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Winter 2005 ECE 766Computer Interfacing and

110 -

Error DetectionError Detection

Data transmission can contain errors ingle!"it

#urst errors of lengt$ n

%n& distance "et'een t$e first and last errors in data

"loc()

*o' to detect errors

If onl+ data is transmitted, errors cannot "e detected

end more information 'it$ data t$at satisfies aspecial relations$ip

-dd redundanc+

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210 -

Error Detection MethodsError Detection Methods

Vertical Redundancy Check (VRC)

-ppend a single "it at t$e end of data "loc(

suc$ t$at t$e num"er of ones is e.en

E.en Parit+ %odd parit+ is similar)011001101100110

011000101100011

/C is also (no'n as Parity Check

Performance&

Detects all odd!num"er errors in a data "loc(

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10 -


Longitudinal Redundancy Check (LRC)

rgani3e data into a ta"le and create a parit+

for eac$ column

11100111 11011101 00111001 10101001

11100111

11011101

00111001

10101001

10101010

11100111 11011101 00111001 10101001 10101010

riginal Data 4C

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10 -


Performance& Detects all "urst errors up to lengt$ n

%num"er of columns)

isses "urst errors of lengt$ n1 if t$ere are n!1

unin.erted "its "et'een t$e first and last "it If t$e "loc( is "adl+ gar"led, t$e pro"a"ilit+ of

acceptance is

Checksum

8sed "+ upper la+er protocols

imilar to 4C, uses one9s complementarit$metic

( )n2

1

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510 -

Cyclic Redundancy CheckCyclic Redundancy Check

Po'erful error detection sc$eme

at$er t$an addition, "inar+ di.ision is

used:inite -lge"ra ;$eor+ %

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610 -


4et us assume kmessage "its andn"its of redundanc+

-ssociate "its 'it$ coefficients of a

pol+nomial

1 0 1 1 0 1 1

1>60>51>1>10>21>1? >6>>1>1

>>>>>>>>>> ++++

( "its n "its

#loc( of lengt$ (n

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710 -


4et %>) "e t$e message olynomial

4et P%>) "e t$e generator olynomial

P%>) is fi>ed for a gi.en CC sc$eme

P%>) is (no'n "ot$ "+ sender and recei.er

Create a "loc( pol+nomial :%>) "ased on

%>) and P%>) suc$ t$at :%>) is di.isi"le "+

P%>)

)(

0)(

)(

)(

xPxQ

xP

xF+=

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@10 -


ending

1A ultipl+ %>) "+ >n

2A Di.ide >n%>) "+ P%>)

A Ignore t$e Buotient and (eep t$e reminder C%>)A :orm and send :%>) ? >n%>)C%>)

ecei.ing

1A ecei.e :9%>)2A Di.ide :9%>) "+ P%>)

A -ccept if remainder is 0, reect ot$er'ise

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10 -

Proo! o! CRC "enerationProo! o! CRC "eneration

)(

)()(

)(

)()(

)(

)()(

)()()()(

)(remainder,

)(

)()(

)(bydivisibleis)()(thatProve

xP

xCxC

xP

xQxP

xP

xCxMx

xCxQxPxMx

xC

xQ

xMxxP

xPxCxMx

n

n

n

n

++=

+

+=

+

emainder 0 emainder 0

#ote$ %inary modular addition is e&ui'alent to

inary modular sutraction C()*C()+0

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1010 -

EamleEamle

end %>) ? 110011>5>>1 %6 "its) P%>) ? 11001>>11 %5 "its, n ? ) "its of redundanc+

:orm >n%>)110011 0000>D>@>5>

Di.ide >n%>) "+ P%>) to find C%>)

100001

1001

11001

10000

11001

110011000011001

? C%>)

end t$e "loc( 110011 1001

ecei.e

00000

11001

11001

11001

110011100111001

o remainder

-ccept

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1110 -

Proerties o! CRCProerties o! CRC

ent :%>), "ut recei.ed :9%>) ? :%>)E%>)

W$en 'ill E%>)FP%>) $a.e no remainder,

iAeA, '$en does CC fail to catc$ an errorG

1, ingle %it ErrorE%>) ? >i

If P%>) $as t'o or more terms, P%>) 'ill not di.ide E%>)

