation digits and(n-k) redundant paritycheck bits

The rate (efficiency) forthis code is given by

(n, k) block codes




The rate (or efficiency) of a code represents the ratioof information digits to the total number of digits in thecodeword

Codes can be systematic or non systematic: systematic codes information digits areexplicitly transmitted together with paritydigits non-systematic codes information bitsare not contained explicitly in the codeword

(n, k) block codes (continued)



Definitions of systematic codes:

stricter definition: the k bits must be transmittedcontiguously as a block

less strict definition: the information digits must

be included in the codeword, but not in acontiguous block

(n, k) block codes (continued)


Single parity check codes

Example of a singleparity check code:

Even parity is morecommon than odd parity

The transmission of anall zero codeword isgenerally avoided

(n-k ) block codes


Single parity check codes (contd.)

(n-k ) block codes


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Another example isthe 10-digit ISBNcodeword used in

libraries andbookshops

It uses a modulo 11,weighted checksum

( ) ( )11mod11119

1 = =i ik iChecksum

( ) ( ) 011mod1110

1

=

=iik i

(n-k ) block codes



Single parity check isalso used on rows andcolumns of simple 2-Ddata arrays

Single errors can bedetected and located viathe corresponding rowand columns parity bits

(n-k ) block codes


(n-k ) block codes

Modulo(2^n 1) checksums are in widespread usefor performing error detection on byte-serial networkconnections

They are usually computed by software during thedata block (or packet construction)

if a message of length w (16 bit) bytes, mw -1 , , m0is to be checksummed, the 1 byte checksum is justthe complement of

Modulo (2^n -1) checksums

=

1

0

)65535(modw

iim


This code consists offour information digitsand three parity checkdigits

The circles indicate howthe information bits

contribute to thecalculation of each paritycheck bit

(7, 4) Block code

(n-k ) block codes


the realization of the encoderusing 3-input modulo-2encoders is:

E.g. calculate the parity checkbits for the next informationbits: 1, 0, 1, 1

(7, 4) Block code (contd.)

(7,4) block code hardwaregeneration of three parity check

digits[from Digital Communications ,

Glover/Grantt et al. 1997]

(n-k ) block codes


(7, 4) Block code (contd.)

The parity checkequations are:

This can beconverted into aparity check matrixH

(n-k ) block codes


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if there are j errors in n digits with a probability oferror given by P e , the probability of j errors in n digitsis:

Probability of error in n-digit codewords

( ) ( ) ( ) jn jn

e j

e jerrors C P P P = 1

where the last term is a binomial coefficient given by:

( )!!!

jn jn

C jn

=



the probability of having more than a given numberof errors in a codeword can be calculated as:

Probability of error in n-digit codewords (contd.)

Statistical stability controls the usefulness of thisequation

A long block effectively embodies a large number oftrials to determine whether or not an error will occur

( ) =

=>'

0

)(1' R

j

j P errors R P



choosing a code that can correct P en errors in ablock of data will ensure that there are very few casesin which the coding system will fail

This is the rationale for long block codes

The attraction of block codes is that they aresucceptible to precise performance analysis

by far the most important and more suceptible set ofblock codes is the linear group codes

Probability of error in n-digit codewords (contd.)



the codewords in a linear group code have a one-to-one correspondence with the elements of amathematical group

Linear group codes contain the all-zeros codewordand have the property referred to as closure

taking any two codewords Cj and Ci then:

Linear group codes

k ji C C C =



Example of a linear group code. The property ofclosure is here demonstrated

Linear group codes (contd.)


