lec5-ErrorDetectionCorrection

8/11/2019 lec5-ErrorDetectionCorrection

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Data Communication &Computer Networks

byDr. Nguyen Minh Hoang

[email protected] [email protected]

April 2013
mailto:[email protected]:[email protected]:[email protected]:[email protected]


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Data Link Layer


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LLC and MAC Sublayers


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Single Bit Error vs. Burst Error


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Error Detection

Error detection uses the concept of redundancy,which means adding extra bits for detecting

errors at the destination.


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Redundancy for Error Detection


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Parity Check


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2-d Parity Check


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2-d Parity CheckBased on bit addition A redundancy of n bits can detect aburst error of length nIf length more than n, can detect witha high probabilitySome exceptions

Any example?


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Cyclic Redundancy Check (CRC)

Based on modulo-2 divisionMost popular


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Binary Division in CRC Generator


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Binary Division in CRC Checker


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Standard PolynomialsName Polynomial Application

CRC-8 x 8 + x 2 + x + 1 ATMheader

CRC-10 x 10 + x 9 + x 5 + x 4 +

x 2 + 1 ATM AAL

ITU-16 x 16 + x 12 + x 5 + 1 HDLC

ITU-32

x 32 + x 26 + x 23 + x 22 + x 16 + x 12 + x 11 +

x 10 + x 8 + x 7 + x 5 + x 4 + x 2 + x + 1

LANs


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Rules to Select A PolynomialShould not divisible by xDetect all burst errors of length less than or equal todegree

Should be divisible by x+1Detect all burst errors affecting an odd number ofbits

E.g., x+1, x2+1

The CRC-12x12 + x11 + x3 + x + 1detect all burst errors affecting an odd number ofbits, will detect all burst errors with a length lessthan or equal to 12, and will detect with 99.97%

probability burst errors with a length of 12 or more.


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Checksum


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Checksum: SendingSuppose the following block of 16 bits is to be sent usinga checksum of 8 bits.

10101001 00111001 The numbers are added using ones complement

10101001

00111001 ------------

Sum 11100010

Checksum 00011101

The pattern sent is 10101001 00111001 00011101


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Checksum: ReceivingNow suppose the receiver receives the pattern sent inprevious Example and there is no error.10101001 00111001 00011101

When the receiver adds the three sections, it will get all 1s, which,after complementing, is all 0s and shows that there is no error.

10101001

00111001

00011101

Sum 11111111

Complement 00000000 means that the pattern is OK.


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Checksum: ErrorA burst error of length 5 that affects 4 bits.

10101 11 1 11 111001 00011101

When the receiver adds the three sections, it gets

10101111

11111001

00011101

Partial Sum 1 11000101

Carry 1

Sum 11000110 => Complement 00111001 !!!??


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Error CorrectionRetransmission

Go back N retransmissionSelective retransmission

Forward Error CorrectionHamming Code

Burst Error Correction


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Forward Error CorrectionPurpose: An FEC (n, m) encoder

Take m -bit original data as input

Add r=n-m check bits to the original data toproduce a n-bit codewordThe receiver can fix any error


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Hamming DistanceCode: set of codewordsHamming distance between 2 codewords isthe number of bit positions where the 2codewords differ

HammingDist(10001001, 10110001) = 3Hamming distance of a code is the minimum Hamming distance between any twocodewords in the code

HammingDist({0000000000,0000011111,1111100000,1111111111}) = 5


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Interesting FindingsTo detect d single-bit errors

Need a distance-( d+1 ) codeAppling d single-bit errors to a codeword must resultin an invalid codeword. ( WHY?)

To correct d single-bit errorsNeed a distance-( 2d+1) code

Given an incorrect codeword, the correspondingcorrect one must be the codeword closest in Hammingdistance ( WHY?)


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How Many Extra Bits is Needed?Msg 1msg x x 1

1-x 1

x2 xm+r

x2 xm+r

x1 1-x 2 xm+r

x1 x2 1-x m+r

m+r+1invalid

Suppose we want to correct single errors# all possible data inputs = 2 m, # possible (m+r)-bit numbers = 2 m+r => 2m(m+r+1) 2m+r =>m+r+1 2 r

# check bits needed = r min


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Data and Redundancy BitsNumber of data bits

mNumber of redundancy bits

rTotal bits

m + r

1 2 32 3 5

3 3 6

4 3 7

5 4 9

6 4 10

7 4 11


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Hamming CodeTo correct single errorsRedundant bits are placed at all power-of-2 positions of the original data bits.

The value of a redundant bit is the parity check bitof a set of data bits.

Considering a data bit at position k , express k as a sum ofpowers of 2.

E.g., 11 = 1+2+8. Thus, data bit at position 11 is checked by

redundant bit at position 1, 2, and 8.


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Redundant bits in Hamming Code


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Correct Errors with HammingCode

Correct: 10011100101

Parity check for r1

Parity check for r2

Parity check for r4

Parity check for r8


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Burst Errors with HammingCode

A sequence of k codewords is arranged as a matrix,1st column sent first, then 2 nd column, then 3 rd , etc.

k-bit burst error => at most 1 bit in each codeword is wrongUse Hamming code to correct such bits


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Question?

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