., . /solated ingle %it Errors%dou"le errors)E%>) ? >i>C, iH

E%>) ? >C%>i!C1)Pro.ided t$at P%>) is not di.isi"le "+ >, a sufficientcondition to detect all dou"le errors is t$at P%>) doesnot di.ide %>t1) for an+ t up to i! %iAeA, "loc( lengt$)

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1210 -


, dd #umer o! %it ErrorsIf >1 is a factor of P%>), all odd num"er of "it errorsare detected

Proo!$-ssume an odd num"er of errors $as >1 as a factorA;$en E%>) ? %>1);%>)AE.aluate E%>) for > ? 1E%>) ? E%1) ? 1 since t$ere are odd num"er of terms

%>1) ? %11) ? 0 %>1);%>) ? %11);%1) ? 0E%>) %>1);%>)

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110 -


2, hort %urst Errors%4engt$ t J n, num"er of redundant "its)E%>) ? >%>t!1K1)4engt$ t, starting at "it position If P%>) $as an >0term and t J n, P%>) 'ill not di.ide E%>)

-ll errors up to lengt$ n are detected3, Long %urst Errors %4engt$ t ? n1)

8ndetecta"le onl+ if "urst error is t$e same as P%>)P%>) ? >n K 1 n!1 "its "et'een >nand >0E%>) ? 1 K 1 must matc$

Pro"a"ilit+ of not detecting t$e error is 2!%n!1)4, Longer %urst Errors%4engt$ t H n1)

Pro"a"ilit+ of not detecting t$e error is 2!n

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110 -


E>ample& CC!12 ? >12>11>>2>1

CC!16 ? >16>15>21

CC!CCI;;? >16>12>51 CC!16 and CC!CCI;; catc$ all

ingle and dou"le errors

dd num"er of "it errors

#ursts of lengt$ 16 or less

A7L of 17!"it error "ursts

A@L of 1@!"it and longer error "ursts

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1510 -

5ard6are /mlementation5ard6are /mlementation

8sual practice&-fter ta(ing ( data "its, n 0s are padded to t$e

stream, t$en di.ided "+ t$e generator

-im& Introduce t$e last n "its of 0s 'it$out reBuiring n e>tra

s$ifts

Eliminate t$e need to 'ait for all data to enter t$es+stem to start generating CC

-pproac$& t n "its of message as all 0s

Correct t$e guess as t$e actual "its arri.e

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1610 -


essage ? 1011011 ( ? 7P%>) ? 1101 ? >1>2>0 n ?

1100101

0101101110000001100

1011

110100000111

00000011

11011100

110100010110111101

Con'entional

Method$

1100101

010110111000100

000001000101

10110010

0001000010001001

00000110

1110110110101010

11011000

00000000001101Circuit$

1

0

1

1

0

1

1

6essage

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1710 -


Data

Input

>erial Muotient %M)

2 1 0>0>

2

7ransmit$Data

1

0

1

1

0

1

1

M

1

1

0

0

1

0

1

#it 2

0

1

1

1

1

1

0

0

#it 1

0

0

1

1

0

0

1

0

#it 0

0

1

1

0

0

1

0

1

6essa

ge

4#

#

CCend # first( s$ifts later, CC is in register

$ift out %'it$out an+ =) in n s$ifts

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1@10 -


Data

Input

>erial Muotient %M)

2 1 0>0>

2

Recei'e$Data

1

0

1

1

0

1

1

0

0

1

M

1

1

0

0

1

0

1

0

0

0

#it 2

0

1

1

1

1

1

0

0

0

1

0

#it 1

0

0

1

1

0

0

1

0

1

0

0

#it 0

0

1

1

0

0

1

0

1

0

0

0

6essa

ge

4#

#

n( s$ifts later, remainder is 0

Data acceptedC0C

4#

#

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110 -


Data ut

8

8

Data In

2

8

8

>

2

1 0

>0

Data N

Control 4ine -&1& a(eF;est CC

0& $ift ut CC

:or ;ransmitting&

-ssert 8true '$ile feeding ( "its of message

-ssert 8false for n cloc( c+cles to output CC

:or ecei.ing&

-ssert 8true '$ile feeding (n "its of message and CC

Ignore Data ut, c$ec( Data 9for correctness

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766-10n.ppt