Linear Group Codes

Group codes can be divided into two groups cyclic: polynomial generated in simple feedback shiftregisters

non-cyclic: others

The polynomial generated group codes can be furtherdivided into subgroups

binary Bose-Chaudhuri-Hoquenghem (BCH) non-binary Reed-Solomon

Members of the group code family


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Linear Group Codes

the minimum Hamming distance between any pair ofcodewords determines the code performance

to determine the overall performance of a block codeall possible codeword pairs would have to be

examined, and their Hamming distances measured

For the case of group codes, however, considerationof each of the codewords with the all-zeros codeword issufficient

Performance prediction


Linear Group Codes

The weight structure of a set of codewords is just a listof weights of all the codewords

The probability Dij of the i th codeword ( C i ) beingmisinterpreted as the j th codeword ( C j ) depends on the

Hamming distance between these two codewords Since this is a linear group code, the distance Dij isequal to the weight of a third codeword Ck which isusually the modulo2 sum of Ci and Cj

The probability of C k being mistaken by C 0 is equal tothe probability of C 0 being misinterpreted as C k

Performance prediction (contd.)


Linear Group Codes

The maximum possible error correcting power, t , of acode is defined by its ability to correct all patterns of t orless errors

t is related to the codes minimum Hamming distanceby:

Error detection and correction capability

=2

1int min

Dt

and

t e D +=1min


Linear Group Codes

Longer codes with larger Hamming distances offergreater detection and correction capabilities byselecting different t and e values

(e.g.) Dmin = 7 can offer: t = 1 bit correctionwith e = 5 bit detection

or t = 2 bit correctionwith e = 4

Error detection and correction capability (contd.)


Linear Group Codes

There is a trade-off between rate (efficiency) and errorcorrection power

- An n = 63, k = 57 BCH code gives R = 0.9, with t = 1

- reducing k to 45 gives R = 0.71, with t = 3

- further reducing k to 24 gives R = 0.4, with t = 7

codes with lower rates present higher error correction

power

Error detection and correction capability (contd.)


Error control coding

The two most important strategies for decoding blockcodes are:

Nearest neighbour Maximum likelihood decoding

if the probability of t errors is much greater than theprobability of t +1 errors, both strategies are equivalent

Nearest neighbour decoding is based on nearestneighbour decoding tables



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Nearest neighbour decoding ofblock codes (contd.)



There is a condition that has to be satisfied for a codeto exist

for a code with codewords of length n, comprising k information digits and having error correcting power t ,the performance upper bound is:

this equation can be derived from the nearestneighbour decoding table for the ( n , k ) code

Hamming bound

t nnn

nk

C C C n +++++

...12

232



Nearest neighbour decoding ofblock codes (contd.)



The problem with decoding of block codes using thenearest neighbour decoding table is the physical sizeof the table for large n

the syndrome decoding technique provides a solutionto this problem

the rows of the generator matrix G are used to derivethe actual transmitted codewords

Syndrome decoding



The generator matrix G can be used to generate theappropriate n-digit codeword from any given k -digitdata sequence

the H and G matrices for the example (7,4) block codeare:

Syndrome decoding (contd.)



example: if the data sequence is 1 0 0 1- to generate the codeword, the data sequenceis multiplied by G using modulo2 arithmetic


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Syndrome decoding

the nearest neighbour decoding table is normally toolarge for practical implementation

a different technique involving a smaller table(called syndrome decoding table) can be used

the syndrome decoding is smaller than the nearestneighbour table by a factor equal to the number ofcodewords in the code set (2^ k )

this is because the syndrome is independent of thetransmitted codeword and only depends on the errorsequence

Syndrome table for error correction


Syndrome decoding

Example: if d is a message of k digits, G is the k x ngenerator matrix and C is the n-digit codewordcorresponding to the message d, the equation for theprevious example can be written as:

d G = c furthermore

H c = 0 Also

r = c + e

Syndrome table for error correction (contd.)


Syndrome decoding

The product H r is referred to as the syndrome vector s

s = H r = H (c + e )= H c + H e = 0 + H e

s is easy to calculate, and if there are no received

errors, s will be equal to the all zero vector (called 0 ) calculating the vector s provides immediate access tothe vector e and hence to the position of the error

Syndrome table for error correction (contd.)


