f8 error correcting codes.ppt

7/24/2019 F8 error correcting codes.ppt

1/43

Lecture 8: Coding

Anders Vstberg

[email protected] 44

!"ides are a se"ection #ro$ the s"ides #ro$

cha%ter 8 #ro$:htt%:&&'i""ia$sta""ings.co$&(ire"ess&(ire"ess)e.ht$"
mailto:[email protected]:[email protected]


2/43

Co%ing 'ith *ata

+rans$ission ,rrors ,rror detection codes

*etects the %resence o# an error

Auto$atic re%eat re/uest A123 %rotoco"s "ock o# data 'ith error is discarded

+rans$itter retrans$its that b"ock o# data

,rror correction codes5 or #or'ard

correction codes 6,C3 *esigned to detect and correct errors


3/43

,rror *etection

robabi"ities *e#initions

Pb: robabi"it o# sing"e bit error ,13

P9: robabi"it that a #ra$e arrives 'ith no bit

errors

P): (hi"e using error detection5 the %robabi"it that

a #ra$e arrives 'ith one or $ore undetected errors

P:: (hi"e using error detection5 the %robabi"it that

a #ra$e arrives 'ith one or $ore detected biterrors but no undetected bit errors


4/43

,rror *etection

robabi"ities (ith no error detection

6 ;


5/43

Error Detection Process


6/43

arit Check

arit bit a%%ended to a b"ock o# data

,ven %aritAdded bit ensures an even nu$ber o# s

=dd %aritAdded bit ensures an odd nu$ber o# s

,>a$%"e5 7-bit character ?000 ,ven %arit ?0000

=dd %arit ?000


7/43

Cc"ic 1edundanc Check

C1C3 +rans$itter

6or a k-bit b"ock5 trans$itter generates ann-k3-bit #ra$e check se/uence 6C!3

1esu"ting #ra$e o# nbits is e>act" divisib"e b%redeter$ined nu$ber

1eceiver

*ivides inco$ing #ra$e b %redeter$inednu$ber

# no re$ainder5 assu$es no error


8/43

C1C using Bodu"o )

Arith$etic ,>c"usive-=1 =13 o%eration

ara$eters: T ; n-bit #ra$e to be trans$itted

D ; k-bit b"ock o# dataD the #irst k bits o# T

F ; (n k3-bit 6C!D the "ast (n k3 bits o# T

P ; %attern o# nkE bitsD this is the %redeter$ineddivisor

Q; 2uotient R; 1e$ainder


9/43

CRC using Modulo 2

Arithmetic For T/Pto have no remainder start !ith

Divide 2n"k

D#$ Pgives %uotient andremainder

&se remainder as FC'

FDT kn += 2

P

RQ

P

Dkn+=

2

RDT kn += 2


10/43

C1C using Bodu"o )

Arith$etic *oes R cause T&P have no re$ainderF

!ubstituting5

act" divisib"e b P

P

R

P

D

P

RD

P

T knkn

+=+

=

22

Q

P

RRQ

P

R

P

RQ

P

T=

++=++=


11/43

C1C using o"no$ia"s

A"" va"ues e>%ressed as %o"no$ia"s

*u$$ variab"eX'ith binar coe##icients

( )( )

( ) ( )( )

( ) ( ) ( )XRXDXXT

XPXRXQ

XPXDX

kn

kn

+=

+=


12/43

C1C using o"no$ia"s

(ide" used versions o#P(X)

C1C)

) E E E ) E E

C1CG G E E ) E

C1C CC++

G E ) E E

C1C ) ) E )G E ) E ))E G E ) E E 0 E8 E 7 E E4 E

) E E


13/43

C1C using *igita" Logic

*ividing circuit consisting o#:

=1 gates

H% to n k =1 gates

resence o# a gate corres%onds to the %resence o#

a ter$ in the divisor %o"no$ia" PX3

A shi#t register

!tring o# -bit storage devices 1egister contains n k bits5 e/ua" to the "ength o#

the 6C!


14/43

Digital (ogic CRC


15/43

(ire"ess +rans$ission

,rrors ,rror detection re/uires retrans$ission

*etection inade/uate #or 'ire"ess

a%%"ications

,rror rate on 'ire"ess "ink can be high5 resu"ts

in a "arge nu$ber o# retrans$issions

Long %ro%agation de"a co$%ared to

trans$ission ti$e


16/43

"ock ,rror Correction

Codes +rans$itter

6or'ard error correction 6,C3 encoder $a%s

each k-bit b"ock into an n-bit b"ock code'ord

Code'ord is trans$ittedD ana"og #or 'ire"ess

trans$ission

1eceiver

nco$ing signa" is de$odu"ated

"ock %assed through an 6,C decoder


17/43

For!ard Error Correction

Process


18/43

6,C *ecoder =utco$es


19/43

"ock Code rinci%"es

Ia$$ing distance #or ) n-bit binar

se/uences5 the nu$ber o# di##erent bits

,.g.5 v9;00D v);000D dv5 v)3;

1edundanc ratio o# redundant bits to data bits Code rate ratio o# data bits to tota" bits

Coding gain the reduction in the re/uired Eb&N0

to achieve a s%eci#ied ,1 o# an error-correctingcoded sste$


20/43

Io' Coding $%roves

er#or$ance


21/43

Ia$$ing Code

*esigned to correct sing"e bit errors

6a$i" o# n5 k3 b"ock error-correcting codes 'ith%ara$eters:

"ock "ength: n ; )m


22/43

Ia$$ing Code rocess

,ncoding: kdata bits E n -k3 check bits

*ecoding: co$%ares received n -k3 bits

'ith ca"cu"ated n -k3 bits using =1

1esu"ting n -k3 bits ca""ed syndrome word

!ndro$e range is bet'een 0 and )n-k3-

,ach bit o# sndro$e indicates a $atch 03 or

con#"ict 3 in that bit %osition


23/43

Cc"ic Codes

Can be encoded and decoded using "inear

#eedback shi#t registers L6!1s3

6or cc"ic codes5 a va"id code'ord c05 c95 J5 cn-

935 shi#ted right one bit5 is a"so a va"id code'ordcn-95 c05 J5 cn-)3

+akes #i>ed-"ength in%ut k3 and %roduces #i>ed-

"ength check code n-k3 n contrast5 C1C error-detecting code acce%tsarbitrar "ength in%ut #or #i>ed-"ength check code


24/43

CI Codes

6or %ositive %air o# integers m and t5 a n5k3 CI code has %ara$eters: "ock "ength: n ; )m


25/43

1eed-!o"o$on Codes

!ubc"ass o# nonbinar CI codes

*ata %rocessed in chunks o# m bits5 ca""eds$bo"s

An n5 k3 1! code has %ara$eters: !$bo" "ength: m bits %er s$bo"

"ock "ength: n ; )m s$bo"s ; m)m 3 bits

*ata "ength: k s$bo"s

!iMe o# check code: n k ; 2t s$bo"s ; m2t3 bits Bini$u$ distance: d$in; 2t E s$bo"s


26/43

)loc* +nterleaving

Data !ritten to and read ,rom memor$ in di,,erentorders

Data #its and corres-onding chec* #its are

inters-ersed !ith #its ,rom other #loc*s At receiver data are deinterleaved to recover original

order

A #urst error that ma$ occur is s-read out over a

num#er o, #loc*s ma*ing error correction -ossi#le


27/43

)loc* +nterleaving


28/43

Convolutional Codes

.enerates redundant #its continuousl$

Error chec*ing and correcting carried outcontinuousl$

0n k K code +n-ut -rocesses k#its at a time

ut-ut -roduces n#its ,or ever$ kin-ut #its

K3 constraint ,actor

kand ngenerall$ ver$ small

n"#it out-ut o, 0n k K code de-ends on4 Current #loc* o, kin-ut #its

Previous K"1 #loc*s o, kin-ut #its


29/43

Convolutional Encoder


30/43

*ecoding

+re""is diagra$ e>%anded encoder diagra$

Viterbi code error correction a"gorith$ Co$%ares received se/uence 'ith a"" %ossib"e

trans$itted se/uencesA"gorith$ chooses %ath through tre""is 'hose codedse/uence di##ers #ro$ received se/uence in the#e'est nu$ber o# %"aces

=nce a va"id %ath is se"ected as the correct %ath5 the

decoder can recover the in%ut data bits #ro$ theout%ut code bits


31/43

+re""is *iagra$


32/43

+re""is *iagra$


33/43

+re""is diagra$


34/43

Viterbi a"gorith$


35/43

Auto$atic 1e%eat

1e/uest Bechanis$ used in data "ink contro" and

trans%ort %rotoco"s

1e"ies on use o# an error detection code

such as C1C3

6"o' Contro"

,rror Contro"


36/43

6"o' Contro"

Assures that trans$itting entit does notover'he"$ a receiving entit 'ith data

rotoco"s 'ith #"o' contro" $echanis$ a""o'

$u"ti%"e *Hs in transit at the sa$e ti$e *Hs arrive in sa$e order theNre sent

!"iding-'indo' #"o' contro" +rans$itter $aintains "ist 'indo'3 o# se/uence

nu$bers a""o'ed to send 1eceiver $aintains "ist a""o'ed to receive


37/43

6"o' Contro"

1easons #or breaking u% a b"ock o# data

be#ore trans$itting:

Li$ited bu##er siMe o# receiver

1etrans$ission o# *H due to error re/uires

s$a""er a$ounts o# data to be retrans$itted

=n shared $ediu$5 "arger *Hs occu%

$ediu$ #or e>tended %eriod5 causing de"asat other sending stations


38/43

Flo! Control


39/43

,rror Contro"

Bechanis$s to detect and correct

trans$ission errors

+%es o# errors:

Lost *H : a *H #ai"s to arrive

*a$aged *H : *H arrives 'ith errors


40/43

,rror Contro"

1e/uire$ents ,rror detection

1eceiver detects errors and discards *Hs

ositive ackno'"edge$ent

*estination returns ackno'"edg$ent o# received5error-#ree *Hs

1etrans$ission a#ter ti$eout !ource retrans$its unackno'"edged *H


41/43

Oo-back-< A12

Ackno'"edg$ents

11 ; receive read no errors occur3

1,P ; reQect error detected3

Contingencies

*a$aged *H

*a$aged 11

*a$aged 1,P


42/43

Oo-back < A12


43/43

Io$e'ork e#ore 6

+a reda %R vad #S"Qande begre%% betder

och gSr en kort beskring #Sr varQe.

O,=

B,=

L,=

hSQdvinke"

f8 error correcting codes.ppt

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