Syndrome decoding

A syndrome table isconstructed by assumingtransmission of the all-zeros codeword andcalculating the syndromevector associated with

each corresponding errorpattern

Syndrome table for errorcorrection (contd.)


Syndrome decoding

Example: Assume that the received vector for the (7,4)code is r = 1 0 0 1 1 0 1. Find the correct transmittedcodeword


Syndrome decoding

Example: For a (6,3) systematic linear block code, thecodeword comprises I1 I2 I3 P1 P2 P3 where the three paritycheck bits P1, P2 and P3 are formed from the information bitsas follows:

Find: a) the parity check matrixb) the generator matrixc) all possible codewordsd) the minimum weighte) the minimum distancef) the error detecting and correcting capability

g) if the received sequence is 1 0 1 0 0 0, calculate thesyndrome and decode the received sequence

211 I I P =312 I I P =323 I I P =


10/17


Cyclic codes are a subclass of group codes which do notpossess the all-zeros codeword

the Hamming code is an example of a cyclic code

The properties and advantages of cyclic codes are: their mathematical structure permits higher order correctingcodes

their code structure can be easily implemented in hardware byusing simple shift registers and XOR gates

cyclic code members are a ll lateral, or cyclical, shifts of oneanother

cyclic codes can be represented as, and derived usingpolynomials

Cyclic Codes



an example of a cyclic code is:

1 0 0 1 0 1 1

0 1 0 1 1 1 0

0 0 1 0 1 1 1

non-systematic codes are obtained by multiplying thedata vector by a generator polynomial with moduloarithmetic

Cyclic Codes (contd.)



cyclic codes encompass BCH codes and the Reed-Solomon non-binary codes

RS codes operate on the multiple bit symbol principle

An important property of the RS codes is their bursterror correction property

The error correcting power of RS codes is

Reed-Solomon non-binary codes


2k n

t =


systematic Cyclic Redundancy Check (CRCs) are usedfor performing error detection, but not error correction, onbit serial channels

they are combines with ARQ for error correction

if a message of length k bits, m k -1, , m 1, m 0 is to betransmitted over a channel, for coding purposes it may beconsidered to represent a polynomial of order k -1:

Polynomial codeword generation


( ) 0111 ... m xm xm x M k k +++=


the message M ( x ) is modified by the generatorpolynomial P ( x ) as follows:

M ( x ) is multiplied, or bit shifted , by the order of P( x )

the extended (or bit shifted) version of M(x) is thendivided by P ( x )

the remainder is then appended to M ( x ) replacing thezeros which were previously added by the bit shiftingoperation

Polynomial codeword generation (contd.)



e.g. generate a polynomial codeword from the datasequences a) 1 1 0 0, b) 0 1 0 1 and c) 1 0 1 1, where theleading bit in each codeword represents the first bit toenter the encoder, for the generator polynomial x^3+x+1




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e.g. generate a polynomial codeword from the datasequence 1 1 0 0

Sol.1) for M ( x ) = 1 1 0 0 (i.e. x 3 + x 2), the bit shifted

sequence is kM ( x ) = 1 1 0 0 / 0 0 0





Sol.1) for M ( x ) = 1 1 0 0 (i.e. x 3 + x 2), the bit shiftedsequence is kM ( x ) = 1 1 0 0 / 0 0 0

2) the result is then divided by the polynomialP ( x ) = 1 0 1 1 (i.e. x 3+x+1)





Sol.

3) The 0s from the shift are replaced by the remainder:the transmitted codeword is 1 1 0 0 0 1 0




An encoding circuit, which is equivalent to the longdivision operation from the previous slide is:




if the generator polynomial had been x 3 + x 2 + 1,the encoder would have used the next structure:




the hardware decoding scheme is the inverse of theencoding scheme




12/17


the polynomial generator is chosen so that almost allerrors will be detected

using a generator of degree k allows the detection ofall burst errors affecting up to k consecutive bits

the generator chosen by the IEEE for ATM CRC (usedextensively for LANs) is:



( ) 1245781011121622232632 ++++++++++++++= x x x x x x x x x x x x x x x M


block codes are widely used in compact disk players.- they employ a powerful concatenated and cross-interleaved RS coding scheme to handle burst errors

burst transmission errors can be minimised by

partitioning data into blocks and then splitting the blocksand interleaving them

using FECC it is possible to correct for a long errorburst (at the expense of the delay required for theinterleaver encoder/decoder function)

Interleaving



in someapplications bit bybit interleaving isemployed tospread burst errorsacross a datablock prior todecoding

Interleaving (contd.)



convolutional codes are generated by passing a datasequence through a shift register which has two or moresets of register taps

each set terminates in a modulo-2 adder

the code output is produced by sampling the output ofall modulo-2 adders once per shift register clock period

Encoding of convolutional codes



the coder output is obtained by the convolution of theinput sequence with the impulse response of the coder

Encoding of convolutional codes (contd.)


Example of arate convolutional

encoder


in this example, the output encoder operation can bedefined by two generator polynomials

the 1 st and 2 nd encoded outputs, P 1( x ) and P 2 ( x ) can bedefined by:

the error correcting power is related to the constraintlength, increasing with longer length or shift registers



21 1)( x x P +=

x x P +=1)(2


13/17


Example: assume the data sequence 1 1 0 1. Calculatethe outputs of the convolutional encoder presented inthe previous slide




The convolutional encoder operation may berepresented by a tree diagram

Tree diagram


the tree diagram isconventionally drawn so that

inputting a zero results inexiting the present state bythe upper path while inputtinga one causes it to exit by thelower path


The Trellis diagram shown here corresponds to theconvolutional encoder of the previous example

Trellis diagram


the horizontal linerepresents time, whilethe states are arrangedvertically

on the arrival of eachnew bit, the tree isextended to the right


In the state transition diagram, the input to theencoder is shown on the appropriate branch, and thecorresponding outputs are shown in brackets outsidethe input

State Transition Diagram


a is the startingstate


There are three main types of decoder: sequential threshold Viterbi

data encoded by modern convolutional coders areusually divided into message blocks for decoding

the convolutional code message ranges between 500

and 10000 bits

Viterbi decoding of convolutional codes



the Viterbi decoding algorithm is used at every stageof progression through the decoding trellis

it retains only the most likely path to a given node

all other possible paths are rejected as their Hammingdistance is larger

this leads to a linear increase in storage requirementwith block length

Viterbi decoding of convolutional codes (contd.)



14/17


the Viterbi decoding algorithm implements a nearestneighbour decoding strategy

it picks the path trough the trellis that assumes theminimum number of errors

conceptually, a decoding trellis similar to the encodingone is used




Example: assume a received data sequence10 10 00 10 10 for the next encoder:

Identify:a) the errorsb) the corresponding transmitted sequence






Received sequence

10 10 00 10 10








there is a practical limitation to the length of datawhich can be retained in the decoder memory whenperforming a Viterbi decoding operation

this limitation is known as the decoding window

in a practical decoder the new distance metrics areadded to the previous path metrics to obtain updatedpath metrics

details of the paths, which correspond to these variousdistances are carried forward in the decoding process

Decoding Window



15/17


the window lengthshould be longenough to cover allbursts of decodingerrors

however longerlengths involvemore computation

Decoding Window (contd.)



the memory requirement for long constraint lengths inthe Viterbi algorithms can be reduced by sequentialdecoding

Sequential decoding directly constructs the sequence

of states by performing a distance measure at each step

sequential decoding proceeds forwards until completedecoding is accomplished or the accumulative distanceexceeds a preset threshold

Sequential Decoding



Viterbi decoding circuit for decoding the trellis of the previous example



Examples of practical block codes are: the BCH (127,64) => t = 10 the RS(16,8) or (64,32) => t = 4 Golay(23,12)=> t = 3

in general RS codes are attractive for bursty errorchannels where extremely low Pb values (e.g. 10^-10)are required

convolutional codes are favoured for Gaussian noisechannels where more moderate Pb values (10^-6) arerequired

Practical coders



Performance of rate codes for DPSK signals in AWGN with hard decision decoding


for a t = 1 BCHcode, the coding gainover the uncodedsystem, at a P b of10^-5 is > 2dB

for the longer t = 3bit BCH (127, 106)code, the coding gainapproaches 4 dB

the t = 4 symbol RScode is inferior to t = 3

bit Golay at high P b


practical convolutional codes often have constraintlengths of n = 7 with a rate coder, which employs 7delay stages

this requires a decoding window in the trellis of 35 toachieve the theoretical coding gain

Such a coder has a Dmin = 10 and its performance isequivalent to the BCH(127,64) block code

in general, reducing the efficiency, or rate, R of thecoder increases the Hamming distance Dmin between

possible paths and improves the error correcting power

Practical coders (contd.)



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Summary


Error rate control is necessary for many systems toensure that the probability of bit error is acceptably low

Bit error rates may be reduced by: increasing the transmitted power

applying various forms of diversity using echo back and retransmission employing ARQ incorporating FECC


Summary (contd.)


channel codes can be systematic and non systematic

the rate R , of a code, is the ratio of informationtransmitted to total bit transmitted

the Hamming distance between a pair of binarycodewords is a measure of how easily one codewordcan be transformed into another

the weight of a codeword is equal to the number ofbinary ones which it contains


Summary (contd.)


block codes divide the precoded data into k bitlengths and add ( n-k ) parity check bits to create apostcoded block with length n bits

an ( n-k ) block has an efficiency or rate:

single parity check codes are block codes whichappend a single digital one, or zero to each codeword inorder to ensure that all codewords have either an evenor odd weight

nk

R =


Summary (contd.)


data can be arranged in 2-D arrays where parity checkbits are added at the end of the rows and the columns

linear group codes are block codes that contain the allzeros codeword and have the closed set property

the error correcting power, t , of a linear group code isgiven by:

the error detecting power, e, of a linear group code, isgiven by

2

1int min

= Dt

1min = t De


Summary (contd.)


group codes are the most important block codes dueto the ease with which their performance can bepredicted

block codewords can be generated using thegenerator matrix

complete codewords are generated by the product ofthe precoded data vector with the generator matrix

block codes can be easily decoded using the nearestneighbour strategy


Summary (contd.)


the nearest neighbour strategy is equivalent tomaximum likelihood decoding providing thatP (t errors) >> P (t + 1 errors)

nearest neighbour decoding can be implementedusing a nearest neighbour decoding table or asyndrome table

syndrome decoding is advantageous when block sizeis large since the syndrome table for all (single) errorpatterns is much smaller than the nearest neighbourdecoding table


17/17


Summary (contd.)


cyclic codes are linear group codes in which thecodeword set consists of cyclical shifts of any onemember of the codeset

cyclic codes are particularly easily generated usingshift registers with appropriate feedback connections

the syndromes of cyclic codes are also easilycalculated using shift register hardware

CRC codes are systematic cyclic codes which arepotentially capable of error correction, but are oftenused for error detection only


Summary (contd.)


a polynomial representation of a block of informationbits is multiplied by the order of a generator polynomialand then divided by the generator polynomial

the remainder of the division is appended to theblock of information bits to form the completecodeword

convolutional codes are unsystematic and operate onlong data blocks

the encoding of convolutional codes can be describedusing trees, trellis or state diagrams


Summary (contd.)


the Viterbi algorithm is usually used to decodeconvolutional codes

the error correction capability of convolutional codes isnot inherently in excess of that of block codes, but theimplementation of encoders and decoders in the formeris simpler

FECC combinations of block and convolutional codesare widely applied to accommodate random and bursterrors


TutorialProblems 10.1, 10.3 10.7Chapter 10, Glover/Grantt


BB Channel Coding

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